ფიზიკა by Vano Tabatadzé

a. gigineiSvili, g. kukulaZe

z o g a d i

f i z i k a

I tomi

meqanikisa da molekuluri fizikis safuZvlebi

winasityvaoba

winamdebare saxelmZRvanelo gankuTvnilia teqnikuri universitetis sainJinro specialobis studentebisaTvis. igi warmoadgens zogadi fizikis kursis samtomeulis

pirvel toms

(meqanikisa da molekuluri fizikis

safuZvlebi). teqnikur universitetSi zogadi fizikis kursi bolo wlebSi iswavleboda ori semestris ganmavlobaSi. Sesabamisad, saxelmZRvaneloc gamoicemoda gadasvlam

tomad.

warmoSva

swavlebis

bunebriv

saxelmZRvanelos

sam

ritmze

tomad

samsemestrianze

gamocemis

saWiroeba.

wignSi Sesulia avtorebis mier adre gamocemuli ortomeulis pirveli tomis masalis nawili, cxadia, garkveuli damatebebiTa da SesworebebiT. saxelmZRvanelos

done

struqtura

iseTia,

rom

students

gauadvildes masze damoukideblad muSaoba. kursSi mocemulia klasikuri da Tanamedrove fizikis mxolod arsebiTi sakiTxebi im moculobiT, rom students gamoumuSavdes damoukideblad azrovnebis saWiro Cveva. wignSi

farTodaa

gankuTvnilia

gamoyenebuli

studentebisaTvis,

petitis

romelnic

teqsti,

moisurveben

romelic ama

sakiTxze damatebiTi informaciis miRebas. avtorebi Tbilisis

gulwrfel

saxelmwifo

fakultetTan institutis

arsebuli direqtors,

madlobas

universitetis nivTierebaTa

uZRvnian zust

wignis da

kvlevis

fizika-maTematikis

recenzentebs,

sabunebismetyvelo samecniero-kvleviTi

mecnierebaTa

doqtors,

mecnierebis damsaxurebul moRvawes, prof. n. kekeliZes da saqarTvelos teqnikuri

universitetis

k. cxakaias

sasargeblo

fizikis

departamentis

rCevebisaTvis,

agreTve

srul akademiur

profesors doqtors,

asistent-profesors qalbaton l. CaxvaSvils wignis gamocemaSi gaweuli daxmarebisaTvis.

-3-

ยง1. Sesavali

fizika

TamaSobs

udides

rols

Tanamedrove

bunebismetyvelebaSi,

Tanamedrove teqnikisa da saxalxo meurneobis yvela dargis ganviTarebaSi. es winaswar gansazRvravs fizikis kursis mniSvnelobas umaRlesi skolis, gansakuTrebiT umaRlesi teqnikuri skolis saswavlo programaSi. mizani, romelsac isaxavs fizikis swavleba umaRles teqnikur skolaSi, mravalmxrivia. umniSvnelovanesi maT Soris aris ZiriTadi fizikuri movlenebisa da maTi praqtikuli gamoyenebis gacnoba, am movlenebis meqanizmsa da kanonzomierebebSi garkveva. amiT eyreba safuZveli momdevno zogadteqnikuri da specialuri disciplinebis swavlebas. fizikuri movlenebis Sesaxeb swori warmodgena gansakuTrebiT mniSvnelovania axali sakiTxebis dayenebisas, romelic yovelTvis SeiZleba wamoiWras inJinris praqtikuli moRvaweobis procesSi. bolos, fizikis swavleba xels uwyobs logikuri azrovnebisa da swori, materialisturi msoflmxvdelobis Camoyalibebas. termini `fizika~ warmodgeba berZnuli sityvisagan `fiuzis~, rac bunebas niSnavs. es miuTiTebs, rom Tavdapirvelad fizika warmoiSva rogorc bunebaSi mimdinare yvela movlenis, yvela procesis Semswavleli mecniereba. magram SemdegSi, codnis TandaTanobiT dagrovebisa da kvlevis gaRrmavebis kvaldakval, aRmoCnda, rom bunebis yvela movlenis erT mecnierebaSi gaerTianeba praqtikulad SeuZlebelia da rigi sabunebismetyvelo mecnierebebi (qimia, astronomia, biologia da sxv.) gamoeyo fizikas da Camoyalibda damoukidebeli disciplinebis saxiT. f i zik a ar is m e c n ie re b a, r om e lic Se is w av l is m ate ri is m oZ r ao b i s kurs,

u m ar tiv e s

da,

a m av e

at om u r -m ole k u lu r s ,

d ro s ,

u zo g ade s

gr av it ac iu l s ,

fo rm e b s :

m e q an i -

e le q tr o m ag ni tu r s ,

S i d a at om u r s , S i da b ir T v u ls d a m aT u r T ie rT g a rd aq m n as . moZraobis

sxvadasxva

f i zik u ri

for m e b i

umartivesi

da,

amave

dros,

uzogadesia imitom, rom isini gvxvdeba materiis yvela sxva, ufro maRali da rTuli saxis moZraobebSi (qimiur, biologiur da sxva procesebSi). maT Seiswavlis sxva sabunebismetyvelo mecnierebebi. magaliTad, organizmis cxovelmoqmedebis process, romelsac Seiswavlis biologia, yovelTvis Tan

-4-

axlavs meqanikuri, eleqtruli, Sidaatomuri da sxva fizikuri procesebi (Tumca ar daiyvaneba maTze). imis gamo, rom fizika Seiswavlis moZraobis umartives da uzogades formebs, mis mier dadgenili kanonzomierebani bunebis yvelaze ufro zogadi da Zireulia da isini unda iZleodnen im kerZo saxis kanonzomierebebis axsnis principul SesaZleblobas, romlebsac ayalibebs danarCeni sabunebismetyvelo mecnierebebi. amis mkafio magaliTia kvanturi meqanikis saSualebiT molekuluri Zalebis bunebisa da qimiuri elementebis TvisebaTa periodulobis axsna. cxadia, ar SeiZleba imis mtkiceba, rom fizikas, ganviTarebis mocemul etapze, SeuZlia nebismieri sabunebismetyvelo mecnierebis nebismieri gadauWreli amocanis gadawyveta, magram fizikis ganviTarebasTan, mis srulyofasTan erTad, aseTi SemTxvevebis ricxvi sul ufro da ufro izrdeba. unda

aRiniSnos,

rom

xSir

SemTxvevebSi

gaZnelebulia

mkveTri

sazRvris gavleba fizikasa da sxva sabunebismetyvelo mecnierebebs (mag., qimias) Soris. mecnierebis ganviTarebasTan erTad maT Soris kavSiri sul ufro Rrmavdeboda, fizika Tavisi TeoriebiTa da kvlevis meTodebiT sul ufro Rrmad `iWreboda~ sxva sabunebismetyvelo mecnierebebSi, rac iwvevda mosazRvre

disciplinaTa

warmoSobas.

ase

warmoiSva

fizikuri

qimia,

romelic Seiswavlis qimiur procesebs fizikuri meTodebiT; astrofizika, romelic geofizika,

Seiswavlis romelic

ciur

sxeulebSi

Seiswavlis

mimdinare

dedamiwis

fizikur

qerqSi

movlenebs;

atmosferoSi

mimdinare fizikur procesebs; biofizika, romelic Seiswavlis cocxal organizmSi mimdinare fizikur procesebs da a.S. fizikisa da kerZod, biologiis mWidro urTierTobaze miuTiTebs is faqtic, rom Tanamedroveobis erT-erTi udidesi aRmoCena biologiaSi _ memkvidruli informaciis matarebeli nivTierebis _ dezoqsiribonukleinis mJavas (dnm) molekulis sivrciTi struqturis dadgena (e.w. ormagi spirali), ramac SesaZlebeli gaxada genetikuri kodis Caweris meqanizmis garkveva, aris biolog jeims uotsonisa da fizikos frensis krikis erToblivi kvlevis Sedegi. aranakleb aris dakavSirebuli fizika teqnikasTan. fizikis aRmoCenebis safuZvelze warmoiSva teqnikis iseTi dargebi, rogorebicaa eleqtroteqnika da radioteqnika, eleqtronika da gamoTvliTi teqnika, birTvuli energetika da kosmosuri teqnika da mravali sxva. fizika aris baza teqni-5-

kis axali dargebis SeqmnisaTvis. teqnika, Tavis mxriv, xels uwyobs fizikis ganviTarebas: jer erTi, igi aiaraRebs fizikas axali, srulyofili xelsawyoebiTa da danadgarebiT, romlebic aucilebelia kvlevis fizikuri meTodebis gaumjobesebisaTvis, awvdis mas eqsperimentisaTvis saWiro masalebs, meorec, ayenebs mis winaSe axal-axal problemebs, rac iwvevs fizikis arsebuli dargebis ganviTarebas da axali dargebisa da Teoriebis warmoSobas. magaliTad, Zvel saberZneTSi meqanikis ganviTareba dakavSirebuli iyo imdroindel samSeneblo da samxedro teqnikis moTxovnebTan, me-19 saukuneSi Termodinamikis swrafi ganviTareba dakavSirebuli iyo siTburi manqanis margi qmedebis koeficientis gazrdis moTxovnasTan da a.S. ZiriTadi cnebebi, romlebsac emyareba fizika, aris materia, sivrce da dro. m at e ri a ar is y v e l a fe ri, r a c re a l u r ad ar s e b ob s b u ne b a Si ( s am y ar o Si ) d a Se i Z le b a aR m o a C in os ad am i anm a grZ nob is o rg a no e b iT a n s pe ci a lu r i xe l s aw y oe b iT . arsebobs materiis ori saxe _ nivTiereba da veli. nivTiereba materiis is saxea, romelsac (velisagan gansxvavebiT) aqvs uZraobis masa. saboloo jamSi, nivTiereba miiReba im elementaruli nawilakebisagan, romelTa uZraobis masa ar udris nuls (ZiriTadad, eleqtronebisagan, protonebisa da neitronebisagan).

yvelaferi, rac Cvens irgvliv arsebobs, aqvs suni, gemo,

SeiZleba xeliT Sexeba da a.S. aris nivTiereba. vels zemoT CamoTvlili Tvisebebi ar gaaCnia. amitom xSirad mas materiis gansakuTrebul formas uwodeben, Tumca masac aqvs energia, impulsi, gavrcelebis siCqare da mravali sxva sidide, romlebic Cveulebrivi materiis _ nivTierebis damaxasiaTebeli sidideebia. adre iTvleboda, rom nivTierebas wyvetili (diskretuli) agebuleba aqvs, veli ki uwyvetia (talRaa), magram fizikis ganviTarebam cxadyo, rom veli uwyvetobasTan erTad wyvetili Tvisebebis matarebelicaa iseve, rogorc nivTiereba, diskretulobasTan erTad uwyvetobis TvisebaTa matarebelia (de-broilis hipoTeza). es, diskretulobisa da uwyvetobis erTianoba, materiis erT-erTi ZiriTadi Tvisebaa. m at e ri a

m u d m iv ,

g an u w y v e te l m o Z r ao b a S i a . am moZraobis arc

Seqmna da arc mospoba ar SeiZleba. amitom moZraobas ganixilaven rogorc

-6-

materiis pirvelad, Tandayolil Tvisebas. ar arsebobs materia moZraobis gareSe iseve, rogorc moZraoba ar arsebobs materiis gareSe. m oZ r a ob a ,

s ity v is

f arT o

( fi lo s o f iu r i)

g a ge b iT ,

gu -

l is x m ob s y v e l a cv l il e b a s , y v e l a pr oce s s , d aw y e b u ls u b r al o g a d a a dg ile b iT d a d am T av re b u ls azr ov n e b is r T u l i pr oce s iT . yoveli movlena, materiis yoveli cvlileba warmoebs sivrcesa da droSi. sivrce da dro ar arsebobs materiis gareSe iseve, rogorc materia ar arsebobs sivrcisa da drois gareSe. marTlac, yovel materialur obieqts aqvs garkveuli moculoba, e.i. misTvis damaxasiaTebelia sivrciTi ganfeniloba. dro ki gamosaxavs erTmaneTis Senacvlebadi procesebis Tanamimdevrobas. s iv r ce d a dr o m a te r ii s y of ie re b is f orm e b i a. fizikuri kvlevis meTodebia dakvirveba da cda, eqsperimenti. dakvirveba aris movlenis Seswavla bunebriv pirobebSi. eqsperimenti aris movlenis Seswavla xelovnur, laboratoriul pirobebSi. es pirobebi mkacrad kontrolirebadia, saSualebas iZleva, Tvalyuri vadevnoT movlenis msvlelobas da aRvadginoT (gavimeoroT) igi yovelTvis, am pirobebis gameorebisas. eqsperiments mTeli rigi upiratesobebi aqvs dakvirvebasTan SedarebiT: igi zogavs dros, radgan aCqarebs movlenis Seswavlis SesaZleblobas (mecnieri ar ucdis, rodis warimarTeba es movlena bunebaSi, aramed xelovnurad qmnis mas laboratoriaSi, saWiro dros), xSir SemTxvevaSi afarToebs movlenis Seswavlis diapazons (mag., bunebaSi temperaturis cvlileba xdeba SedarebiT mcire intervalSi, laboratoriaSi ki SeiZleba

miRebul

iqnas

rogorc

Zalian

maRali,

ise

Zalian

dabali

temperatura) da a.S. dagrovili eqsperimentuli monacemebis safuZvelze iqmneba hipoTeza. hipoTeza _ samecniero varaudia, romelic gamoiTqmeba ama Tu im faqtis an movlenis asaxsnelad. gamoTqmuli hipoTezis siswore mowmdeba Sesabamisi cdebis CatarebiT. Tu hipoTeza da misgan gamomdinare Sedegebi ar ewinaaRmdegeba arc erT cdiseul faqts, maSin igi yalibdeba fizikur Teoriad. fizikuri Teoria aris ZiriTadi ideebis sistema, romelic ganazogadebs cdiseul faqtebs da asaxavs bunebis obieqtur kanonzomierebebs.

-7-

§2. fizikuri sidideebis erTeulebi. erTeulTa sistemebi fizikis kanonTa umravlesoba warmodgenilia formulebis saxiT, romlebic amyarebs damokidebulebas sxvadasxva fizikuri sidideebis ricxviT mniSvnelobebs Soris. am mniSvnelobebis misaRebad saWiroa fizikuri sidideebis gazomva. fizikuri sididis gazomva niSnavs mis Sedarebas misive gvaris fizikur sididesTan (sigrZis _ sigrZesTan, masis _ masasTan da a.S.), romelic pirobiTad erTeulad aris miCneuli. principSi, SeiZleboda, yoveli fizikuri sididisaTvis Tavisi, damoukidebeli erTeulis SerCeva, magram bevri erTeulis gamoyeneba, Tu isini ar arian erTmaneTTan dakavSirebuli, praqtikulad

mouxerxebelia.

amitom

iqcevian

Semdegnairad:

nebismierad

arCeven ramdenime, erTmaneTisgan damoukidebeli fizikuri sididis erTeuls da miiCneven ZiriTadad, yvela danarCeni fizikuri sididis erTeuli ki miiReba ZiriTadi erTeulebisagan, saTanado formulebis gamoyenebiT, romlebic akavSireben am sidideebs ZiriTad sidideebTan. maT warmoebuli erTeulebi ewodeba. Z ir i T ad i d a w a rm oe b u li e rT e u le b is e rT ob l iob as e w o de b a e r T e u lT a s is te m a . 1960 wels, zoma-wonis XI generalurma konferenciam SemoiRo erTeulTa s a e r T a So ri s o s is te m a . es sistema, romelic dRReisaTvis sayovelTaod

miRebulia

rogorc

upiratesi

sistema

mecnierebis,

teqnikisa

saxalxo meurneobis yvela dargSi, mokled aRiniSneba ori asoTi SI (System International). am sistemas safuZvlad udevs Svidi ZiriTadi (sigrZis erTeuli _ metri, masis erTeuli _ kilogrami, drois erTeuli _ wami, temperaturis erTeuli _ kelvini, nivTierebis raodenobis erTeuli _ moli, denis Zalis



fizikuri sidide fizikuri obieqtis (sxeuli, sistema, agreTve sistemis mdgomareobebi da masSi mimdinare procesebi_ zomva di Tviseba, maxasiaTebelia, romelic Tvisobrivi TvalsazrisiT saerToa mravali fizikuri obieqtisaTvis, magram raodenobrivi TvalsazrisiT gansxvavebuli, individualuria TiToeuli maTganisTvis. fizikuri sidideebia: sigrZe, masa, simkvrive wneva, temperatura, siCqare, simZlavre da a.S. mag. fizikuri sidide _ simkvrive Tvisobrivad axasiaTebs mraval fizikur sxeuls (wyali, vercxliswyali, almasi, rkina da sxv.), magram raodenobrivi TvalsazrisiT gansxvavebulia yoveli maTganisaTvis. fizikuri sidide SeiZleba iyos sk ala ruli an veqtor uli . skalaruli ewodeba iseT fizikur sidides, romelic mxolod ricxviTi mniSvnelobiT xasiaTdeba. aseTebia masa, temperatura, dro, muxti da sxv. skalaruli sidide SeiZleba iyos rogorc dadebiTi, aseve uaryofiTi da maTi Sekreba xdeba algebrulad. veqtoruli sidide ewodeba iseTs, romelic ricxviTi mniSvnelobis garda mimarTulebiTac xasiaTdeba. aseTebia siCqare, aCqareba, Zala, impulsi da sxva. maTi Sekreba xdeba geometriulad. -8-

erTeuli _ amperi, sinaTlis Zalis erTeuli _ kandeli) da ori damxmare erTeuli (brtyeli kuTxe _ radiani da sxeulovani kuTxe _ steradiani). m e tr i (m ) aris sigrZe, romelic udris kripton 86-is atomis 2p10 da 5d5 doneebs Soris gadasvlis Sesabamisi gamosxivebis talRis 1650763,73 sigrZes vakuumSi. asjer nakleb erTeuls santimetri ewodeba, aTasjer naklebs _ milimetri, xolo aTasjer mets _ kilometri. k ilo g ram i

(kg)

aris

masa

etalonuri

cilindrisa,

romelic

damzadebulia platinisa da iridiumis Senadnisagan da romlis simaRle fuZis diametris tolia. igi inaxeba zoma-wonis saerTaSoriso biuroSi sevrSi, parizis maxloblad. am etalonis meaTasedi nawilia grami (g), xolo memilionedi _ miligrami (mg). 100 kg-s ewodeba centneri, xolo 1000 kg-s _ tona. w am i (wm) aris drois Sualedi, romelic udris cezium 133-is atomis ZiriTadi mdgomareobis or zefaqiz dones Soris gadasvlis Sesabamisi gamosxivebis 9 192 631 770 periods. k e lv i ni (K) aris wylis sammagi wertilis Sesabamisi temperaturis 1/273,16 nawili. m ol i aris nivTierebis raodenoba, romelic Seicavs avogadros ricxvis

toli

raodenobis

nawilakebis

(molekulebis,

atomebis,

ionebis)

ricxvs. a m pe r i (a) aris denis Zala, romelic gadis vakuumSi moTavsebul, erTi metriT daSorebul or usasrulod grZel da wvril wrfiv gamtarebSi da sigrZis yovel metrze iwvevs 2.10-7 niutonis tol urTierTqmedebis Zalas. k an de li * (kd) aris sinaTlis Zala, romelsac asxivebs specialuri sinaTlis

etalonis**

ganivkveTis

1/600000 m2

farTobi

ganivkveTis

perpendikularuli mimarTulebiT. r a di a ni (rad) aris kuTxe wrewiris or radiuss Soris, roca kuTxis momWimavi rkalis sigrZe radiusis tolia. s te r a di a ni (sr) aris sxeulovani kuTxe, wveroTi sferos centrSi, roca kuTxis momWimavi zedapiri sferos radiusis kvadratis tolia. *

misi adrindeli dasaxelebaa sanTeli. saerTod sityva kandela berZnuli warmoSobisaa da

niSnavs sanTels. **

specialuri

sinaTlis

etaloni

damzadebulia

absoluturad

Savi

sxeulis

saxiT.

gamosxiveba xdeba platinis gamyarebis temperaturisa (2046,6 K) da 101325 pa wnevis dros. -9-

SI sistemis SemoRebamde fizikaSi iyenebdnen (da rig SemTxvevebSi dResac iyeneben) erTeulTa ramdenime sistemas. esenia: absoluturi CGS (ce-Je-es) sistema, romelic meqanikuri gazomvebisaTvis emyareba sam ZiriTad erTeuls _ sigrZis, masis da drois erTeulebs. sigrZis erTeulia santimetri, masis erTeuli _ grami, xolo drois erTeuli _ wami. eleqtruli da magnituri sidideebis gasazomad gamoiyeneboda CGS sistemis bazaze agebuli absoluturi eleqtrostatikuri sistema, romelic aRiniSneba CGSEE simboloTi da absoluturi eleqtromagnituri sistema _ CGSM sistema. metad gavrcelebuli iyo e.w. simetriuli CGS anu gausis sistema, romelSic magnituri erTeulebi toli iyo CGSM sistemis erTeulebis, xolo eleqtruli _ CGSE sistemis erTeulebis. CGS

sistemis

erTeulebi,

simciris

gamo,

yoveldRiur

cxovrebaSi

naklebad

gamoiyeneba, vinaidan am SemTxvevaSi saqme gveqneboda Zalian did ricxvebTan. magaliTad, aravin

gamosaxavs

oTaxis

farTobs

kvadratul

santimetrebSi.

mxriv,

ufro

mosaxerxebelia erTeulTa praqtikuli anu MKS (em-ka-es) sistema. meqanikuri gazomvebisTvis am sistemaSic ZiriTad sidideebad miRebulia isev sigrZe, masa da dro, magram sigrZisa da masis erTeulebi gazrdilia. saxeldobr, sigrZis erTeulad miRebulia metri, xolo masis erTeulad _ kilogrami. drois erTeuli aqac (iseve, rogorc yvela sxva sistemaSi) wamia. eleqtruli da magnituri sidideebis gazomvebisaTvis gamoiyeneba MKSA sistema, romelic miiReba MKS sistemaze meoTxe ZiriTadi erTeulis _ amperis damatebiT. rogorc vxedavT, MKS da MKSA sistemebi arian Semadgeneli nawilebi saerTaSoriso SI sistemisa. teqnikaSi

jer

kidev

iyeneben

erTeulTa

teqnikur

si stem as .

sistema

principulad gansxvavdeba yvela danarCeni sistemisagan imiT, rom meore ZiriTad sidided miCneulia Zala da ara masa. amrigad. am sistemis ZiriTadi erTeulebia: sigrZis erTeuli _ metri, Zalis erTeuli _ kilogram-Zala (kgZ), drois erTeuli _ wm. kgZ aris Zala, romelic 1 kg masis mqone sxeuls aniWebs 9,80665 m/wm2 aCqarebas. es sistema ar gamodgeba eleqtruli da magnituri sidideebis erTeulebis dasadgenad.

gamosaxulebas, romelic gviCvenebs ama Tu im fizikuri sididis warmoebuli erTeulis damokidebulebas ZiriTadi sidideebis erTeulebze, ewodeba am sididis g a n zom il e b a . fizikuri sididis ganzomileba aRiniSneba am sididis kvadratul frCxilebSi CasmiT, xolo misi dadgena xdeba am sididis ganmsazRvreli formulis saSualebiT. magaliTad, siCqaris, aCqarebis da Zalis ganzomilebebisaTvis gveqneba: 1    S   L  LT 1 ; a     LT t  T t  T F   m a  MLT 2

 LT 2 ;

sadac L, T da M , Sesabamisad, sigrZis, drois da masis ganzomilebebia: s  L;

t   T

da m  M . - 10 -

pirveli nawili

me qanikis safuZvlebi I Tavi

wertilis kinematika ยง3. meqanikuri moZraoba. moZraobis kinematikuri gantolebebi meqanika aris fizikis nawili, romelic Seiswavlis meqanikur moZraobas. s xe u lis m e q anik u ri m oZ raob a e w ode b a am s xe u lis m de b a r eo b is Se c v las s xv a s xe u le b is m im arT . meqanikuri moZraoba moZraobis umartivesi saxea da ar arsebobs bunebaSi sxeuli, mas rom ar asrulebdes. mag., mTa, xe, saxli, romlebic uZravia dedamiwis mimarT, monawileoben masTan erTad brunviT moZraobaSi da icvlian mdebareobas mzisa da sxva planetebis mimarT. amrigad, ar arsebobs absoluturi uZraoba. y ov e li u Z r a ob a, is e v e rogorc m oZ raob a, fardob iT ia. meqanika sami nawilisagan Sedgeba: kinematika, dinamika da statika. k ine m atik a aris meqanikis nawili, romelic Seiswavlis sxeulebis moZraobas maTi masebisa da moZraobis gamomwvevi mizezebis gauTvaliswineblad. din am ik a Seiswavlis sxeulebis moZraobas maTi masebisa da moZraobis gamomwvevi mizezebis gaTvaliswinebiT. s tatik a Seiswavlis sxeulis wonasworobis pirobebs. Tu cnobilia moZraobis kanonebi, maTgan SeiZleba wonasworobis kanonebis miReba. amitom statikis kanonebi dinamikis kanonebisagan gancalkevebiT ar ganixileba. meqanikuri moZraobis yvelaze martivi SemTxvevaa wertilis moZraoba, magram zogjer, amocanis gamartivebis mizniT, SeiZleba mTeli sxeuli ganxilul iqnas rogorc erTi wertili. amisaTvis SemoaqvT nivTieri wertilis cneba. niv T ie ri anu m ate ria lu ri w e rtili e w ode b a is e T s xe u ls , r om lis

zom e b i

Se iZ le b a

u gu le b e lvy oT

m oce m u li

am ocanis

g anxilv is as . am ganmartebidan gamomdinareobs, rom erTi da igive sxeuli zog SemTxvevaSi SeiZleba ganvixiloT rogorc nivTieri wertili, zogSi ki _ ara. magaliTad, traeqtoriis formis dadgenisas TviTmfrinavi SeiZleba ganvixiloT rogorc nivTieri wertili, TviTmfrinavis haeris nakadTan urTierTqmedebisas ki mxedvelobaSi unda miviRoT misi rogorc zomebi, aseve forma.

sxeulis moZraoba cnobilia, Tu cnobilia misi mdebareoba drois nebismier momentSi. sxeulis mdebareobis gansazRvra ki SeiZleba mxolod romelime sxva sxeulis mimarT. sxeuls (an sxeulTa jgufs), romlis mimarTac

ganixileba moZraoba,

a T v lis

s xe u li

ewodeba.

aTvlis

sxeuls,

rogorc wesi, ukavSireben ama Tu im sakoordinato sistemas. garda amisa, procesi unda iyos qronometrizebuli, e.i. momaragebuli unda iyos drois sazomi xelsawyoTi, romlis saSualebiTac ganisazRvreba moZravi sxeulis sivrceSi sxvadasxva mdebareobebis Sesabamisi drois momentebi. aT v lis s xe u ls , m as T an d ak av Sire b u l s ak oordin ato s is t e m as d a drois as aT v le l xe ls awy os e rT ob liob aSi e w ode b a aT v lis s is te m a. yvelaze ufro gavrcelebulia dekartis marTkuTxa koordinatTa sistema. am sistemaSi M nivTieri wertilis mdebareoba, romlis radius-veqto ria r , ganisazRvreba sami koordinatiT _ x, y, z (sur. 3.1). sxeulis moZraoba cnobilia, Tu cnobilia x -is, y -is da z -is mniSv-

nelobebi drois nebismieri momentisaTvis, anu Tu cnobilia Semdegi funqciuri damokidebulebani:

x  xt , y  yt , z  zt  .

M z

 r

(3.1)

(3.1) gantolebebs ewodeba moZraobis k ine m ay

x y

sur. 3.1

ti k u ri gantolebebi. moZraobis kinematikuri gantoleba SeiZleba Caiweros veqtoruladac.   r x, y, z   r t  .

(3.2)

$4. wrfivi Tanabari da Tanabarcvladi moZraobebi. maTi gantolebebi da grafikebi yoveli sxeuli moZraobisas

aRwers

wirs,

romelsac

traeqtoria

ewodeba. traeqtoria SeiZleba iyos wrfe an mrudi wiri. amis mixedviT ansxvaveben wrfiv da mrudwirul moZraobebs. traeqtoriis formisagan damoukideblad moZraoba SeiZleba iyos Tanabari an araTanabari. moZraobis 

traeqtoriis gaswvriv erTi mimarTulebiT gavlil manZils gza ewodeba, xolo veqtors, romelic sxeulis sawyis mdebareobas aerTebs momdevnosTan _ gadaadgileba. cxadia, gadaadgilebis sidide Tanxvdeba gzas mxolod wrfivi moZraobis dros. 12

umartivesi saxea wrfivi Tanabari moZraoba. es iseTi moZraobaa, romlis traeqtoria aris wrfe da romlis drosac sxeuli drois nebismier tol SualedebSi toli sigrZis gzebs gadis. moZraobis dasaxasiaTeblad fizikaSi SemoRebulia siCqaris cneba. vTqvaT, sxeulma t droSi gaiara s manZili. maSin siCqare s  .

(4.1) t Tu (4.1) formulaSi davuSvebT, rom t  1 , maSin   s , e.i. s iCq ar e e r T e u lov an droSi g av lili m anZ ilis toli a. siCqaris erTeulis misaRebad (4.1) formulaSi unda CavsvaT manZilisa da drois erTeulebi. SI sistemaSi siCqaris erTeulia m/wm. Tanabari moZraobis ganmartebidan gamomdinareobs, rom am moZraobis siCqare mudmivi sididea:

  const .

(4.2)

Tanabari moZraobis dros gavlili gza gamoiTvleba (4.1) formulidan: (4.3) s t . vinaidan   const , amitom s ~ t , e.i. Tanabari moZraobis dros gavlili gza drois pirdapirproporciulia. (4.1) da (4.3) gantolebebs ewodeba Tanabari moZraobis gantolebebi. am gantolebebis grafikebs aqvs Semdegi saxe:



  const

s t S

 o

(4.2)

suraTebidan

sur. 4.2

sur. 4.1 (4.1)

 

Cans,

rom

Tanabari

moZraobis

siCqaris

grafiki warmoadgens droTa RerZis paralelur wrfes, xolo gzis grafiki koordinatTa saTaveSi gamavali wrfea. (4.2) suraTidan s  t  tg , saidanac tg 

s   . e.i. gzis grafikis droTa RerZisadmi daxris  kuTxe damokidet

bulia moZraobis siCqareze _ rac metia siCqare, miT metia daxris kuTxe. 13

wrfivi Tanabarcvladi moZraoba. moZraobaTa umravlesoba, romelTac

Cven vxvdebiT bunebaSi da yoveldRiur cxovrebaSi, aris araTanabari anu cvladi.

dros

sxeulis

siCqare

ganuwyvetliv

icvleba.

moZraobis dasaxasiaTeblad SemoRebulia s a Su alo

araTanaabri

da m y is i

siCqaris

cnebebi: vTqvaT, sxeuli wrfeze araTanabrad moZraobisas, t droSi aRmoCnda A wertilidan B wertilSi, gaiara ra AB  S manZili (sur. 4.3.) saSualo

siCqare AB ubanze ganisazRvreba tolobiT: S . t siCqare gzis

saS. 

e.i.

saSualo

(4.4)

S

mocemul ubanze

t

izomeba am ubnis sigrZis fardobiT im dros-

sur.4.3

Tan, ra droSic igi iqna gavlili. cxadia, rom sazogadod, erTi ubnisTvis gamoTvlili saSualo siCqare sxva ubnebisTvis vargisi

iqneba.

araTanabari

moZraobis

dasaxasiaTeblad

mxolod

saSualo siCqaris codna sakmarisi ar aris, vinaidan igi ar gvaZlevs informacias moZraobis cvlilebis xasiaTze t araTanabari formuliT: m y is i

moZraobis

dros

drois ganmavlobaSi.

gavlili

manZili

an S  saS.  t , S  saS.  t . s iCq are e w ode b a s iCq are s , rom e lic

gamoiTvleba

s xe u ls

(4.5) gaa Cnia

tr ae q tor iis m oc e m u l w e rtilSi an, r ac i giv e a, drois m oce m u l m om e ntSi .

rogor

gavigoT

igi? vTqvaT,

gvinda gavigoT

myisi

siCqare

traeqtoriis A wertilSi. cxadia, igi gansxvavdeba saSualo siCqarisagan AB ubanze, magram isic cxadia, rom rac ufro mcirea AB ubnis sigrZe anu

moZraobis t dro, miT ufro mcire iqneba es gansxvaveba (mcire droSi ver moeswreba siCqaris Secvla). amitom, A wertilSi myisi siCqaris misaRebad unda movnaxoT zRvari, romliskenac miiswrafvis saSualo siCqare, roca t miiswrafvis nulisaken. e.i. S . * t 0 t 0 t maTematikidan cnobilia, rom es zRvari warmoadgens manZilis pirveli

  lim saS.  lim

rigis warmoebuls droiT. amitom SeiZleba davweroT: * limes laTinuri sityvaa da niSnavs sazRvars, zRvars.

dS , (4.6) dt e.i. wrfivi moZraobis dros m y is i s i Cq are tolia m anZ ilis p irv e li



r igis w arm oe b u lis a droiT . (4.6) formula gansazRvravs myisi siCqaris sidides. rac Seexeba mimarTulebas, cxadia, rom wrfivi moZraobis dros siCqaris veqtoris mimarTuleba Tanxvdeba moZraobis mimarTulebas. rogorc aRvniSneT, araTanabari moZraobis dros siCqare ganuwyvetliv icvleba. s i Cq aris

cv lile b is

s is w rafis

das axas i aT e b lad

Se m o -

aq v T aCq are b is cne b a. zogad SemTxvevaSi, wrfivi araTanabari moZraobis dros icvleba ara marto siCqare, aramed aCqarebac. amitom siCqaris msgavsad, aqac unda SemoviRoT saSualo da myisi aCqarebebi: Tu t drois ganmavlobaSi sxeulis siCqare Seicvala  sididiT, maSin saSualo aCqareba

 , (4.7) t e.i. saSualo aCqareba tolia siCqaris cvlilebis fardobisa im drosTan, ra asaS . 

droSic es cvlileba moxda. myisi aCqarebis misaRebad unda moviqceT ise, rogorc moviqeciT myisi siCqaris miRebisas _ gadavideT zRvarze, roca t miiswrafis nulisaken. miviRebT, rom myisi aCqareba  d , (4.8)  t 0 t 0 t dt e.i. myisi aCqareba tolia siCqaris pirveli rigis warmoebulisa droiT. Tu a  lim asaS .  lim

am ukanasknel formulaSi SevitanT siCqaris mniSvnelobas (4.6) formulidan, miviRebT:

d d  dS  d 2 S    . (4.9) dt dt  dt  dt 2 amrigad, m y is i a Cq are b a toli a s iCq ar is pirv e li ri gis w arm oe b u a

lis a dr oiT , an m anZ ilis m e ore rigis w arm oe b u lis a droiT . araTanabari moZraobis umartivesi saxea Tanabarcvladi moZraoba. es iseTi moZraobaa, romlis drosac siCqare drois nebismier tol SualedebSi toli sididiT icvleba. am SemTxvevaSi      0 , t  t  t 0 da aCqarebisaTvis gveqneba:

a sadac  0

  0 t  t0

aris sxeulis siCqare sawyis t 0

(4.10) momentSi, xolo 

_ siCqare

drois t momentSi. Tu simartivisaTvis davuSvebT, rom t 0  0 gveqneba:

a

  0 .

(4.11) t (4.11) formula gviCvenebs, rom aCqareba ricxobriv ad tolia siCqaris cvlilebis a drois e rTeulSi . SI sistemaSi aCqarebis erTeulia m/wm2. Tanabarcvladi moZraobis ganmartebidan gamomdinareobs, rom am moZraobis dros aCqareba mudmivi sididea: a  const . Tanabarcvladi moZraoba SeiZleba iyos TanabaraCqarebuli da TanabarSenelebuli. pirvel SemTxvevaSi siCqare Tanabrad izrdeba, amitom aCqareba dadebiTia

a  0 ,

meore SemTxvevaSi siCqare Tanabrad mcirdeba da

aCqareba uaryofiTia a  0 . TanabaraCqarebuli moZraobis siCqaris formulas miviRebT (4.11)-dan:

   0  at .

(4.12) TanabaraCqarebuli moZraobis gzis formulis misaRebad gamoviyenoT is faqti, rom, vinaidan am dros siCqare Tanabrad izrdeba, saSualo siCqare tolia sawyisi da saboloo siCqareebis naxevarjamisa:

saS. 

0   .

2 Tu saSualo siCqaris am mniSvnelobas SevitanT (4.5) formulaSi, mivi-

RebT TanabaraCqarebuli moZraobis gzis formulas:

0  

 0   0  at

at 2 . (4.13) 2 2 2 Tu (4.12) da (4.13) formulebidan gamovricxavT dros, miviRebT damokiS  saS.  t 

t 

 t  0t 

debulebas gavlil manZilsa da siCqares Soris:

 2   02 . S

(4.14) 2a (4.12) da (4.13) gantolebebs ewodeba TanabaraCqarebuli moZraobis gantolebebi. usawyiso siCqariT moZraobis SemTxvevaSi buli moZraobis gantolebebi Semdegnairad Caiwereba:

at 2 . 2 am gantolebebis grafikebs aqvs Semdegi saxe:

  at;

S

 0  0 ,

TanabaraCqare-

 S

   0  at

S  0t 

at 2 2 S

  at

0

at 2 2

sur. 4.1

sur. 4.2

sxeulTa Tavisufali vardna. wrfivi TanabaraCqarebuli moZraobis ker-

Zo saxes warmoadgens sxeulis Tavisufali vardna. T a v is u f a li v ar dn a e w o de b a s im Z im is Z al iT g am ow v e u l m oZ r a ob a s u h a e ro s iv r ce S i . Tavisufali vardnis Tavisebureba (kanoni) mdgomareobs SemdegSi: dedamiwis mocemul adgilas (ganedze) yvela sxeuli erTnairi aCqarebiT vardeba. am aCqarebas ewodeba Tavisufali vardnis aCqareba an simZimis Zalis aCqareba da aRiniSneba g asoTi. rogorc aRvniSneT, g mocemul ganedze mudmivi sididea, ganedis mixedviT ki icvleba; kerZod, izrdeba ekvatoridan polusisken

moZraobisas.

g -s

normalur

mniSvnelobad

miCneulia

misi

mniSvneloba geografiul 45 ganedze. aq g daaxloebiT 9,8 m/wm2-is tolia. vinaidan Tavisufali vardna warmoadgens TanabaraCqarebul moZraobas, mis gantolebebs miviRebT TanabaraCqarebuli moZraobis gantolebebidan, Tu a -s SevcvliT g -Ti, s -s- ki _ h -iT (simaRle, rogorc wesi, h -iT aRiniSneba), gveqneba:

   0  gt , h   0t 

Tu  0  0 -s:

gt 2 . 2

  gt , h

gt 2 . 2

vertikalurad zeviT asrolili sxeulis moZraoba. rogorc aRvniSneT,

TanabarSenelebuli

moZraobis

dros

aCqareba

gantolebebs eqneba Semdegi saxe:

   0  at ; S  0t 

at 2 . 2

uaryofiTia.

amitom

mis

TanabarSenelebuli moZraobis kerZo saxes warmoadgens vertikalurad zeviT asrolili sxeulis moZraoba. am dros sxeulze moqmedebs simZimis Zala, romelic aniWebs mas simZimis Zalis g aCqarebas. amitom, am moZraobis gantolebebs

miviRebT,

TanabarSenelebuli

moZraobis

gantolebebSi

a -s SevcvliT g -Ti, xolo s -s _ h -iT. gveqneba:

   0  gt ,

(4.15)

gt . (4.16) 2 gamoviyenoT es gantolebebi da amovxsnaT martivi amocana: gavigoT h  0t 

vertikalurad asrolili sxeulis zeviT moZraobis maqsimaluri dro ( tmaqs. .) da vertikalurad zeviT moZraobis dros gavlili maqsimaluri manZili (asvlis maqsimaluri simaRle). vertikalurad zeviT asrolili sxeulis siCqare TandaTan mcirdeba da bolo wertilSi xdeba nulis toli. amitom tmaqs. -s miviRebT, Tu (4.15) tolobaSi davuSvebT, rom   0 :

0   0  gtmaqs. , saidanac

tmaqs. 

hmaqs. -is misaRebad (4.16)-Si t

; g unda SevcvaloT

0

miRebuli

0



tmaqs. -iT:

2 0 2

 02  02  02 g hmaqs.   0      . g 2 g 2g 2g Sedegebi gviCvenebs, rom rogorc

tmaqs. ,

aseve

hmaqs.

damokidebulia sawyis (asrolis)  0 siCqareze.

§5. mrudwiruli moZraobis siCqare vTqvaT,

nivTieri

wertili

moZraobs

sivrceSi

raime

mrud

wirze

(sur.5.1). SeviswavloT es moZraoba koordinatTa oxyz sistemis mimarT. maSin, wertilis nebismieri mdebareoba wirze ganisazRvreba Sesabamisi radiusveqtoriT.

wertilis

radius-veqtori

drois t momentSi, roca igi imyofeboda A mdgomareobaSi, aRvniSnoT r -iT, xolo t  t momentSi, roca  wertili aRmoCnda B mdgomareobaSi, aRvniSnoT _ r1 -iT. naxazidan Cans, rom    r1  r  AB . 18

(5.1)



sadac

z A  r

 r

gadaadgilebis

veqtoria.*

(5.1)-dan

    AB  r1  r  r amrigad, gadaadgilebis veqtori

radius-veqtoris nazrdis tolia t droSi. si-

 r1

didiT is udris

qordas da gansxvavdeba

sxeulis mier gavlili S gzisagan ( AB rka-

lisagan),

magram

gansxvaveba

miT

ufro

mcirea, rac ufro mcirea drois t Sualedi.

sur. 5.1

maSasadame, sakmaod mcire t drois SemTxvevaSi AB rkali (anu S ) SeiZleba Seicvalos AB qordiT ( r -iT). amiT mrudwirul moZraobas daviyvanT wrfiv moZraobaze

AB mimarTulebiT. wrfivi

moZraobis saSualo siCqare ki udris gavlili gzis Sefardebas Sesabamis drosTan. amitom, wertilis saSualo siCqare AB ubanze iqneba:   r . (5.2) saS  t myisi siCqaris misaRebad drois t momentSi (anu A wertilSi) unda gadavideT zRvarze, roca t miiswrafvis nulisken: 



  lim saS. t 0

  r dr ,  lim  t 0 t dt

maSasadame,

 dr , (5.3)  dt anu m r u dw i ru l i m o Z r aob is dr os m y is i s i Cq a re t oli a ra d iu s 

v e q to r is pirv e l i r i gi s w arm oe b u l is a d ro iT . ganvsazRvroT mrudwiruli moZraobis siCqaris mimarTuleba: saSualo   siCqares aqvs r -is anu AB veqtoris mimarTuleba. zRvarSi, roca t  0 ,



anu B wertili usasrulod uaxlovdeba A -s, AB icvlis mimarTulebas da emTxveva A wertilSi gavlebuli mxebis mimarTulebas. maSasadame, m ru dw i r u li m oZ ra ob i s d ros s i Cq ari s v e q to rs m ru di s y ov e l w e r t il S i a q v s am w e r t il S i g av le b u li m xe b is m im a rT u le b a. rac Seexeba siCqaris sidides, mas davadgenT im mosazrebidan, rom, roca t *

mcirea,

ase ewodeba veqtors, romelic wertilis sawyis mdebareobas aerTebs mis momdevno

mdebareobasTan ($4). 19

r -sa da Sesabamis

gansxvaveba

S

rkals Soris mcirea da miiswrafis

nulisaken, roca t  0 . amitom, siCqaris sididisaTvis (5.3) gvaZlevs: dS , (5.4) dt e.i. mrudwiruli moZraobis siCqare sididiT tolia manZilis pirveli rigis



warmoebulisa droiT (iseve, rogorc gvqonda wrfivi moZraobis dros). (5.3) veqtoruli tolobis dagegmileba koordinatTa RerZebze gvaZlevs siCqaris mdgenelebs:

x 

dx ; dt

y 

dy ; dt

z 

dz . dt

siCqaris sidide

   x2   y2   z2 .

§6. mrudwiruli moZraobis aCqareba. aCqarebis mxebi da normaluri mdgenelebi aCqareba SemoRebulia siCqaris cvlilebis dasaxasiaTeblad. siCqare veqtoruli

sididea,

igi

xasiaTdeba

rogorc

ricxviTi

mniSvnelobiT

(sididiT), ise mimarTulebiT. mrudwiruli moZraobis dros, sazogadod, icvleba siCqaris rogorc sidide, aseve mimarTuleba. amitom, cxadia, rom am moZraobisas daaxasiaTebs

aCqarebas siCqaris

unda

gaaCndes

sididis

ori

mdgeneli,

cvlilebas,

xolo

romelTagan meore

erTi

siCqaris

mimarTulebis cvlilebas. ganvixiloT es sakiTxi dawvrilebiT. jer

D B

) C



SemoviRoT

s im ru dis

w ri s

cneba.

simrudis wre mrudis mocemul A wertilSi ewodeba im wrewiris zRvrul mdebareobas, romelic A wertilis garda gadis kidev B da C werti-

lebSi im pirobiT, rom B  A da C  A (sur.6.1). am wris simrudis radiuss ewodeba mrudis simrudis radiusi A wertilSi rA  , xolo wris centrs

sur.

6.1

_ mrudis simrudis centri A wertilisaTvis. gavavloT B da C

wertilebSi BD da CE mxebi.

maTi normalebi BO da CO warmoadgenen wris radiusebs da gadaikveTebian mis O centrSi. kuTxe radiusebs Soris aRvniSnoT  -Ti. cxadia, igi BD da CE mxebs Soris kuTxis tolia. BC rkalis sigrZe aRvniSnoT S -iT. 20

naxazidan Cans, rom S  r   . saidanac r  S  r rom iyos simrudis radi usi mrudis A wertilisaTvis, pirobis Tanaxmad saWiroa, rom S miiswrafodes nulisaken, an, rac igivea,   0 . amrigad, gveqneba: S .  0  gadavideT axla rA  lim



1 )

 



aCqarebis

 1 F

 2

sakiTxze.

(6.1) uSualod davuSvaT,

mrudwiruli moZraobisas, raRac t

sur. 6.2

droSi, sxeuli aRmoCnda

wertilidan, sadac misi siCqare  iyo 1 , B wertilSi, sadac misi

 siCqare gaxda  2

(sur. 6.2). rogorc naxazidan Cans, am dros Seicvala  siCqaris rogorc sidide, ise mimarTuleba. A wertilidan gadavzomoT  2   veqtoris toli da paraleluri monakveTi AC da 1 veqtoris wvero Seva  erToT AC veqtoris wverosTan. miRebuli veqtori aRvniSnoT  -Ti. veq       torTa Sekrebis wesis Tanaxmad 1     2 saidanac    2  1 e.i.  aris

siCqaris

veqtoris

cvlileba

t

droSi.

saSualo

aCqareba

t

droSi,

ganmartebis Tanaxmad, tolia    , asaS.  t xolo myisi aCqarebis misaRebad unda gadavideT zRvarze, roca t  0     d . a  lim  t 0 t dt Tu (6.3) formulaSi SevitanT siCqaris (5.3) mniSvnelobas, miviRebT:

e.i.

myisi

aCqareba

tolia

 d 2r a 2 . dt radius-veqtoris

(6.2)

(6.3)

(6.4) meore

rigis

warmoebulisa

droiT. (6.4)-dan aCqarebis gegmilebisTvis miviRebT Semdeg mniSvnelobebs:

d2x d2y ax  2 ; a y  2 ; dt dt aCqarebis sididisaTvis ki –

d2z az  2 . dt

a  a x2  a y2  a z2 .  gadavzomoT AC monakveTze 1 veqtoris toli monakveTi AF

(6.5)

 da 1

veqtoris wvero SevaerToT F wertilTan. miRebuli veqtorebi aRvniSnoT 21

 1 -iT da

  2 -iT. naxazidan Cans, rom

    1   2   .  –s mniSvneloba

SevitanoT (6.3)-Si:       1   2  1  2 .   a  lim  lim  lim  lim t 0 t t 0 t 0 t t 0 t t

(6.6)

miRebuli toloba gviCvenebs, rom mrudwiruli moZraobis dros aCqareba marTlac ori mdgenelisagan Sedgeba. pirveli mdgeneli gaCnda imitom, rom siCqarem Seicvala mimarTuleba, amitom igi axasiaTebs siCqaris mimarTulebis cvlilebas. meore mdgeneli gaCnda imitom, rom siCqarem Seicvala sidide, amitom igi axasiaTebs siCqaris sididis cvlilebas. pirvel mdgenels ewodeba norm alu ri anu ce ntris k enu li aCqareba da aRiniSneba a n –iT:  1 ,  (6.7) a n  lim t 0 t xolo meore mdgenels ewodeba m xe bi anu ta nge n cialu ri aCqareba da  aRiniSneba a t -iT:   2 .  (6.8) at  lim t 0 t a n -s ewodeba normaluri anu centriskenuli aCqareba imitom, rom zRvarSi, roca   0 , kuTxe AEF miiswrafvis 90 gradusisken, ris gamoc   1 da maSasadame a n -ic aRmoCndeba 1 siCqaris marTobuli da mimarTuli simrudis centrisken.  a t -s ewodeba mxebi anu tangencialuri aCqareba imitom, rom roca      0 ,  2 -is da maSasadame a t -s mimarTuleba emTxveva 1 -is mimarTulebas, romelsac A wertilSi gavlebuli mxebis mimarTuleba aqvs. davadginoT normaluri aCqarebis sidide. naxazidan Cans, rom 1  1 .  1 -is es mniSvneloba SevitanoT (6.7) formulaSi da tolobis marjvena nawili gavamravloT da gavyoT S -ze:

  S  a n  lim 1  . t 0 S t  magram namravlis zRvari udris TanamamravlTa zRvrebis namravls. garda amisa, roca t  0 , maSin S  0 . Tu 1 -s, rogorc mudmiv sidides, gamovitanT zRvris niSnis gareT, gveqneba: S   lim . t 0 t S 0 S

a n  1 lim

 1 S  ,  1 , xolo (6.1)-is Tanaxmad lim S 0 S t 0 t rA

magram lim amitom

an 

 at

amrigad,  a

 an

12

. rA sabolood

(6.9) SeiZleba

davaskvnaT:

mrudwiruli moZraobis dros aCqarebas gaaCnia ori mdgeneli: 1. norm alu ri anu ce ntris k e nu li aC  q are b a a n , rom e lic a xas i aT e b s si Cq aris m i m ar T u leb is cv lile b as . misi sidide a n   , sar 2

sur.

6.3

dac 

siCqaris mniSvnelobaa traeqtoriis im wer-

tilSi, sadac simrudis radiusi aris r , da 2. m xe b i anu t ange ncialu r i a Cq ar e b a, rom e lic axas i aT e b s s i Cq aris s idid is cv lile b as . misi sidide

vinaidan

normaluri

  d . at  lim  t 0 t dt da mxebi aCqarebebi

urTierTmarTobni

arian

(sur. 6.3), amitom sruli aCqarebis sidide 2

  2   d  . a  a  a        r   dt  2 n

2 t

II Tavi

m y a r i s xe u l i s k i n e m a t i k a §7. myari sxeulis gadataniTi moZraoba myari sxeulis moZraobis ganxilvisas, sakiTxis gamartivebis mizniT, SemoaqvT absoluturad myari sxeulis cneba. ab s olu tu rad m y ari s xe u li e w ode b a is e T s xe u ls , rom lis ne b is m ie r or w e rtils So r is m an Z ili ar icv le b a ne b is m ie ri gare g ani ze m oq me de b is dros . sxvanairad, es aris absoluturad aradeformadi sxeuli. cxadia, bunebaSi absoluturad myari sxeulebi ar arsebobs. yoveli sxeuli Zalis moqmedebiT deformirdeba _ icvlis formas an moculobas da maSasadame, icvleba manZili mis wertilebs Soris, magram myari sxeulebis umravlesobis 23

deformacia*, Tu maTze didi Zalebi ar moqmedebs, imdenad umniSvneloa, rom SeiZleba

misi

ugulebelyofa

maTi

CaTvla

absoluturad

myar

sxeulebad. absoluturad myari sxeulis nebismieri moZraoba daiyvaneba mis gadataniT da brunviT moZraobaze. m y ar i s xe u lis ga data niT i m oZ raob a e w ode b a is e T m oZ raob as , B

sur. 7.1 r om lis

dros ac s xe u lSi

gav leb u li ne b is m ie ri

w rfe

m oZ -

r ao b is as r Ce b a T av is T av is parale lu ri (sur. 7.1). gadataniTi moZraobis dros myari sxeulis yvela wertili erTnairad moZraobs _ aqvT erTnairi traeqtoriebi, siCqare da aCqareba. amitom myari sxeulis gadataniTi moZraobis dasaxasiaTeblad sakmarisia misi erT-erTi wertilis moZraobis Seswavla.

§8. myari sxeulis brunviTi moZraoba m y ar i s xe u lis b ru nv iT i m oZ raob a e w ode b a is e T m oZ raob as , r om lis dros ac m is i y v e la w e rtili Se m os we rs parale lu r w r e w ir e b s , r om e lT a ce ntre b i e rT s w or xazze (w rfe ze ) m de b are ob s . mas brunvis RerZi ewodeba*. myari sxeulis brunviTi moZraoba xasiaTdeba kuTxuri siCqariTa da aCqarebiT. ganvixiloT es sidideebi. davuSvaT, myari sxeuli brunavs OO' RerZis garSemo da raRac droSi misi romeliRac wertili aRmoCnda A mdebareobidan B

t

mdebareo-

baSi. am dros wertilis centrTan SemaerTebeli radiusi Semobrunda  kuTxiT (sur. 8.1). cxadia, rom fardoba

 axasiaTebs wertilis (sxeulis) t

brunvis siswrafes. mas ewodeba kuTxuri siCqare da aRiniSneba  asoTi: *

sityva deformacia laTinuri warmoSobisaa da niSnavs damaxinjebas.

* Cven ganvixilavT brunviTi moZraobis umartives SemTxvevas, roca brunvis RerZi uZravia.



 . t

(8.1)

(8.1) formulidan Cans, rom k u T xu ri s iC q a re ric xob riv ad e rT e u lov an droS i ra di A

u s is Se m obr u ne b is k u T xis tolia. SI siste-

 B

maSi misi sazomi erTeulia radiani/wm, magram vinaidan radiani kuTxis ganyenebuli sazomia, am erTeuls ubralod aRniSnaven 1/wm-iT.

o sur.

Tu sxeulis brunva araTanabaria, (8.1) for8.1

mula gamosaxavs saSualo kuTxur siCqares

t

droSi. myisi kuTxuri siCqaris misaRebad unda gadavideT zRvarze, roca t  0 :

 d .  t 0 t dt kuTxuri siCqare tolia

  lim

maSasadame,

myisi

(8.2) radiusis

Semobrunebis

kuTxis pirveli rigis warmoebulisa droiT. araTanabari brunvisas kuTxuri siCqare drois mixedviT icvleba. am cvlilebis dasaxasiaTeblad SemoaqvT kuTxuri aCqarebis cneba. vTqvaT, t droSi kuTxuri siCqare Seicvala  sididiT. maSin, kuTxuri aCqareba  . (8.3) t (8.3) formula gviCvevnebs, rom kuTxuri aCqarebis sazomi erTeuli SI



sistemaSi aris 1/wm2. nebismieri araTanabari brunvis dros, roca  ganuwyvetliv icvleba, (8.3) formula gansazRvravs kuTxuri aCqarebis saSualo mniSvnelobas t droSi. myis kuTxur aCqarebas miviRebT, Tu gadavalT zRvarze, roca t  0 :

  lim

t 0

 d  t dt .

(8.4)

e.i. myisi kuTxuri aCqareba tolia kuTxuri siCqaris pirveli rigis warmoebulisa

droiT.

(8.4)-Si

SevitanT

 -s

mniSvnelobas

(8.2)-dan,

gveqneba:

d d  d  d 2   . (8.5)  dt dt  dt  dt 2 maSasadame, myisi kuTxuri aCqareba udris kuTxuri siCqaris pirveli



rigis warmoebuls droiT, an kuTxis meore rigis warmoebuls droiT. brunviTi moZraoba xasiaTdeba agreTve periodiTa da sixSiriT. 25

pe r io di e w ode b a dros , rom lis g anm av lob aSic s xe u li e r T srul

b r u ns

as ru le b s .

aR ini Sne b a

as oT i,

izom e b a

w am e b iT .

e r T w am Si Se s ru le b u l b ru nT a rao de nob as e w ode b a b ru nT a r i c xv i an s ix Sire . aRiniSneba v -Ti. cxadia, rom T 

1 . marTlac, Tu sxeuv

li erT wamSi v bruns asrulebs, maSin erTi brunis Sesasruleblad saWiro dro anu periodi iqneba v -jer naklebi. SI sistemaSi v -s sazomi erTeulia 1/wm. sixSiris erTeuls SI sistemaSi ewodeba herci. es iseTi sixSirea, romlis drosac erT wamSi sruldeba erTi sruli rxeva. davamyaroT damokidebuleba  -sa da v -s Soris. amisaTvis gamoviyenoT is faqti, rom periodis ganmavlobaSi radiusis mier Semowerili kuTxe 3600is anu 2 radianis tolia:  2   2v . t T wertilebs, garda kuTxuri



mbrunavi gaaCniaT

sxeulis

xazovani

anu

wiriTi

.

siCqare

(8.6) siCqarisa,

davamyaroT

cxadia,

maT

Soris

damokidebuleba. (8.1) suraTze AB rkalis sigrZe aRvniSnoT S -iT, maSin 1 S . r  -s es mniSvneloba SevitanoT (8.2) formulaSi:

S  r   ,

saidanac

 

 1 S 1 dS 1  lim    , t 0 t r t 0 t r dt r

  lim saidanac

  r .

(8.7) maSasadame, wiriTi siCqare kuTxuri siCqarisa da radiusis namravlis tolia. myari sxeulis brunvisas misi yvela wertilis kuTxuri siCqare erTnairia, wiriTi siCqare ki im wertilebisaa meti, romlebic ufro metad arian daSorebulni brunvis RerZidan. davamyaroT

damokidebuleba

wiriT

kuTxur

aCqarebebs

Soris.

amisaTvis (8.7) toloba gavawarmooT droiT: d d r . dt dt

d d  at , xolo   , amitom dt dt at   r . (8.8) e.i. tangencialuri aCqareba kuTxuri aCqarebisa da radiusis namravlis magram

tolia.

normaluri

aCqareba

agreTve 26

SeiZleba

gamovsaxoT

brunviTi

moZraobis damaxasiaTebeli sidideebis saSualebiT. amisaTvis normaluri aCqarebis

gamomsaxvel

formulaSi

SevitanoT

 -s

mniSvneloba

(8.7)

formulidan:

an 

2



 2r 2

  2r .

r r Tu am formulaSi SevitanT  -s mniSvnelobas (8.6) formulidan, gveqneba:

an 

4 2 r  4 2 v 2 r . T2

(8.9)

§9. kuTxuri siCqaris veqtori sxeulis brunviTi moZraobis sruli daxasiaTebisaTvis marto kuTxuri siCqaris sididis codna ar aris sakmarisi. saWiroa vicodeT kidev romeli RerZis irgvliv warmoebs brunva da am RerZis irgvliv sxeulis brunvis mimarTuleba. samive es maxasiaTebeli gansazRvruli iqneba, Tu kuTxur  siCqares ganvixilavT rogorc veqtors (  ), romlis sidide gamoiTvleba (8.2) formuliT, SevuTavsebT am veqtors brunvis RerZs da mis mimarTulebas

 A









 C

 r

 B sur. sur.

9.2

9.1

davukavSirebT sxeulis brunvis mimarTulebas. es dakavSireba xdeba marjvena burRis wesiT: Tu burRis saxeluris brunvis mimarTuleba emTxveva sxeulis brunvis mimarTulebas, maSin burRis gadataniTi moZraobis mimarTuleba gviCvevnebs kuTxuri siCqaris veqtoris mimarTulebas. radgan kuTxuri siCqare veqtoria, (8.7) toloba SeiZleba daiweros veqtoruli saxiT. amisaTvis gamoviyenoT veqtoruli namravlis cneba, rac fa  rTod gamoiyeneba veqtorul analizSi: ori, A da B veqtoris veqtoruli  namravli ewodeba iseT C veqtors, romlis sidide ganisazRvreba tolobiT

C  A  B  sin  . 27

(9.1)

sadac  aris kuTxe C da B veqtorebs Soris. rac Seexeba mimarTulebas,    C yovelTvis marTobia A da B veqtorebze gavlebuli sibrtyisa da aqvs im burRis gadataniTi moZraobis mimarTuleba, romlis saxelursac vabru  nebT A -dan B -sken (sur.9.1). veqtoruli namravli Semdegnairad aRiniSneba*:

 

   (9.2) C  A B . mivaqcioT yuradReba im garemoebas, rom veqtoruli namravlis ganmartebidan

gamomdinare,

mamravlebis

gadasmis

Sedegad

veqtoruli

namravli

icvlis niSans:

A  B   B  A  .

   wertilis wrewirze brunvis dros  ,  da r veqtorebis urTierTganlageba mocemulia (9.2) suraTze. amitom SeiZleba davweroT:        r . marTlac, (9.1)-is Tanaxmad

  90  ,

    r  sin     r , rac

Tanxvdeba

(8.7)-s.

garda





amisa,

(9.3)



marTobia





 r

veqtorebze

gavlebuli sibrtyisa da aqvs im burRis gadataniTi moZraobis mimarTuleba,   romlis saxelursac vabrunebT  -dan r –isken. kuTxuri aCqarebac, iseve, rogorc kuTxuri siCqare, veqtorul sidides warmoadgens. amitom (8.4) formulas SeiZleba mivceT veqtoruli saxe:  d . (9.4)  dt   d   0 ,  veqtoris mimarTuleba emTxveTu kuTxuri siCqare izrdeba   dt   va  -s mimarTulebas da piriqiT _ Tu kuTxuri siCqare mcirdeba     d   0 , maSin  -s aqvs  -s sawinaaRmdego mimarTuleba.   dt 

* aqve SevniSnoT, rom arsebobs veqtorebis skalarulad gadamravlebis wesic:

  A da B

veqtoqrebis skalaruli namravli ewodeba iseT C skalarul sidides, romelic ganisazRvreba Semdegi tolobiT: C  A  B  cos  sadac





aris kuTxe

 

skalaruli namravli Semdegnairad aRiniSneba C  A  B



  A da B veqtorebs Soris.

III Tavi

n i v T i e r i we r t i l i s d i n a mi k a §10. klasikuri meqanikis gamoyenebis sazRvrebi viciT (§3), rom meqanika Sedgeba sami nawilisagan _ kinematika, dinamika da statika. amaTgan dinamika, romelic Seiswavlis sxeulebis moZraobas maTi masebisa da moZraobis gamomwvevi mizezebis gaTvaliswinebiT, meqanikis mTavari nawilia. dinamikas safuZvlad udevs niutonis sami kanoni. aRsaniSnavia, rom es kanonebi cdiseuli kanonebia _ niutonma daadgina isini mravalricxovani dakvirvebebis da cdebis safuZvelze. m e q anik as ,

rom e ls ac

s afu Z v lad

u de v s

niu tonis

k anone b i,

k l a s ik u r i m e q a nik a ewodeba. xangrZlivi drois ganmavlobaSi iTvleboda, rom klasikuri meqanika iZleva meqanikuri movlenebis zust da amomwurav aRweras, magram Semdeg gairkva, rom es ase ar aris. klasikuri meqanika vargisia mxolod iseTi sxeulebis moZraobis asawerad, romelTa  siCqare gacilebiT naklebia vakuumSi sinaTlis gavrcelebis c siCqareze   c  . im SemTxvevaSi

ki,

roca

sxeuli

moZraobs

sinaTlis

siCqaris

maxlobeli

siCqariT, klasikuri meqanika xdeba uvargisi da mis adgils iWers relativisturi meqanika anu ainStainis fardobiTobis Teoria. ganvixiloT magaliTi. klasikur meqanikaSi sxeulis masa mudmivi sididea, ar aris damokidebuli moZraobis siCqareze. relativistur meqanikaSi ki masa damokidebulia moZraobis siCqareze da es damokidebuleba mocemulia formuliT*:

m

m0 1

2

(10.1)

c2 sadac m0 aris uZravi sxeulis masa (mas uZraobis masa ewodeba), xolo m _

 siCqariT moZravi sxeulis masa. c _ sinaTlis gavrcelebis siCqarea. (10.1) formula gviCvenebs, rom m  m0 , kerZod m  m0 da miT ufro, rac ufro 2 axloa  sinaTlis c siCqaresTan, magram Tu   c , maSin   0 da m  m0 , c2

* aRsaniSnavia, rom (10.1) formula, romelic gamoiyvaneba fardobiTobis TeoriaSi, am

Teoriis

gamoqveynebamde

(1905 w)

ramdenadme

adre,

kerZod,

me-19

saukunis

eqsperimentulad iqna dadgenili swrafi eleqtronebis moZraobis Seswavlisas. 29

bolos

anu vimyofebiT klasikuri meqanikis gamoyenebis sazRvrebSi. amrigad, klasikuri meqanikis gamoyenebis kriteriums warmoadgens piroba:

  c . klasikuri

meqanikis

gamoyenebis

meore

(10.2) dakavSirebulia

SezRudva

sxeulis masasTan. XX saukunis dasawyisSi aRmoCnda, rom Zlier mcire masis nawilakebis, e.w. mikronawilakebis (eleqtronebi, protonebi, neitronebi, atomebi,

mcire

zomis

molekulebi) moZraobis

zusti aRwera klasikur

meqanikas ar SeuZlia. mikronawilakebis moZraobas ganixilavs kvanturi meqanika. kvanturi meqanika dafuZnebulia frangi fizikosis de-broilis hipoTezaze imis Sesaxeb, rom korpuskulur-talRuri dualizmi damaxasiaTebelia

ara

broilis

marto

sinaTlisaTvis,

aramed

Tanaxmad,

mikronawilaks,

romlis

mikronawilakebisTvisac. masaa

de-

moZraobs



siCqariT, eTanadeba talRa, romlis sigrZe gamoiTvleba formuliT: h , m aris plankis



sadac

h  6,62  10 34 jouli.wm

talRuri

bunebidan

gamomdinareobs,

rom

(10.3) mudmiva. `klasikuri

mikronawilakis nawilakisagan~

gansxvavebiT*, SeuZlebelia misi mdebareobisa ( x koordinatis) da siCqaris erTdrouli da zusti gansazRvra _ rac ufro zustad ganvsazRvravT nawilakis mdebareobas, miT ufro did Secdomas davuSvebT misi siCqaris gansazRvrisas da piriqiT. aRmoCnda, rom nawilakis x koordinatisa da  siCqaris erTdrouli gansazRvrisas maTi gazomvis x da  cdomilebebis (uzustobebis,

ganuzRvrelobebis)

namravli

SeiZleba

sidideze

naklebi iyos:

x    h (10.4)

Tanafardobas

ewodeba

. m h aize nb e rgis

(10.4) ganu zR v re lob aT a

T an afar d ob a (princi pi). igi gviCvenebs, rom rac ufro mcirea x , miT metia  da piriqiT. mniSvnelovania aRiniSnos, rom mikronawilakis koordinatisa da siCqaris erTdrouli da zusti gansazRvris SeuZlebloba gamowveulia ara xelsawyoebisa da gazomvis meTodebis arasrulyofilebiT, aramed amis principuli SeuZleblobiT. es principuli SeuZlebloba asaxavs mikronawilakTa * ase ewodeba nawilaks, romlis moZraoba aiwereba klasikuri meqanikis kanonebiT.

obieqtur Tvisebebs, kerZod imas, rom maT ormagi (dualisturi) buneba aqvT _ erTdroulad gaaCniaT nawilakis (korpuskulis) Tvisebebic da talRisac. vinaidan plankis mudmiva metad mcire sididea, amitom didi masis sxeulebisaTvis h m  0 . aseT SemTxvevaSi x da  ganuzRvrelobebi erTdroulad SeiZleba umniSvnelod mcire sididis iyos. marTlac, didi masis sxeuli klasikuri sxeulia da, rogorc viciT, misTvis ar arsebobs haizenbergis SezRudva. misi koordinata da siCqare erTdroulad da zustad izomeba. mikronawilakebisaTvis, maTi masebis simciris gamo, fardoba h m aSkarad metia nulze, amitom ar SeiZleba, rom x da  sidideebi erTdroulad aRmoCndes Zalian mcire. naTqvamis naTelsayofad ganvixiloT magaliTi. davuSvaT, ganixileba birTvis moZraoba, romlis masa m  10 3 kg. Cveulebriv, birTvis mdebareoba SeiZleba ganisazRvros milimetris meaTedi da measedi sizustiT. yovel SemTxvevaSi, azri ara aqvs vilaparakoT mdebareobis ufro meti sizustiT gazomvaze, vidre atomis zomaa. davuSvaT amitom, rom x  10 10 m. maSin (10.4) ganuzRvrelobaTa Tanafardobidan viRebT:  

h 6,62  10 34  10  10 20 m/wm. 3 x  m 10  10

x da  sidideebis erTdrouli simcire aris swored dadastureba

(mtkiceba) klasikuri meqanikis gamoyenebadobisa. sul sxvanairadaa saqme, roca laparakia atomur movlenebze _ metad mcire masis nawilakebis moZraobaze

sivrcis

eleqtronis eleqtronis rigisaa,

metad

moZraoba masa

aseTive

mcire

wyalbadis

m  9  10 31 kg. iqneba

moculobaSi. atomSi.

radgan

eleqtronis

ganvixiloT, misi

siCqare

wyalbadis

mdebareobis

atomis

magaliTad,

  10 6 m/wm. zoma

10 10 m

ganuzRvrelobac,

e.i.

x  10 10 m. maSin siCqaris ganuzRvrelobisTvis (10.4)-dan miviRebT: h 6,62  10 34   7  10 6 m/wm. 31 10 m  x 9  10  10 eleqtronis siCqaris ganuzRvreloba  

gamodis,

rom

daaxloebiT

7-jer

aRemateba TviT siCqares. es ki imas niSnavs, rom atomis eleqtronis mdeba-

reobisa da siCqaris erTdroulad da zustad gansazRvra SeuZlebelia da, maSasadame, dauSvebelia misTvis klasikuri meqanikis kanonebis gamoyeneba*. amrigad, haizenbergis principi iZleva klasikuri meqanikis gamoyenebis meore

kriteriums:

klasikuri

meqanikis

gamoyeneba

SesaZlebelia

SemTxvevaSi, rodesac x da  sakmaod mcirea, anu nawilakis mdebareoba da siCqare SeiZleba gaizomos erTdroulad da zustad. Tu SevajamebT mocemuli paragrafis Sedegebs, SeiZleba iTqvas, rom k las i k u r i m e q anik a aris m cire s iCq are e b is (s inaT lis s i Cq a r e s T an

Se da r e b iT )

di di

m as e b is

(atom is

m as as T an

Se dare b iT )

m e q anik a.

§11. niutonis pirveli kanoni. inerciuli sistemebi wina paragrafSi aRvniSneT, rom niutonis kanonebi cdiseuli kanonebia. es kanonebi Sedegia genialuri ganzogadebisa im dakvirvebis Sedegebis da cdiseuli monacemebis, romlebic miRebuli iyo niutonamde da TviT niutonis mier. dinamikas pirveli kanonis saxiT niutonma safuZvlad daudo kanoni, romlis arsi Camoyalibebuli iyo jer kidev galileis mier: y ov e l i

s xe u l i

i na r Cu n e b s

u Z ra ob is

w rf iv i

T a n ab ar i

m oZ r ao b is m d gom ar e ob as , s an am s x v a s xe u lT a m oq m e de b a a r g am o i y v a ns m as am m dg om are ob i d an. sxeulis Tvisebas, SeinarCunos uZraobis an wrfivi Tanabari moZraobis mdgomareoba, inercia ewodeba. amitom niutonis pirvel kanons xSirad

aq mxedvelobaSi unda miviRoT erTi garemoeba. kerZod, eleqtroni atomSi da igive

eleqtroni

Tavisufal

mdgomareobaSi

gansxvavebul

Tvisebebs

avlens.

haizenbergis

principi atomis eleqtronisaTvis yovelTvis sruldeba (atomi kvanturi sistemaa), magram rodesac eleqtroni Tavisufalia, magaliTad, amofrqveulia liTonidan Termoeleqtruli emisiis Sedegad da moZraobs sxivur-eleqtronul an sxva raime milakSi, igi moZraobs rogorc klasikuri nawilaki. 

sagulisxmoa, rom niutonma dinamikis kanonebi, iseve, rogorc msoflio mizidulobis

kanoni da optikuri kvlevis Sedegebi, agreTve diferencialuri da integraluri aRricxvis safuZvlebi, Camoayaliba sruliad axalgazrdam, 25 wlis asakSi. sainteresoa aRiniSnos, rom ainStainma fardobiTobis Teoria imave asakSi Seqmna. 

warmodgeba laTinuri sityvisgan inertia – umoqmedoba. 32

inerciis kanons uwodeben, xolo sxeulis wrfiv Tanabar moZraobas, roca masze sxva sxeulebi ar moqmedeben _ inerciiT moZraobas. rogorc uZraobis, aseve wrfivi Tanabari moZraobis dros, sxeulis siCqare mudmivia, e.i. aCqareba nulis tolia. maSasadame, Tu sxeulze sxva sxeulebi ar moqmedebs, igi moZraobs aCqarebis gareSe. inerciis kanonis niutoniseuli formulireba Seicavs gaurkvevlobas, vinaidan araferia naTqvami aTvlis sistemaze. meore mxriv, yoveldRiuri dakvirvebebi gviCvvenebs, rom es kanoni yvela aTvlis sistemis mimarT ar aris samarTliani. vTqvaT, aTvlis sistema dakavSirebulia vagonTan. Tu vagoni uZravia, an moZraobs wrfivad da Tanabrad, niutonis pirveli kanoni sruldeba, magram sakmarisia vagonma daiwyos moZraoba aCqarebulad, rom kanoni dairRveva _ aqamde uZravad myofi sxeulebi daiwyeben moZraobas maTze sxva sxeulebis moqmedebis gareSe. s is te m e b s , rom le b Si c s ru l de b a n iu t on is p irv e li k an on i, i ne r c iu l i s is te m e b i e w od e b a . vinaidan niutonis pirveli kanoni araa marTebuli aTvlis yvela sistemisaTvis, mas xSirad ayalibeben Semdegi saxiT: a r s e b ob s aT v l is is e T i s is te m e b i , r om e l T a m im ar T a c s xe u l i in a r Cu ne b s u Z r aob is a n w r fi v i T an ab ar i m oZ r a ob i s m d go m a r e ob as , T u m as ze s xv a s x e u le b i a r m oq m e d e b e n a n m aT i m oq m e de b a u r T ie rT k om pe n s i re b u l i a. unda aRiniSnos, rom niutonis pirveli kanoni sulac ar aris iseTi cxadi, rogorc es erTi SexedviT SeiZleba mogveCvenos. odiTganve egonaT, rom Tu sxeulze sxva sxeulebi ar moqmedeben, an es moqmedeba urTierTkompensirebulia, igi mxolod uZravi SeiZleba iyos, rom uZraoba sxeulis bunebrivi mdgomareobaa. es Sexeduleba TiTqosda dasturdeba yoveldRiuri gamocdilebiT. magaliTisaTvis ganvixiloT raime sxeuli, romelic Zevs horizontalur zedapirze. igi uZravia, vinaidan dedamiwis moqmedeba kompensirdeba

zedapiris

(sayrdenis)

reaqciiT.

imisaTvis,

rom

sxeulma

imoZraos mudmivi siCqariT, saWiroa mudmivad veweodeT an vawvebodeT mas. italielma mecnierma galileim pirvelma miuTiTa, rom wrfivi Tanabari moZraoba sxeulebis iseTive bunebrivi mdgomareobaa, rogorc uZraoba da Tu sxeulis mudmivi siCqariT asamoZraveblad mudmivad unda vawvebodeT mas, imiT aixsneba, rom sxeulis moZraobisas zedapiri ara marto akompensirebs dedamiwis zemoqmedebas, aramed qmnis sxeulze damatebiT moqmedebas, 33

romelsac xaxuni ewodeba. misi zemoqmedeba, vinc sxeuls awveba, saWiroa xaxunis kompensaciisaTvis. rac ufro gluvia zedapiri, miT mcirea xaxuni da miT ufro didxans moZraobs sxeuli zemoqmedebis Sewyvetis Semdeg mudmivi siCqariT. amitom, askvnis galilei, xaxuni rom srulebiT ar yofiliyo, sxeuli mudam imoZravebda wrfivad da Tanabrad. Cven vnaxeT, rom aCqarebulad moZrav vagonSi niutonis pirveli kanoni ar sruldeba. igi a r ai ne rci u l i sistemaa. dedamiwac, brunvis gamo, arainerciuli

sistemaa,

magram

brunvis

kuTxuri

siCqaris

simciris

gamo,

movlenaTa Zalian didi wrisaTvis, dedamiwis brunva gavlenas ar axdens da igi praqtikulad iqceva ise, rogorc inerciuli sistema. sakmaod didi sizustiT inerciul sistemas warmoadgens e.w. heliocentruli (kopernikis) sistema.

sistemis

saTaved

miCneulia

mzis

centri,

xolo

RerZebi

mimarTulia Soreuli `uZravi~ varskvlavebisaken.

ยง12. niutonis meore kanoni. moZraobis diferencialuri gantolebebi niutonis pirveli kanonidan gamomdinareobs, rom sxeulis aCqarebuli moZraobis mizezi aris am sxeulze sxva sxeulebis moqmedeba. erTi sxeulis meoreze moqmedebis dasaxasiaTeblad fizikaSi SemoRebulia Zalis cneba. Z a la

a r is

f iz ik u r i

s i di de ,

ro m e l ic

w a rm o a d ge n s

m oc e -

m u l s x e u l ze s xv a s xe u lis (s xe u le b i s ) m e q a nik u ri m oq m e de b i s z o m as . sxeulze moqmedebs Zala, niSnavs, rom am sxeulze moqmedebs sxva sxeuli (sxeulebi). Zala xasiaTdeba sididiT (moduliT), mimarTulebiT da modebis wertiliT. es niSnavs, rom Zala veqtoria, romlis dasawyisi modebis wertilSia. Zalis meqanikuri moqmedeba sxeulze marto aCqarebis gamowveviT ar Semoifargleba. mas SeuZlia gamoiwvios sxeulis deformaciac*. amitom, Zalis gasazomad unda gaizomos misi moqmedebiT gamowveuli aCqarebis an deformaciis sidide. Cveulebriv, Zalas zomaven mis mier gamowveuli deformaciis sididiT. * unda aRiniSnos, rom sxeulis deformaciis drosac adgili aqvs misi Semadgeneli

nawilakebis aCqarebul moZraobas. amitom Zalas xSirad ganmartaven rogorc aCqarebis gamomwvev mizezs. 34

niutonis meore kanoni amyarebs damokidebulebas sxeulze moqmed Zalas, am Zalis moqmedebiT gamowveul aCqarebas da sxeulis masas Soris. kerZod, mravalricxovani cdebis safuZvelze niutonma daadgina: a C q a re b a,

rom li T a c

m oZ r a ob s

s x e u l i,

pi rd a p ir p r o p or -

c iu l i a s xe u lze m oq m e d i Z a li s a d a u k u pr o por ciu li a s xe u l is m as is a, a nu F. (12.1) m koeficientis ricxviTi mniSvneloba damokidebulia ak

proporciulobis

formulaSi Semavali sidideebis erTeulebis SerCevaze. Tu am sidideebs gavzomavT erTi da imave sistemis erTeulebiT, maSin k  1 da a

F. m

(12.2)

saidanac (12.3) F  ma . e.i. sxeulze moqmedi Zala masisa da am Zalis moqmedebiT gamowveuli aCqarebis namravlis tolia. yoveli m masis sxeulze moqmedebs simZimis Zala P , romelic aniWebs mas simZimis Zalis g aCqarebas. niutonis meore kanonis Tanaxmad: masa

fizikaSi

P  mg . Semotanilia rogorc

sxeulis

sakuTari

(12.4) Tvisebis

damaxasiaTebeli sidide. niutonma igi ganmarta, rogorc n i v T i e re b is r a o d e n ob a m o ce m u l s xe u lS i . masis es ganmarteba srulyofili araa Tundac imitom, rom sxeulis masa, nivTierebis raodenobis garda, damokidebulia moZraobis siCqareze (§10). garda amisa, Tu masis niutoniseuli ganmarteba ase Tu ise misaRebia makrosxeulisaTvis, igi sruliad gaugebaria elementaruli nawilakebisaTvis. masis

ganmartebas

ukavSireben

sxeulis

inercias.

(12.2)

formula

gviCvenebs, rom rac metia sxeulis masa, miT naklebia mocemuli ZaliT gamowveuli sxeulis aCqareba, anu miT naklebad icvleba sxeulis siCqare. es imas niSnavs, rom rac metia sxeulis masa, miT met winaaRmdegobas uwevs igi Tavisi uZraobis an wrfivi Tanabari moZraobis mdgomareobis Secvlas, anu

rogorc

amboben

xolme,

miT

ufro

inertulia

igi.

masa

aris

s xe u li s i ne rtu lob is zom a. sazogadod, masa sxeulis ramdenime Tvisebis maxasiaTebelia. garda zemoT CamoTvlilisa, masa gansazRvravs sxeulebis gravitaciis (mizidvis) unarsac _ sxeulebs Soris mizidvis Zala maTi 35

masebis proporciulia. dabolos, rogorc SemdgomSi vnaxavT, masa gansazRvravs sxeulis energiis srul maragsac. cdebi gviCvenebs, rom sxeulis aCqarebas yovelTvis aqvs masze moqmedi Zalis mimarTuleba. amitom (12.3) gantoleba SeiZleba daiweros veqtoruli saxiT:

  F  ma .

(12.5)  Tu sxeulze moqmedebs ramdenime Zala, maSin F aris am Zalebis veqtoruli jami. (12.5) gantolebas ewodeba moZraobis gantoleba. (12.5) veqtoruli gantolebis

dagegmilebiT

koordinatTa RerZebze

miviRebT

sam

skalarul

gantolebas:

d 2x , dt 2 d2y , (12.6) Fy  m 2 dt d 2z Fz  m 2 . dt (12.6) gantolebebs ewodeba m oZ ra ob is di fe re n ci al u r i g an to le b e b i . Fx  m

am gantolebebis saSualebiT (saTanado sawyisi pirobebis gaTvaliswinebiT), Tu mocemulia sxeulis masa da Zalis Fx , Fy , Fz mdgenelebi, gavigebT x, y da z -s rogorc drois funqciebs (kinematikuri gantolebebi), anu amovxsniT meqanikis ZiriTad amocanas. viciT (§2), rom SI sistemaSi masis erTeulia kilogrami (kg), xolo aCqarebis _ 1 m/wm2. maSin (12.3) formulis Sesabamisad, Z a li s e r T e u l i am s is t e m a Si iq ne b a is e T i Z a l a, ro m e l ic 1 k g m as i s m q one s xe u ls m ia ni W e b s 1 m / w m 2 a Cq are b a s . am e rT e u ls e w o de b a ni u t on i (n ). e.i. 1 n = 1 kg . 1 m/wm2 teqnikur sistemaSi Zalis erTeulia kilogramZala (§2). ganmartebis Tanaxmad: 1 kgZ = 1kg . 9.80665 m/wm2 maSasadame

1 kgZ = 9,80665 n  9,81 n.

§13. niutonis mesame kanoni Zala meqanikaSi SemoRebulia erTi sxeulis moqmedebis dasaxasiaTeblad meoreze, magram am moqmedebas calmxrivi xasiaTi ki ara aqvs, aramed adgili

aqvs

urTierTqmedebas:

roca

erTi

sxeuli

moqmedebs

meoreze,

meorec moqmedebs pirvelze, anu qmedeba iwvevs ukuqmedebas. niutonis mesame kanoni amyarebs damokidebulebas qmedebisa da ukuqmedebis Zalebs Soris: Z a le b i, m im ar T u li a

rom le b i T a c

o ri

e rT i

ga s w v ri v ,

w r fi s

s xe u l i

e r T im e ore ze

s id i diT

m oq m e d e b s ,

(m o du l iT )

t ol ia

d a m im ar T u le b i T _ s ap ir is p iro; a nu q m e de b a iw v e v s t ols a d a

 F12

s aw i n a aR m de go u k u q m e de b as .  F12 aris Zala, romliTac B sxeuli  moqmedebs A -ze, xolo F21 _ Zala, romli-

 F21

sur. 3.1

Tac A sxeuli moqmedebs B -ze (sur. 3.1).

niutonis III kanonis Tanaxmad:   F12   F21 (13.1) amrigad, Zalebi y ov e lT v i s Cn de b i an w y v i l -w y v i la d. q m e de b i s a

d a u k u q m e de b is Z a le b i e r T m ane T s ar as d ros ar aw on a s w or e b e n, v in ai d an m o de b u l ni a ri an s xv ad a s xv a s xe u le b ze . unda aRiniSnos, rom qmedebisa da ukuqmedebis Zalebi mudam erTnairi bunebisaa. magaliTad, Tu

erT

sxeulze

meoris

mxridan

moqmedebs

drekadobis

msoflio

mizidulobis Zala, maSin pirveli sxeulic meoreze agreTve drekadobis an msoflio mizidulobis ZaliT imoqmedebs. aRvniSnoT urTierTqmedi A da B sxeulebis masebi da aCqarebebi Sesabamisad

m1 ,

m2 ,

a 2 -iT. maSin, niutonis meore kanonis Tanaxmad

    F12  m1 a1 da F21  m2 a2 . Zalebis es mniSvnelobebi SevitanoT (13.1)-Si:   m1a1  m2a2 .

an sididiT saidanac

m1a1  m2 a2 . a1 m2  . a2 m1

(13.2)

e.i. a Cq are b e b i, rom le b s a c i R e b e n s x e u le b i u rT i e rT q m e d e b i s as , m aT i m a s e b is u k u pr o po rci u l i a: is sxeuli iRebs met aCqarebas, romlis masa naklebia.

ganvixiloT ramdenime magaliTi. niutonis mesame kanonis Tanaxmad: ra ZaliTac dedamiwa izidavs nebismier sxeuls, vTqvaT, qvas, imave ZaliT qva izidavs

dedamiwas.

urTierTqmedebis

Sedegad,

orive

sxeuli

iRebs

maTi

masebis

ukuproporciul aCqarebebs da vinaidan dedamiwis masa gacilebiT metia nebismieri qvis masaze, igi praqtikulad uZravia, qva ki vardeba masze g aCqarebiT. ra ZaliTac dedamiwa izidavs mTvares, imave ZaliT mTvare izidavs dedamiwas. dedamiwis

moqmedebiT

mTvare

moZraobs

mis

irgvliv

mrudwirul

orbitaze,

mTvaris

moqmedeba ki, kerZod, iwvevs dedamiwaze periodulad ganmeorebad zRvebisa da okeaneebis moqcevasa da miqcevas. ra ZaliTac cxeni eweva etls, imave ZaliT etli eweva cxens da mxolod am Siga Zalebis moqmedebiT sistema `cxeni-etli~ verasdros ver daiZvreboda adgilidan. imisaTvis, rom ori urTierTqmedi sxeuli amoZravdes erTi mimarTulebiT, saWiroa mesame sxeulis arseboba. aseTi mesame sxeulia dedamiwa. cxeni, romelic eweva etls, eyrdnoba dedamiwas. maSasadame, cxenze dedamiwis mxridan moqmedebs horizontaluri gare Zala _ xaxunis Zala, romelsac mohyvas moZraobaSi es sistema. Tu xaxunis Zala moispo (moyinuli gza), sistema ver amoZravdeba.

gamoviyenoT niutonis mesame kanoni mrud wirze moZravi sxeulisaTvis. rogorc viciT ($6), mrud wirze moZrav sxeuls gaaCnia normaluri anu centriskenuli aCqareba an   , romelic mimarTulia simrudis centrisken. 2

Tu simrudis r radiusi mudmivia, sxeuli moZraobs wrewirze. am dros wiriTi

siCqare

 r

an   2 r.

aCqarebas, ce nt ri s k e n u l i

Zalas,

romelic

iwvevs

centriskenul

Z a l a ewodeba. cxadia, es Zala mimarTulia

centrisken. niutonis meore kanonis Tanaxmad igi tolia: Fn  man 

m 2  m 2 R r

niutonis mesame kanonis Tanaxmad, wrewirze moZravi sxeuli, Tavis mxriv, moqmedebs im sxeulze, romelic iwvevs misi traeqtoriis gamrudebas, centriskenuli Zalis toli da sawinaaRmdegod mimarTuli ZaliT. am Zalas c e n tr i d an u l i Z a l a ewodeba. magaliTad, Tokze gamobmuli qvis wriuli moZraobis dros centriskenuli Zala moqmedebs qvaze, xolo centridanuli _ Tokze. mTvarisa da dedamiwis SemTxvevaSi centriskenuli Zala moqmedebs mTvareze, xolo centridanuli _ dedamiwaze da a.S. dabolos, sxvadasxva

aRvniSnoT,

buneba

hqondes.

rom

centriskenul

magaliTad,

Zalas

dedamiwis

SeiZleba

irgvliv

sul

mTvaris

moZraobisas mTvareze moqmed centriskenul Zalas warmoadgens dedamiwis mizidulobis centriskenul

Zala, Zalas

Tokze

gamobmuli

warmoadgens

qvis

Tokis 38

moZraobisas

daWimulobis

qvaze Zala,

moqmed birTvis

irgvliv eleqtronis brunvisas eleqtronze moqmed centriskenul Zalas warmoadgens eleqtruli mizidvis Zala, mbrunav magidaze uZravad myof sxeulze moqmed centriskenul Zalas warmoadgens xaxunis Zala da a.S.

$14. fardobiTobis meqanikuri principi. galileis gardaqmnis formulebi viciT ($11), rom inerciuli sistema ewodeba iseT sistemas, romelSic sruldeba niutonis pirveli kanoni anu romelSic izolirebuli wertilis siCqare mudmivia. ismis kiTxva: ramdeni inerciuli sistema arsebobs bunebaSi? am kiTxvaze pasuxis gasacemad ganvixiloT ori sistema oxyz anu S sistema, romelic inerciulia da o' x' y' z ' anu S ' sistema, romelic mis mimarT 

moZraobs mudmivi 0

siCqariT. simartivisaTvis davuSvaT, rom

sistemebis Sesabamisi RerZebi urTierTparaleluria da moZraoba warmoebs

t  0

x RerZis gasvwriv (sur.14.1). davuSvaT agreTve, rom sawyis momentSi

koordinatTa o' saTave emTxveva o saTaves. vTqvaT,

drois

momentSi

moZravi wertili imyofeba A mdgoA

mareobaSi. veqtorTa Sekrebis wesis

 r

Tanaxmad, SeiZleba davweroT    r  r ' oo' ,

z=z'



sadac r

da r ' wertilis radius-

veqtorebia, Sesabamisad sistemaSi. S'

(14.1)

gadaadgildeba

sistemis

y=y'

drois ganmavlobaSi

koordinatTa

 0

 r'

x y

y' sur. 14.1

saTave

mdgomareobidan

 o' mdgomareobaSi. vinaidan es gadaadgileba xdeba mudmivi  0 siCqariT, ami



tom oo'  0t . SevitanoT (14.1)-Si:    r  r '0t .

gavawarmooT es toloba droiT:

  dr dr '  .   0 dt dt

(14.2)



magram dr   aris wertilis siCqare S sistemaSi, xolo dr '   ' _ siCqare S ' sistemaSi. amitom 

 

   '0 .

formula,

romelic

gamosaxavs

(14.3) siCqareTa

Sekrebis

k l as ik u r



w e s s , gviCvenebs, rom Tu siCqare  mudmivia, anu Tu S sistema inerciulia, 

maSin  ' -ic mudmivi iqneba, anu

sistemac inerciuli iqneba. amrigad,

miviReT, rom yoveli sistema, romelic moZraobs inerciulis mimarT mudmivi siCqariT, aris agreTve inerciuli, rac amtkicebs, rom bunebaSi uamravi inerciuli sistemaa. rogorc vnaxeT, inerciis kanoni ar icvlis saxes erTi inerciuli sistemidan meoreSi gadasvlisas. axla vaCvenoT, rom niutonis meore kanonic ar icvlis saxes S sistemidan S ' sistemaSi gadasvlis dros. gavawar

mooT (14.3) toloba droiT, da gaviTvaliswinoT, rom 0  const . miviRebT:   a  a' . (14.4) maSasadame, siCqarisagan gansxvavebiT, aCqareba absoluturia _ igi erTnairia yvela inerciul sistemaSi. garda amisa, mocemul wertilze moqmedi Zalebic orive sistemaSi erTnairia   F  F' ,

(14.5)

viniaidan Zala gansazRvravs mocemul wertilze garemomcvel sxeulTa moqmedebas, rac koordinatTa sistemis SerCevaze ar aris damokidebuli*. klasikur fizikaSi sxeulis masa mudmivi sididea. Tu Cveni wertilis masas aRvniSnavT m -iT, (14.4) da (14.5) Tanafardobebidan miviRebT:

F F'   m, a a'

(14.6)

e.i. niutonis meore kanons orive inerciul sistemaSi erTnairi saxe aqvs. aseTive Tviseba aqvs meqanikis sxva kanonebsac. amrigad, m e q a nik is k a n on e b i ar i c v l i an s ax e s e rT i ine rci u l i s is te m i d a n m e ore S i g ad a s v l is as . am daskvnas ewodeba fardobiTobis meqanikuri principi an galileis fardobiTobis principi. am princips xSirad Semdegnairad ayalibeben: *

sazogadod, sxeulebs Soris urTierTqmedebis Zala damokidebulia am sxeulebis urTierTganlagebaze da erTmaneTis mimarT moZraobis siCqareze. (14.2) da (14.3) Tanafardo    bebidan gamomdinareobs, rom nebismieri ori nivTieri wertilisaTvis r2  r1  r2 'r1 ' da



2  1  2 '1 ' maSasadame, Zalebi, romlebic gamosaxaven mocemul wertilze (sxeulze) sxva 



sxeulebis moqmedebas, yvela sistemaSi erTnairia. e.i. F  F ' . 40

m e q an ik u r i

m ov le ne b i

yvela

( i g iv u r a d)

m im di na re o b s .

in e rc iu l principis

s is te m aS i

fizikuri

arsi

e rT na ir a d mdgomareobs

imaSi, rom inerciuli sistemebi savsebiT tolfasni arian maTSi meqanikuri movlenebis mimdinareobis TvalsazrisiT. davagegmiloT (14.2) gantoleba koordinatTa RerZebze da gaviTvaliswinoT, rom klasikuri fizikis Tanaxmad dro yvela inerciul sistemaSi erTnairad mimdinareobs t  t ' , anu dro absoluturia. maSin aTvlis orive inerciul sistemaSi wertilis koordinatebsa da dros Soris myardeba Semdegi kavSiri: x  x'0t ,

y  y' ,

(14.7)

z  z' , t  t' .

am formulebs ewodeba g alile is g ard aq m nis form u le b i da klasikuri fizikis Tanaxmad maTi saSualebiT xdeba erTi inerciuli sistemidan meoreze gadasvla. unda aRiniSnos, rom galileis gardaqmnis formulebSi drois absoluturobis garda `Cadebulia~ sivrcis absoluturoba, romlis Tanaxmadac monakveTis sigrZe ar aris damokidebuli aTvlis sistemaze. winaaRmdeg SemTxvevaSi, gaugebari iqneboda kavSiri radius-veqtorebsa 

(formula 14.2) da koordinatebs Soris (14.7), vinaidan r



da r ' , iseve, ro-

gorc x, y, z da x' , y' , z' sxvadasxva sistemaSi izomeba. romelime sididis an algebruli gamosaxulebis ucvlelobas raime gardaqmnis mimarT inv ari antob a  ewodeba, TviT sidides an gamosaxulebas ki

inv ar iantu li

principi

asec

inv ar i antu lni

(ab s olu tu ri) . SeiZleba

amitom

Camoyalibdes:

(s a xe u cv le lni)

ar ian

galileis m e q anik is g alile is

fardobiTobis k anone b i gar daq m nis

for m u le b is m im arT .



warmodgeba laTinuri sityvisagan invariantis – ucvleli. sidideebs, romlebic icvleba erTi inerciuli sistemidan meoreze gadasvlisas variantuli (fardobiTi) sidideebi ewodeba. invariantuli sidideebia: dro, manZili aCqareba, Zala, potenciuri energia (vinaidan es ukanaskneli aris sxeulTa Soris manZilis funqcia, manZili ki invariantuli sididea). variantuli sidideebia radius-veqtori, siCqare, impulsi, kinetikuri energia. sagulisxmoa, rom kanonebi, romlebic invariantulia galileis gardaqmnis formulebis mimarT (mag. impulsis da energiis mudmivobis kanonebi), Seicaven variantul sidides (siCqares, kinetikur energias). 

$15. impulsi. impulsis cvlilebisa da mudmivobis kanonebi sxeulis moZraobis raodenoba anu impulsi ewodeba misi masisa da siCqaris namravls*   P  m .

(15.1) impulsi veqtoruli sididea, mas aqvs siCqaris mimarTuleba. misi erTeuli SI sistemaSi aris kgm/wm. impulss aqvs erTi metad mniSvnelovani Tviseba, romelic mxolod ramdenime fizikuri sididisTvisaa damaxasiaTebeli. esaa mudmivobis anu Senaxvis Tviseba**. gamoviyvanoT impulsis mudmivobis kanoni. niutonis meore kanonis Tanaxmad,   F  ma ,



 d , a dt

magram

(15.2)



 d . F m dt

amitom

klasikur fizikaSi masa mudmivi sididea, amitom igi SeiZleba SevitanoT warmoebulis niSnis qveS:   d m  dP . F  dt dt

(15.3)

e.i. sxeulze moqmedi Zala misi impulsis cvlilebis siCqaris tolia. (15.3) gantolebac niutonis meore kanons gamosaxavs. amasTan, igi warmoadgens am kanonis maTematikur formulirebas zogad SemTxvevaSi, vinaidan (marTalia, Cven igi miviReT masis mudmivobis gaTvaliswinebiT) mtkicdeba, rom marTebulia relativisturi anu cvladi masisTvisac. (15.2) formuliT gamosaxuli niutonis

meore kanoni ki marTebulia mxolod ararelativisturi anu

mudmivi masis SemTxvevaSi. fardobiTobis Teoriis Tanaxmad, relativisturi impulsi ganisazRvreba gamosaxulebiT:  P

 m 1

sxeulis

masisa

siCqaris

namravls

2

(15.4)

niutonma

uwoda

moZraobis

raodenoba.



dasaxeleba moZvelda da amJamad m -s uwodeben sxeulis impulss. ** impulsis garda, mudmivobis Tviseba gaaCnia energias da impulsis moments. am sidideebis

mudmivobis kanonebi gamoiyeneba meqanikis amocanebis amosaxsnelad im SemTxvevebSi, roca sxeulze moqmedi Zalebis mniSvnelobaTa gansazRvra Zalian Znelia. magram am sidideebisa da maTi mudmivobis Tvisebis gansakuTrebuli mniSvneloba isaa, rom isini did rols TamaSoben ara marto meqanikaSi, aramed fizikis sxva nawilebSic. 42

klasikuri meqanikis gamoyenebis sazRvrebSi, roca   c

(15.4) formula

gadadis (15.1)-Si. amrigad, (15.2) gantoleba warmoadgens (15.3)-is kerZo saxes im SemTxvevaSi, roca masa mudmivia. (15.3) formulidan  dP

  dP  Fdt

aris sxeulis impulsis cvlileba, xolo

(15.5)  Fdt -s anu Zalis namravls

qmedebis droze ewodeba Zalis impulsi. m aS as a d am e , s xe u l is im pu l s is c v li le b a

m c ire

dro S i

to lia

am av e

dr oS i

m as z e

m oq m e di

Z a lis im pu ls is a. drois sasrulo

0, t  SualedSi sxeulis impulsis cvlileba toli

iqneba: t    P   Fdt . Tu F  const, maSin P  Ft , 0

anu

   m  m0  Ft .

(15.6)

davubrundeT (15.3) formulas. davuSvaT, sxeuli izolirebulia, maSin F  0 da   m  P  const .

(15.7) maSasadame, i zol ir e b u l i s xe u lis im pu ls i m u dm iv i a . am daskvnas ewodeba impulsis mudmivobis kanoni. vinaidan masa mudmivia, (15.7) tolobidan 

  const ,

rac

warmoadgens

inerciis

kanons.

amrigad,

erTi

sxeulis

SemTxvevaSi

impulsis mudmivobis kanoni inerciis kanonze daiyvaneba da arafers axals ar iZleva. davuSvaT axla, rom mocemulia sxeulTa jgufi anu sistema. sistemis Semadgeneli

sxeulebis

urTierTqmedebis

Zalebs

Sin aga ni

Z ale b i

ewodeba, xolo Zalebs, romliTac gareSe sxeulebi moqmedeben sistemis Semadgenel sxeulebze _ gare

Z ale b i . simartivisaTvis davuSvaT, rom

sistema Sedgeba ori sxeulisagan, romelTa masebia m1 da m2 , xolo   siCqareebi 1 da  2 . davweroT maTTvis moZraobis gantoleba anu niutonis meore kanoni:    dP1 d m11     F12  F1 . dt dt

   dP2 d m22     F21  F2 . dt dt



sadac F12 da F21 sxeulebs Soris moqmedi urTierTqmedebis Zalebi anu Sina



gani Zalebia, xolo F1 da F2 _ Sesabamisad, pirvel da meore sxeulze moqmedi gare Zalebi. SevkriboT es gantolebebi da gaviTvaliswinoT, rom   niutonis mesame kanonis Tanaxmad F12   F21 gveqneba:





    d P1  P2 d m11  m22    F , dt dt

(15.8)

   sadac F  F1  F2 sistemaze moqmedi gare Zalebis geometriuli jamia.

vTqvaT,

sistema

izolirebulia.

m eq anik u r

s is te m as

e w ode b a

izolir e b u li anu C ak e tili, romlis Se m ad ge ne li s xe u le b i m xo lod e r T m a ne T T an u rT ie rT q me deb e n *. am ganmartebis Tanaxmad, gareSe sxeulebTan urTierTqmedebas ara aqvs adgili,  tolqmedi F  0 da (15.8)-is Tanaxmad   d  p1  p2   0, dt rac niSnavs, rom   p1  p2  const

amitom

gare

Zalebis

(15.9)

e.i. C ak e ti li s is t e m i s s ru l i im p u l s i m u dm iv i s i d id e a . urTierTqmedebis Sedegad calkeuli sxeulebis impulsebi icvleba (vinaidan icvleba siCqareebi) _ erTis izrdeba, meoris ki mcirdeba, anu adgili aqvs im p u l s is

ga d ac e m as

erTi

sxeulidan

meoreze,

magram

ise,

rom

maTi

veqtoruli jami anu sistemis sruli impulsi rCeba mudmivi. cxadia, rom impulsis mudmivobis kanonis Sinaarsi mdgomareobs rogorc saerTo impulsis mudmivobaSi, ise impulsis gadacemaSi erTi sxeulidan meoreze. saqme isaa, rom (15.9) toloba Sesruldeba maSinac, sxeulebs Soris urTierTqmedebas rom ar hqondes adgili. am SemTxvevaSi maTi siCqareebi iqneba mudmivi (inerciis kanonis Tanaxmad), mudmivi iqneba maTi impulsebi da maSasadame, maTi jamic. e.i. impulsis gadacemis gareSe sistemis impulsis mudmivobis kanonic daiyvaneba inerciis kanonze. *

SevniSnoT, rom es ganmarteba gansxvavdeba imisagan, rac TermodinamikaSia miRebuli.

kerZod, TermodinamikaSi izolirebuli da Caketili sistema erTi da igive ar aris. Termodinamikur sistemas ewodeba izolirebuli, Tu igi ar awarmoebs garemosTan arc energiis da arc nivTierebis cvlas da Caketili, Tu igi awarmoebs garemosTan mxolod energiis cvlas. 44

impulsis mudmivobis kanoni advilad SeiZleba gavrceldes zogad SemTxvevaze, roca Caketili sistema Seicavs n raodenobis sxeuls (nivTier wertils): n



 mii   pi  const . i 1

impulsis

mudmivobis

(15.10)

i 1

kanoni

bunebis

fundamenturi

kanonia.

igi

sruldeba ara marto niutoniseul, aramed relativistur da kvantur meqanikaSic. Teoriul fizikaSi mtkicdeba, rom es kanoni Sedegia sivrcis erTgvarovnebis. sivrcis erTgvarovneba vlindeba imiT, rom Caketili sistemis meqanikuri Tvisebebi ar icvleba misi, rogorc mTlianis, paraleluri gadataniT sivrceSi. Zalebi meqanikaSi bunebaSi

arsebuli

yvela

cnobili

Zala

(simZimis,

drekadobis,

birTvuli da a.S.) ramdenime saxis urTierTqmedebis gamovlenaa. dReisaTvis miRebulia, rom ars e b ob s

n aw i l ak e b s

So ris

f u n d am e ntu r

urTi -

e r T q m e de b aT a oT x i s axe : Z l ie r i , e le q tr om ag ni tu r i, s u s t i d a g r av i t ac iu l i* . Zalebi, romlebic gvxvdeba meqanikaSi (msoflio mizidulobis, drekadobis, xaxunis) mxolod ori saxis urTierTqmedebis gamovlenaa _ gravitaciulis da eleqtromagnituris. gravitaciuli urTierTqmedebis gamovlenaa msoflio mizidulobis Zala, eleqtromagnituris ki _ drekadobisa da xaxunis Zalebi. amboben, rom drekadobisa da xaxunis Zalebs aqvT eleqtromagnituri buneba. ganvixiloT mokled es Zalebi.

$16. drekadobis Zala. hukis kanoni y ov e l i s xe u li Z a lis m oq m e de b iT i cv l is f orm a s a da z o m e b s an u ga n ic dis d e fo rm a ci as . deformacia aris sxeulis Semadgeneli nawilakebis urTierTgadaadgilebis Sedegi, ris gamoc icvleba maT Soris manZili. deformacias ewodeba d re k a di , Tu Zalis moqmedebis Sewyvetis Semdeg sxeuli mTlianad aRidgens formasa da zomebs da pl a s t ik u r i , Tu es aRdgena mTlianad ar xdeba anu Tu saxezea narCeni deformacia. * 

fundamentur urTierTqmedebaTa Sesaxeb dawvrilebiTaa saubari kursis mesame tomSi. unda aRiniSnos, rom xaxunis Zalis buneba bolomde Seswavlili ar aris. 45

amis mixedviT ansxvaveben drekad da plastikur sxeulebs. pirobebis SecvliT drekadi sxeuli SeiZleba gadavides plastikurSi. magaliTad, drekadi foladis zambara maRal temperaturaze xdeba plastikuri. agreTve, Tu drekad sxeulze modebuli Zala aRemateba garkveul sidides, romelsac d r e k a dob is zR v a ri ewodeba, deformacia xdeba plastikuri. sxeulis drekadi Tvisebebi ganpirobebulia mis Semadgenel nawilakebs Soris mizidvisa da ganzidvis Zalebis arsebobiT. es Zalebi ZiriTadad eleqtruli bunebisaa (gamowveulia nawilakebSi Semavali eleqtruli muxtebis urTierTqmedebiT) da moqmedebs metad mcire – TviT nawilakis zomis rigis manZilze. manZilis gadidebiT mcirdeba rogorc mizidvis, aseve ganzidvis Zalebi, magram ganzidvis Zalebis Semcireba ufro swrafad xdeba, vidre mizidvis. amitom Zalian mcire manZilze Warbobs ganzidva, SedarebiT ufro did manZilze ki mizidva. molekulebs Soris garkveul r0 manZilze (romelsac w on as w oru li m a nZ ili ewodeba) mizidvisa da ganzidvis Zalebi awonasworeben erTmaneTs. rodesac drekadi sxeuli aradeformirebulia, misi nawilakebi imyofebian erTamneTisagan r0 manZilze – mizidvisa da ganzidvis Zalebi urTierTgawonasworebulia. sxeulis gaWimvisas izrdeba manZili nawilakebs Soris da mizidvis Zalebi gadaaWarbebs ganzidvisas. es iwvevs sxeulSi e.w. drekadobis Zalis aRZvras, romelic ewinaaRmdegeba gare gamWimav Zalas. Tu am ukanasknels

movxsniT,

drekadobis

Zalebi

daabruneben

sxeuls

sawyis

mdgomareobaSi. sxeulis SekumSvisas xdeba piriqiT – mizidvis Zalebi ufro nela izrdeba, vidre ganzidvisa da axla ukve ganzidvis Zalebi asrulebs drekadobis Zalebis rols. drekadobis Zala modebulia deformaciis gamomwvev sxeulze. arsebobs deformaciis sxvadasxva saxe: calmxrivi gaWimva (an kumSva), yovelmxrivi gaWimva (an kumSva), Runva, grexa, Zvra (gadaweva). yvela isini dawvrilebiT ganixileba specialur sainJinro kursebSi (masalaTa gamZleoba, samSeneblo

meqanika

sxv.).

Cven

SevCerdeT

drekadi

Reros

sigrZiv

gaWimvaze*.

* deformaciis umravlesoba, amocanis gamartivebis mizniT, SeiZleba dayvanil iqnes or

ZiriTad saxeze: gaWimvisa da kumSvis deformaciaze. magaliTad, Runvis deformaciis dros aRiZvreba kumSvis deformacia sxeulis (Reros) Cazneqil elementebSi da gaWimvis deformacia_ amozneqilSi. Zvris dros sxeulis elementebi erTdroulad ikumSebian da iWimebian ori urTierTmarTobi mimarTulebiT. 46

sazogadod, nebismieri deformaciis dros, drekadobis farglebSi (anu mcire deformaciisas) moqmedebs hukis cdiseuli kanoni: d e fo rm aci is S e d e g a d s xe u l Si a R Z ru l i

d r e k ad ob i s Z a -

l a p r o p orc iu l i a x d e f orm a ci is s i d id is a: F  kx .

(16.1)

proporciulobis koeficients ewodeba sxeulis sixiste. ricxobrivad igi im Zalis tolia, romelic aRiZvreba sxeulSi erTeulovani deformaciisas. misi mniSvneloba damokidebulia sxeulis zomebsa da nivTierebis gvarobaze. zomebze damokidebulebis gamo sixiste mouxerxebeli sididea, ris gamoc saWiro xdeba sxva koeficientis Semoyvana, romelic mxolod nivTierebis gvarobaze iqneba damokidebuli. vTqvaT,

erTgvarovani

Rero

zeda

fuZeTi

uZravadaa damagrebuli, qveda fuZeze ki moqmedebs  normaluri Fn Zala, romelic iwvevs mis gaWimvas.

 F l

(sur. 16.1)

Reros

sawyisi

sigrZe

aRvniSnoT

l -iT,

ganivkveTis farTi _ S -iT, xolo gaWimvis Sedegad l

 Fn

sur. 16.1

Reros wagrZeleba anu Reros deformacia _ l -iT. gaWimvis Sedegad ReroSi aRiZvreba F

drekadobis

Zala, romelic sididiT Fn gamWimavi Zalis tolia

da mimarTulia mis sawinaaRmdegod. drekadobis Zalis Sefardebas ganivkveTis farTobTan meqanikuri Zabva ewodeba:

F . (16.2) S sazogadod, deformaciis zomas warmoadgens fardobiTi deformacia ^б 

(  ). f a r d ob iT i de fo r m ac i a e w ode b a s xe u l is zom is cv l il e b i s f a r do b as s aw y is zom as T a n. gaWimvis deformaciis SemTxvevaSi l . l deformaciis



cdiT meqanikuri

dadgenilia, Zabva

rom

proporciulia

misi

Sedegad

fardobiTi

sxeulSi

aRZruli

deformaciisa

(hukis

kanonis meorenairi formulireba): sadac



drekadobis

^ б   , modulia. calmxrivi

gaWimvis

(an

(16.3) kumSvis)

SemTxvevaSi  -s aRniSnaven E asoTi da mas iungis moduls uwodeben: 47

^ б  E . (16.4) iungis modulis fizikuri arsis gasagebad (16.4)-Si davuSvaT, rom   1 , б E.

maSin

maSasadame,

iungis

moduli ricxobrivad im

Zabvis

tolia,

romelic aRiZvreba sxeulSi erTeulovani fardobiTi deformaciis dros. magram   Z ab v is

l  1 niSnavs, rom l  l . e.i. iu n gi s m o du li r ic xob riv a d im l

t ol ia ,

rom e li c

a R iZ v r e b a

s xe u l Si

m is i

s i gr Z is

g a or m a ge b is as . am ganmartebis nakli mdgomareobs imaSi, rom aseTi didi deformaciebis

dros

hukis

kanoni

TiTqmis

yvela

sxeulisTvis

xdeba

umarTebulo: sxeuli an iSleba (wydeba), an irRveva proporciuloba Zabvasa da deformacias Soris. iungis moduli damokidebulia mxolod nivTierebis gvarobaze da misi mniSvnelobebi cxrilSia mocemuli. kavSirs k -sa da E -s Soris advilad davamyarebT Tu (16.4)-Si SevitanT б da  -is mniSvnelobebs da amovxsniT F -s: F 

SE l. es gantoleba faqtiurad emTxveva (16.1)-s _ x l S E. l

Secvlilia l -iT. maTi Sedareba gvaZlevs: k 

SI sistemaSi iungis modulis erTeulia paskali. 1 paskali=n/m2.

$17. xaxunis Zala x a xu ni s

Zala

u r T ie r T q m e de b is

aR iZ v re b a Se d e g a d

S e m xe b i

s xe u le b is

y ov e lT v i s

S e xe b is

n aw i lak e b is z e d a pi r is

g a s w v r iv aa m im a rT u l i (drekadobis Zalisagan gansxvavebiT, romelic am zedapiris marTobulia). arsebobs xaxunis ori saxe _ g a r e g a ni da S i n a g an i. garegani xaxuni aRiZvreba ori Semxebi myari sxeulis urTierTgadaadgilebisas (srialis xaxuni) an aseTi gadaadgilebis gamowvevis mcdelobisas (uZraobis xaxuni). Sinagani xaxuni aRiZvreba erTi da imave sxeulis sxvadasxva nawilebs Soris, magaliTad, siTxis an airis sxvadasxva fenebs Soris, romlebic moZraoben sxvadasxva siCqariT. asxvaveben agreTve m Sr al

svel

an u

b l an t xaxuns. mSrali

xaxuni aRiZvreba myari sxeulis zedapirebs Soris SezeTvis gareSe (anu roca zedapirebs Soris ar aris siTxis an airis fena). sveli xaxuni aRiZvreba myari sxeulis zedapirsa da Txevad an airad garemos Soris, romel48

Sic is moZraobs, agreTve SezeTil zedapirebs Soris maTi urTierTgadaadgilebisas*. mSrali xaxuni SeiZleba iyos srialis da gorvis. davuSvaT, Zelaki, romelic moTavsebulia magidis horizontalur ze dapirze, awveba am zedapirs Fn normaluri wnevis ZaliT (sur.17.1). vimoqme doT am Zelakze Semxebi zedapiris paraleluri mcire F ZaliT da TandaTan vzardoT misi sidide (moduli). Tavdapirvelad Zelaki uZravia, rac

 N  Fxax

Fxax

 F

 Fn

 sur. 17.1

sur. 17.2

 miuTiTebs imaze, rom masze, F ZalasTan erTad, moqmedebs kidev erTi Zala    ( F xax.), romelic moduliT F -is tolia. F Zala kompensirdeba (gawonaswo rebulia) raRac F xax-is ZaliT, romlic mimarTulia Semxebi zedapiris gas  wvriv da aqvs F -is sawinaaRmdego mimarTuleba. F xax Zalas ewodeba u Z r a o b is x a xu n is Zala. igi ganpirobebulia Zelakze sayrdenis moqmedebiT da  `avtomaturad~ iRebs mniSvnelobebs, romlebic sididiT TandaTan mzardi F Zalis mniSvnelobebis tolia. magram uZraobis xaxunis Zalis zrda xdeba    garkveul F0 mniSvnelobamde. rodesac F Zalis sidide gaxdeba F0 -ze meti, Zelaki amoZravdeba _ uZraobis xaxuni Seicvleba srialis xaxuniT. amrigad,  F0 gansazRvravs uZraobis xaxunis Zalis maqsimalur mniSvnelobas. TviT  uZraobis xaxunis Zala, modebuli F Zalis sididis Sesabamisad, iRebs erT erT mniSvnelobas 0-dan F0 -mde.  srialis xaxunis Zalis sidide daaxloebiT F0 -is tolia da damokidebulia siCqareze. 17.2 suraTze mocemulia srialis xaxunis Zalis siCqareze

* sveli xaxuni faqtobrivad Sinagani xaxunia, vinaidan siTxeSi CaSvebul sxeuls ekvris

siTxis fena da sxeulis moZraobisas xaxuni aRiZvreba am fenasa da danarCen Txevad garemos Soris. 49

damokidebulebis grafiki, saidanac Cans, rom siCqaris zrdiT srialis xaxu nis Zala ( F xax) Tavdapirvelad umniSvnelod mcirdeba, Semdeg ki iwyebs zrdas. srialis xaxunis Zala yovelTvis mimarTulia moZraobis sawinaaRmdegod da, maSasadame, iwvevs sxeulis fardobiTi siCqaris Semcirebas. amonton-kulonis empiriuli kanonis Tanaxmad, u Z r a ob i s x a xu n is  m aq s im alu ri Z a la F0 a r a ri s d am ok i de b u l i s xe u le b i s S e x e b is  f a r T ob ze d a p ro po rc iu l i a Fn n or m a lu r i w ne v is Z a lis s i di di s a: F0  0 Fn .

(17.1) uganzomilebo sidide  0 -s ewodeba uZraobis xaxunis koeficienti. igi damokidebulia moxaxune zedapirebis bunebaze da mdgomareobaze. analogiur damokidebulebas aqvs adgili srialis xaxunis ZalisaTvis:

Fxax   Fn ,

(17.2) sadac  srialis xaxunis koeficientia. igi, moxaxune zedapirebis masalisa da mdgomareobis garda, ramdenadme aris damokidebuli moZraobis siCqareze. marTalia, uZraobis xaxunis Zala xels uSlis moZraobis dawyebas, magram rig SemTxvevebSi swored igia moZraobis mizezi. magaliTad, siarulisas swored lanCaze moqmedi uZraobis xaxunis Zala aniWebs adamians aCqarebas. es Zala rom ar arsebobdes, lanCa isrialebda da siaruli SeuZlebeli iqneboda. lanCac, Tavis mxriv, moqmedebs dedamiwaze toli da sawinaaRmdego ZaliT da aniWebs mas aCqarebas. amgvaradve xdeba, magaliTad, avtomobilis Tvlebis dedamiwidan ukubiZgeba, elmavlis Tvlebis relsebidan ukubiZgeba da a.S. rac

Seexeba

srialis

xaxunis

Zalis

uaryofiT

gavlenas,

mis

Sesamcireblad mimarTaven SezeTvas, magram ufro radikaluri saSualebaa srialis Secvla gorviT _ iyeneben borblebs, burTulasakisrebs da sxv. gorvis xaxunis Zala mgoravi sxeulis radiusis ukuproporciulia: Fn . R koeficientia, Fxax  K

gorvis

xaxunis

(17.3) romelic

ukve

aRar

aris

uganzomilebo sidide. igi damokidebulia masalis gvarobaze. blant garemoSi (siTxeSi, airSi) moZrav sxeulze moqmedebs moZraobis damamuxruWebeli

Zala.

Zala

warmoiqmneba 50

sveli

xaxunis

Zalisgan

(romelic, rogorc ukve aRvniSneT, aRiZvreba sxeulze garSemortymuli siTxis fenasa da danarCen fenebs Soris) da winaaRmdegobis Zalisagan. winaaRmdegobis Zala Cndeba imitom, rom moZravi sxeulis win wneva ufro metia, vidre ukan, ris gamoc wnevis Zalebis tolqmedi mimarTulia moZraobis sawinaaRmdegod. didi siCqaris SemTxvevvaSi garemos winaRobis Zala SeiZleba bevrad meti aRmoCndes sveli xaxunis Zalaze.

jamur Zalas,

romelic Sedgeba sveli xaxunis Zalisa da garemos winaRobis Zalisagan, pirobiTad

uwodeben

xaxunis

Zalas.

aseTnairad

gansazRvruli

xaxunis

ZalisaTvis damaxasiaTebelia is, rom igi siCqaresTan erTad xdeba nulis toli (anu ar arsebobs uZraobis xaxuni). naTqvamis sailustraciod SeiZleba gamoviyenoT Semdegi cda. TiTqmis horizontalur liTonis filaze devs parafinis naWeri. filis gaTbobisas parafinis gamdnari Txeli fena qmnis sveli xaxunis pirobas da parafinis naWeri daiwyebs moZraobas filaze, Tumca mamoZravebeli Zala (simZimis Zalis horizontaluri mdgeneli) iqneba metismetad mcire. mcire siCqaris SemTxvevaSi xaxunis Zala siCqaris proporciulia da mimarTulia mis sawinaaRmdegod:   Fxax   K1

siCqaris

zrdasTan

erTad

(17.4)

wrfivi

(17.4) damokidebuleba

TandaTan

gadadis kvadratulSi:     Fxax   K1 2   K 2



(17.5)

K1 da K 2 koeficientebi, agreTve im siCqareTa mniSvnelobebi, romlis drosac xdeba gadasvla wrfivi damokidebulebidan kvadratulSi, damokidebulia sxeulis formasa da zomebze, zedapiris mdgomareobaze da garemos Tvisebebze (siblanteze).

$18. msoflio mizidulobis Zala dedamiwis zedapiridan raime simaRleze atanil sxeuls Tu xels gavuSvebT, igi iwyebs vardnas. sxeuli vardeba imitom, rom masze moqmedebs dedamiwis mizidulobis Zala. niutonis mesame kanonis Tanaxmad, sxeulic izidavs dedamiwas toli da sawinaaRmdegod mimarTuli ZaliT. 1667 wels niutoni mivida daskvnamde, rom mizidulobis Zala moqmedebs ara marto 51

dedamiwasa da misi zedapiris maxloblad mdebare sxeuls Soris, aramed sazogadod bunebis (samyaros) yvela sxeuls Soris. am Zalas man msoflio m iz i du lob i s Z a la uwoda. astronomiuli da sxva monacemebis analizis safuZvelze niutonma Camoayaliba m s of l io m iz i du l ob i s k ano n i , romelic zogadad ase gamoiTqmis: o r i n iv T i e r i w e rti li e r T m ane T s i zi d av s Z al iT , rom e li c p i r d a pi r p r op or ciu li a c iu l i a

m aT

So ris

m aT i

m a nZ il is

m a s e b is

n am r av l is a,

k v a dr at is a

u k u p r o po r -

m im arT u li a

m aT i

c e n tr e b is Se m a e rT e b e li w r f is g a s w v r iv : F 

m1  m2 , r2

(18.1)

sadac m1 da m2 nivTier wertilTa masebia, xolo r _ maT Soris manZili.

 proporciulobis koeficients ewodeba gravitaciuli mudmiva. misi fizikuri arsis gasagebad (18.1)-Si davuSvaT, rom m1  m2  1 kg, r  1 m. maSin F   e.i.

gravitaciuli

mudmiva

ricxobrivad

Zalis

tolia,

romliTac

miizidebian erTmaneTisaken TiTo kilogrami masis sxeulebi, roca maT Soris

manZili

erTi

metria.

 –s

ricxviTi

mniSvneloba

pirvelad

gansazRvra kavendiSma grexiTi sasworis meTodiT, 1797 wels. SemdgomSi  mravalgzis

iqna

gansazRvruli

sxvadasxva

meTodiT.

dadgenilia,

rom

 =6,67.10-11 n.m2/kg2. rogorc

vxedavT,

gravitaciuli

mudmiva

Zalian

mcire

sididea.

amitomaa, rom msoflio mizidulobis Zala Cveulebriv sxeulebs Soris, romlebic cxovrebiseuli TvalsazrisiT did sxeulebad iTvlebian, metismetad mcirea (magaliTad, mizidulobis Zala or erTgvarovan sferos Soris, romelTa masebia TiTo tona, xolo centrebs Soris manZili 1 m, tolia 7.10-5 niutonis). es Zala STambeWdav sidides aRwevs mxolod astronomiuli sxeulebisaTvis da warmoadgens e rT a d e rT

Zalas, romelic marTavs am

sxeulebis moZraobas. msoflio mizidulobis kanoni Camoyalibebulia nivTieri wertilebisaTvis, e.i. im SemTxvevisaTvis, rodesac sxeulTa zomebi gacilebiT naklebia maT Soris manZilze. es piroba kargad sruldeba astronomiuli obieqtebisaTvis _ planetebisaTvis, TanamgzavrebisaTvis, kometebisaTvis, varskvlavebisaTvis. irkveva agreTve, rom am kanoniT sargebloba SeiZleba maSinac, rodesac sxeulebi erTgvarovan sferoebs warmoadgenen. am Sem52

TxvevaSi r aris manZili sferoebis centrebs Soris. dabolos, (18.1) formuliT SeiZleba gamoiTvalos mizidulobis Zala didi radiusis sferosa da mis

zedapirTan axlos

mdebare

nebismieri formis

metad mcire

zomis

(sferos radiusTan SedarebiT) sxeuls Soris. es piroba kargad sruldeba dedamiwis

SemTxvevaSi.

dedamiwisken

SeiZleba

amitom

yovelgvari

gamoiTvalos

(18.1)

sxeulis

mizidulobis

formuliT,

romelSic

Zala unda

CavTvaloT, rom r dedamiwis sferos radiusia. sityva

`gravitacia~

laTinurad

mizidulobas

niSnavs.

amitom

mizidulobis Zalas xSirad gravitaciul Zalas uwodeben. gravitaciuli urTierTqmedeba xorcieldeba gravitaciuli velis saSualebiT. gravitaciuli veli iqmneba yoveli sxeulis irgvliv da misi arseboba vlindeba imiT, rom am velSi Setanil sxva sxeulze moqmedebs misi masis proporciuli Zala. veli, iseve rogorc nivTiereba, materiis arsebobis formaa. veli imdenive saxisaa, ramdeni saxis urTierTqmedebac arsebobs. sxeulis masa Sedis or ZiriTad da erTmaneTisagan damoukidebel kanonSi _ niutonis meore kanonSi da msoflio mizidulobis kanonSi. niutonis

meore

kanonis ganxilvisas

masa ganvsazRvreT

rogorc

sxeulis

inertulobis zoma, vinaidan am kanonidan gamomdinareobs, rom rac metia sxeulis masa, miT nakleb aCqarebas iRebs igi mocemuli Zalis moqmedebiT. msoflio mizidulobis kanonSi masa inertulobasTan aRar aris dakavSirebuli. aq igi axasiaTebs sxeulTa gravitaciis anu mizidvis unars _ rac metia sxeulTa masebi, miT ufro didi ZaliT izidaven isini erTmaneTs. SeiZleba mogveCvenos, rom am or kanonSi saqme gvaqvs or sxvadasxva fizikur sididesTan

inertul

masasTan

gravitaciul

masasTan,

magram

mtkicdeba, rom inertuli da gravitaciuli masebi Tanatolni anu igivurni arian. amitom, fizikaSi laparakoben ubralod masaze.

$19. simZimis Zala. wona. uwonoba s im Z im i s

Z al a

e w o de b a

Z al a s ,

r om e l ic

s xe u ls

an iW e b s

s im Z im i s Z a li s g a Cq ar e b a s :

P ď&#x20AC;˝ mg .

(19.1)

ď ˛ es Zala gansxvavdeba gravitaciuli mizidvis P0 Zalisagan, romelic

mimarTulia dedamiwis centrisaken da gamoiTvleba (18.1) formuliT: 53

P0   sadac m da M

mM , R2

(19.2)

Sesabamisad sxeulisa da dedamiwis masebia, xolo R _

dedamiwis radiusi. es gansxvaveba gamowveulia dedamiwis brunviT, magram igi imdenad mcirea (ar

aRemateba 0,35%-s),

rom umravlesi praqtikuli

amocanebis gadawyvetisas mas ugulebelyofen da Tvlian, rom sxeulze moqmedi simZimis Zala tolia am sxeulis dedamiwisken mizidvis Zalisa:

P  P0  

mM . R2

(19.3)

simZimis ZaliT gamowveul moZraobas uhaero sivrceSi Tavisufali vardna ewodeba, xolo am dros sxeulis mier SeZenil aCqarebas _ Tavisufali vardnis aCqareba an simZimis Zalis aCqareba

g  .

niutonis meore kanonis

Tanaxmad g

P M  2 m R

(19.4)

(19.4) formulaSi ar Sedis vardnili sxeulis masa da, maSasadame, igi gamosaxavs cdiseul faqts, rom dedamiwis mocemul adgilas yvela sxeuli erTnairi aCqarebiT vardeba. mkacrad Tu ganvsjiT, (19.4) formula, iseve, rogorc niutonis meore kanoni, maSinaa marTebuli, rodesac Tavisufali vardna ganixileba aTvlis inerciuli sistemis mimarT. dedamiwis brunvis gamo, mis zedapirTan dakavSirebuli aTvlis sistema ar aris inerciuli. inerciulia mxolod polusebTan dakavSirebuli aTvlis sistemebi, vinaidan isini monawileobas ar iReben dReRamur brunvaSi. danarCen wertilebTan dakavSirebuli aTvlis sistemebi ki arainerciulia da miT metad, rac ufro daSorebulia

wertili

polusidan.

amitomaa,

rom

cdiT

gazomili

g -s

mniSvneloba dedamiwis sxvadasxva adgilze (ganedze) sxvadasxvaa*. polusebTan igi daaxloebiT tolia 9,83 m/wm 2-is, ekvatorze _ 9,78 m/wm2-is, xolo 45o ganedze _ 9,81 m/wm2-is. sainteresoa, rom (19.4) formuliT gamoTvlili g Tanxvdeba g -s cdiseul mniSvnelobas polusebze. rogorc vxedavT, gansxvaveba g -s mniSvnelobebs Soris metismetad mcirea. amitom, Tu gamoTvlebi Zalian didi sizustiT ar tardeba, ugulebelyofen g -s cvlilebis zemoT *

am gansxvavebis meore, naklebmniSvnelovani mizezi isaa, rom dedamiwa polusebTan

ramdenadme

Sebrtyelebulia,

ekvatorze

gamoberili,

ris

gamoc

misi

radiusi

sxvadasxva adgilas sxvadasxvaa. kerZod, polusze R  6357 km, xolo ekvatorze R  6378 km. 54

CamoTvlil mizezebs da Tvlian, rom Tavisufali vardnis aCqareba yvelgan erTnairia.

mniSvneloba

g =9,80665

m/wm2

miCneulia

g -s

n orm a lu r

( s ta n da r tu l ) mniSvnelobad.

 G s xe u li s w o n a e w o de b a Z al a s , r om l iT ac s xe u l i de d am i w is a dm i m iz id u l ob is S e d e g a d m oq m e de b s u Z r av h or iz on t alu r s ay r de n z e an v e rt ik a lu r s ak ide lze . amrigad, s xe u l is s i m Z i m is Z al a d a w on a y ov e lT v is s xv ada s xv a s x e u l ze a m o de b u l i. sxeulis wona aris Zala, romelic gamowveulia misi deformaciiT sayrdenTan an sakidelTan urTierTqmedebisas. simZimis Zalisagan gansxvavebiT, sxeulis wona icvleba sayrdenis (an sakidelis) vertikaluri mimarTulebiT aCqarebuli moZraobisas. ganvixiloT magaliTi. vTqvaT, uZravi liftis iatakze devs sxeuli, romlis masaa  m (sur. 19.1). masze moqmedebs simZimis Zala P  mg . sayrdenTan urTierTqmedebisas sxeuli deformirdeba da masSi aRiZvreba drekadi Zala, romli-

 a

 N  P

Tac sxeuli imoqmedebs sayrdenze. esaa sxeulis wona  G . urTierTqmedebis Sedegad deformirdeba sayrde nic. masSic aRiZvreba drekadi Zala N , romliTac is  N imoqmedebs sxeulze. d re k ad ob i s Z a la s , r o m l iT ac s xe u l ze m oq m e de b s s ay rd e n i (a n s a k id e l i ),

 G

re aq c i is

Z a la

e w o de b a

(sakidelis Sem-

TxvevaSi mas xSirad daWimulobis Zalas uwodeben).

niutonis mesame kanonis Tanaxmad   sur. 19.1 G  N (19.5)  Tu lifti vertikalurad qveviT imoZravebs a aCqarebiT, maSin sxeulis moZraobis gantoleba iqneba:       P  N  ma an P  G  ma , saidanac      G  P  ma  mg  a  .

(19.6)

(19.6) toloba gviCvenebs, rom sxeulis wona damokidebulia misi vertikalu     rad moZraobis a aCqarebaze. Tu lifti gaCerebulia, a  0 da G  P  mg , e.i. sxeulis wona da simZimis Zala erTmaneTis tolia.

 rodesac lifti moZraobs zeviT, aCqareba uaryofiTia ( a <0) da      G  mg  a   P  ma . e.i. sxeulis wona aRemateba simZimis Zalas _ sxeuli

mZimdeba, an rogorc amboben xolme, adgili aqvs gadatvirTvas. Sesabamisad, gaizrdeba sayrdenis mxridan ukuqmedebis Zalac. am mdgomareobas kargad icnoben

kosmonavtebi,

vinaidan

gadatvirTvis

dros

izrdeba

sxeulis

nawilebisa da Sinagani organoebis urTierTdawola. startisas kosmosuri  xomaldi moZraobs iseTi a aCqarebiT, romelic ramdenjerme aRemateba simZimis Zalis aCqarebas, ris gamoc kosmonavti ganicdis did gadatvirTvas. amitom, xomaldis startisas kosmonavtebi woliT mdgomareobaSi arian, raTa xerxemali ar dauziandeT.  roca lifti moZraobs qveviT, aCqareba dadebiTia ( a >0), da      G  mg  a   mg  ma . maSasadame, am dros sxeulis wona mcirdeba. vTqvaT,

lifti Tavisuflad vardeba, e.i. masze moqmedebs mxolod simZimis Zala,    maSin a  g da G  0 _ sxeulis wona xdeba nulis toli (igi aRar urTierTqmedebs sayrdenTan, aRar awveba mas). am mdgomareobas ewodeba u w o nob is m d gom ar e ob a. e . i. T u s xe u lze m oq m e d e b s m x olo d s im Z im is Z al a a nu T u is s im Z im is Z al is g a C q a re b iT m oZ ra ob s , m aS in i gi u w o n ob is m d go m a re ob a S ia .

amrigad, sxeulze simZimis Zala moqmedebs yo-

velTvis, wona ki vlindeba mxolod maSin, roca sxeulze garda simZimis Zalisa

moqmedebs

sxva

Zalac

(Zalebic),

ris

gamoc

sxeuli

moZraobs

 Tavisufali vardnis aCqarebisagan gansxvavebuli a aCqarebiT.

dedamiwis xelovnur Tanamgzavrze mxolod simZimis Zala moqmedebs. am SemTxvevaSi

simZimis

Zala

warmoadgens

centriskenul

Zalas,

romelic

aiZulebs Tanamgzavrs imoZraos dedamiwis irgvliv wriul orbitaze. e.i. Tanamgzavris centriskenuli aCqareba Tavisufali vardnis aCqarebis tolia. amitom sxeulebi TanamgzavrSi imyofebian uwonobis mdgomareobaSi _ isini aRar

awvebian

SexebaSi

myof

sxeulebs.

kerZod,

kosmonavtis

Sinagani

organoebi aRar awvebian maT qveviT moTavsebul organoebs, xolo mTlianad sxeuli aRar awveba ConCxis Zvlebs. es ganapirobebs uwonobis specifikur fiziologiur SegrZnebas. gamoviTvaloT pi rv e l i k os m os u ri s i Cq ar e

1  .

ase ewodeba umci-

res siCqares, romelic unda mivaniWoT sxeuls, rom igi gaxdes dedamiwis xelovnuri Tanamgzavri. 1 siCqares wriul siCqaresac uwodeben, radgan 56

aseTi siCqare gaaCnia xelovnur Tanamgzavrs, romelic brunavs dedamiwis irgvliv

wriul

orbitaze

(atmosferuli

winaaRmdegobis

ararsebobis

SemTxvevaSi). Tanamgzavris centriskeni aCqareba Tavisufali vardnis aCqarebis tolia:

12 Rh

 g sadac R dedamiwis radiusia, xolo h _ dedamiwidan

wriuli orbitis daSoreba. h  R , amitom 1  g saidanac 1  gR SevitaR 2

noT ricxviTi mniSvnelobebi:. 1  9,81  10 3  6370

km wm

 7,9

km wm

. 1

siCqareze

meti siCqaris SemTxvevaSi sxeuli moZraobs ara wriul, aramed elifsur orbitaze. m e or e k os m os u r i s iCq a re (  2 ) ewodeba umcires siCqares, romelic unda mivaniWoT sxeuls, rom man daZlios dedamiwis mizidulobis Zala da ibrunos mzis irgvliv, anu gaxdes xelovnuri planeta. mas parabolur siCqaresac uwodeben, vinaidan igi Seesabameba sxeulis parabolur traeqtorias dedamiwis mizidulobis velSi. 2  11,2km/wm. m e s a m e k os m os u r i s i Cq ar e (3 ) ewodeba umcires siCqares, romelic unda mivaniWoT dedamiwis zedapiridan gaSvebul sxeuls, rom man daZlios mzis mizidulobis Zala, e.i. datovos mzis sistema da imoZraos galaqtikisaken anu gadaiqces xelovnur varskvlavad. 3 =16,7 km/wm.*

* aRsaniSnavia, rom aseTi siCqare sakmarisia mxolod im SemTxvevaSi, Tu gaSveba xdeba

dedamiwis orbitis mxebi mimarTulebiT, vinaidan am mimarTulebiT sxeuls ukve gaaCnia m z i s m i m a r T orbituli siCqare 29,8 km/wm. 57

IV Tavi

arainerciuli sistemebi $20. Sesavali Cven

ganvixileT

inerciuli

sistemebi

vnaxeT,

rom

galileis

fardobiTobis principis Tanaxmad, meqanikis kanonebi ar icvlian saxes erTi inerciuli sistemidan meoreSi gadasvlisas ($14). sul sxvanairadaa saqme, roca inerciuli sistemidan gadavdivarT arainerciul (e.i. aCqarebiT moZrav)

sistemaSi.

ukve

gantolebebSi

Cndeba

axali,

gansxvavebuli

Sinaarsis matarebeli wevrebi. arainerciuli sistemebis Seswavlas didi mniSvneloba aqvs, vinaidan sistemaTa umravlesoba, romelTac Cven vxvdebiT teqnikasa da yoveldRiur cxovrebaSi, arainerciulia. TviT dedamiwac, brunvis gamo, warmoadgens arainerciul

sistemas.

dedamiwis

arainerciuloba

gamowveulia

misi

rogorc orbituli, aseve dReRamuri brunviT, magram vinaidan orbituli brunvis kuTxuri siCqare gacilebiT naklebia dReRamuri brunvis kuTxur siCqareze, amitom mas, rogorc wesi, mxedvelobaSi ar iReben da Tvlian, rom dedamiwis

arainerciuloba

ganpirobebulia

misi

dReRamuri

brunviT.

dReRamuri brunvis kuTxuri siCqarec metismetad mcirea:

 2 6,28 1    7,27  10 5 , t T 24  3600wm wm magram, miuxedavad amisa, igi ganapirobebs mTeli rigi saintereso movle-



nebis arsebobas dedamiwaze. zogadi (nebismierad moZravi) arainerciuli sistemis

ganxilva

sakmaod

rTulia.

Cven

ganvixilavT

umartives

SemTxvevas: 1. rodesac arainerciuli sistema moZraobs inerciuli sistemis  mimarT mudmivi a0 aCqarebiT (e.i. wrfivad da TanabaraCqarebulad) da 2. rodesac arainierciuli sistema brunavs  mudmivi  kuTxuri siCqariT.

inerciuli sistemis mimarT

unda aRiniSnos, rom dedamiwaze mimdinare yvela meqanikuri movlena SeiZleba aixsnas inerciuli sistemis TvalsazrisiT (e.i. iseTi sistemis TvalsazrisiT, romlis mimarTac dedamiwa brunavs). arainerciuli sistemebis Seswavlas, garda SemecnebiTi mniSvnelobisa, is azri aqvs, rom rig SemTxvevebSi aTvlis sistemad arainerciuli sistemis arCeva mniSvnelovnad amartivebs movlenebis aRwerasa da axsnas. 58

$21. wrfivad da TanabaraCqarebulad moZravi arainerciuli sistema davuSvaT,



oxyz

anu S

sistemis mimarT moZraobs  mudmivi a0 aCqarebiT arainerciu-

 r

inerciuli

 r' o'

o' x' y' z '

sistema ( S '

sistema).

iseve, rogorc $14-Si, simartivisa-

 a0

x, x '

Tvis davuSvaT, rom S da S ' sistemebis Sesabamisi RerZebi urTierTparaleluria

moZraoba

xdeba X RerZis gaswvriv (sur.21.1).

avirCioT

sur. 21.1

romelime

wertili,

romlis masa aris m , xolo radi  us-veqtorebi S da S ' sistemebSi, Sesabamisad, r da r ' . Tu am wertilze mo-

 qmedebs F Zala, maSin niutonis meore kanonis Tanaxmad, inerciul sistemaSi   (21.1) ma  F ,   d 2r sadac a  2 aris wertilis aCqareba S sistemis mimarT. davadginoT, dt rogor saxes iRebs (21.1) gantoleba arainerciul sistemaSi. amisaTvis am   gantolebaSi a unda SevcvaloT a ' -iT, anu wertilis aCqarebiT S ' sistemis  mimarT; rac Seexeba F Zalas, igi gansazRvravs mocemul wertilze garemomcvel

sxeulTa

moqmedebas

icvleba

erTi

sistemidan

meoreSi

gadasvlisas. (21.1) suraTidan Cans, rom    r  r 'oo ' . miRebuli toloba gavawarmooT orjer:    d 2 r d 2 r ' d 2 oo ' .  2  (21.2) dt 2 dt dt 2 2 magram d r2 '  a ' aris A wertilis aCqareba S ' sistemis mimarT, xolo dt  d 2 oo '   a0 aris S ' sistemis aCqareba S sistemis mimarT. amitom gveqneba: dt 2    a  a 'a0 .  a -s miRebuli mniSvneloba SevitanoT (21.1)-Si: 59

   ma 'ma0  F ,

saidanac (21.3)    ma '  F  ma0 . aseT saxes iRebs (21.1) gantoleba arainerciul sistemaSi. es gantole ba gviCvenebs, rom arainerciul sistemaSi sxeulze, garda `Cveulebrivi~ F Zalisa, romelic gamowveulia masze sxva sxeulebis moqmedebiT, damatebiT  moqmedebs  ma0 Zala. es uCveulo Zalaa, radgan igi ar aris sxeulebis urTierTqmedebis Sedegi _ ar arsebobs sxeuli, romlis moqmedebiTac igi iqneboda gamowveuli. mas ara hyavs `Qpartniori~, dawesebuli niutonis mesame kanoniT. igi `gaCnda~ mxolod imitom, rom S ' sistema arainerciulia. am Zalas, romelic gamowveulia sistemis arainerciulobiT da maSasadame, moqmedebs mxolod arainerciul sistemaSi, ewodeba inerciis Zala:   Fin  ma0

(21.4)

rogorc vxedavT, igi mimarTulia sistemis aCqarebis sawinaaRmdegod. inerciis Zalis gamovlenas Cven vxvdebiT yoveldRiur cxovrebaSi. kerZod, avtobusi damuxruWebisas warmoadgens arainerciul sistemas, romlis aCqareba mimarTulia moZraobis sawinaRmdegod, ris gamoc mgzavrebze moqmedebs moZraobis gaswvriv mimarTuli inerciis Zala. amitomaa, rom avtobusis swrafi damuxruWebisas mgzavrebi win gadaixrebian. avtobusis swrafi aZvrisas xdeba piriqiT _ aCqarebas aqvs moZraobis mimarTuleba, mgzavrebze ki moqmedebs moZraobis sawinaaRmdegod mimarTuli inerciis Zala. amitom, avtobusis swrafi amoZravebisas mgzavrebi ukan gadaixrebian. $19-Si Cven ganvixileT liftis vertikaluri moZraoba da avxseniT uwonobisa da gadatvirTvis mdgomareoba. ganvixiloT igive sakiTxebi arainerciuli sistemis TvalsazrisiT (e.i. aCqarebulad moZrav liftSi myofi  damkvirveblis TvalsazrisiT). vTqvaT, lifti moZraobs qveviT a0 aCqarebiT. maSin am liftSi moTavsebul m masis sxeulze, garda vertikalurad qveviT    mimarTuli P  mg Zalisa, moqmedebs zeviT mimarTuli inerciis  ma0 Zala. amitom sxeulze moqmedi sruli Zala, romelic tolia simZimisa da inerciis Zalebis veqtoruli jamisa, mcirdeba. maSasadame, mcirdeba sxeulis dawolac sayrdenze, rac iwvevs misi wonis Semcirebas. rodesac lifti   Tavisuflad vardeba, inerciis Zala Fin  mg akompensirebs simZimis Zalas, sxeulze moqmedi sruli Zala xdeba nulis toli, ris gamoc sxeuli aRar 60

awveba sayrdens da igi xdeba uwono. liftis aCqarebulad zeviT moZraobisas, inerciis Zala mimarTulia vertikalurad qveviT. igi emateba simZimis Zalas, ris gamoc sxeulze moqmedi sruli Zala izrdeba. amis gamo izrdeba sxeulis dawola sayrdenze. es ki iwvevs sxeulis wonis gazrdas, rasac Cven gadatvirTva vuwodeT. rogorc vxedavT, uwonobisa da gadatvirTvis movlenas advilad xsnis arainerciuli damkvirvebelic, oRond amisaTvis mas uxdeba inerciis Zalis gamoyeneba. (21.4)-is Tanaxmad, (21.3) gantoleba Semdegnairad Caiwereba:    (21.5) ma '  F  Fin . miRebuli gantoleba gamosaxavs dal am b e r is princips, romelic Semdegnairad gamoiTqmis: urTierTqmedebis yvela Zalis da inerciis Zalis veqtoruli jami tolia sxeulis masis namravlisa mis aCqarebaze arainerciuli sistemis mimarT. (21.5) gantoleba garegnulad savsebiT msgavsia niutonis meore kanonisa. msgavseba mxolod garegnulia, sinamdvileSi ki niutonis meore kanoni (iseve, rogorc danarCeni kanonebi) ar aris marTebuli arainerciul sistemaSi, vinaidan masis namravli aCqarebaze ar udris sxva sxeulebis moqmede   biT gamowveul F Zalas. rodesac F  0 , (21.5)-is Tanaxmad, a ' 0 , e.i. sxeulma SeiZleba SeiZinos aCqareba sxva sxeulebis moqmedebis gareSe. dabolos, mudmivobis

aRvniSnoT,

kanonebi,

rom

vinaidan

arainerciul sistemaSi

sistemaSi myof

sruldeba

sxeulTa

nebismieri

jgufisaTvis inerciis Zala warmoadgens gareSe Zalas, ris gamoc aq ar arsebobs izolirebuli (Caketili) sistema.

$22. Tanabrad mbrunavi arainerciuli sistema

davuSvaT, mocemuli gvaqvs inierciuli oxyz sistema ( S sistema) da arainerci



uli o' x' y' z ' sistema ( S ' sistema), romelic

↺

z'  r

brunavs S sistemis mimarT z RerZis ir gvliv mudmivi  kuTxuri siCqariT. avirCioT romelime A wertili, romlis masa  aris m xolo radius-veqtori r (sur.

y' sur. 22.1

A

 22.1). Tu am wertilze moqmedebs F

Zala, maSin, niutonis meore kanonis

Tanaxmad, inerciul sistemaSi:

  ma  F ,

(22.1)

 sadac a aris aCqareba S sistemis mimarT. vnaxoT, rogor saxes iRebT (22.1)

gantoleba mbrunav arainerciul sistemaSi. amisaTvis, iseve, rogorc wina    paragrafSi, am gantolebaSi a unda SevcvaloT wertilis a ' aCqarebiT S ' sistemis mimarT.

 davuSvaT, A wertili gadaadgildeba S ' sistemaSi  0 siCqariT. davadginoT am wertilis siCqare S

sistemis mimarT:

A wertili rom uZravi

yofiliyo S ' sistemaSi, maSin igi mxolod ibrunebda am sistemasTan erTad S sistemis mimarT da misi siCqare, rogorc mbrunavi wertilis siCqare, S   sistemis mimarT toli iqneboda veqtoruli namravlis  r . magram vinaidan   A wertili gadaadgildeba S ' sistemaSi  0 siCqariT, amitom misi sruli 

siCqare

inerciuli

sistemis

mimarT

toli

iqneba

ori

siCqaris

veqtoruli jamisa: 



 

   0   r  .

(22.2)

mivaqcioT yuradReba im garemoebas, rom A wertilis radius-veqtori orive  sistemaSi erTi da igivea ( r ), xolo radius-veqtoris cvlileba aris sxvadasxva, vinaidan siCqareebia sxvadasxva. aRvniSnoT radius-veqtoris cvli  dr d'r leba S sistemaSi –Ti, xolo S ' sistemaSi _ -Ti. maSin (22.2) toloba dt dt ase daiwereba:

aseTia

kavSiri

  dr d ' r      r  . dt dt radius-veqtoris cvlilebebs

(22.3) Soris

sistemebis mimarT. SeiZleba Cveneba, rom (22.3) formula, romelic Cven miviReT radius-veqtoris cvlilebebisaTvis kerZo magaliTis ganxilvis  safuZvelze, samarTliania nebismieri veqtorisaTvis. gamoviyenoT igi  0 veqtorisaTvis. gveqneba:   d0 d '0      0  . dt dt gavawarmooT (22.2) toloba S sistemis mimarT: 62

(22.4)

   d d0   dr     . dt dt  dt    dr d 0 SevitanoT am tolobaSi -s da -s mniSvnelobebi (22.4) da dt dt tolobebidan:    d d '0     d'r        0        r  . dt dt  dt   d 'r  magram  0 , amitom dt   d d '0        2 0     r . dt dt   d  d '   a aris wertilis aCqareba S sistemis mimarT, xolo  a' dt dt wertilis aCqareba S ' sistemis mimarT. amrigad, gveqneba:        a  a '2 0     r  .

(22.5) (22.3)

(22.6) aris

(22.7)

saidanac

       a '  a  2 0     r  . miRebuli toloba gviCvenebs, rom aCqareba arainerciuli sistemis

mimarT tolia aCqarebisa inerciuli sistemis mimarT plus aCqarebebi:    ak  2 0  . da     ac    r . pirvels ewodeba k or io li s is a Cq are b a , meores _ c e ntr i da n u l i a Cq a reba.  SevitanoT a (22.7)-dan (22.1) gantolebaSi:        ma '2m 0   m  r   F , saidanac        (22.8) ma '  F  2m 0   m  r  . aseT saxes iRebs (22.1) gantoleba Tanabrad mbrunav arainerciul  sistemaSi. aq F aris Zala, romelic gamowveulia mocemul sxeulze sxva

sxeulebis moqmedebiT. es Zala moqmedebs sxeulze inerciul sistemaSic da rogorc (22.8) toloba gviCvenebs, Tanabrad mbrunav arainerciul sistemaSi, garda am Zalisa, sxeulze damatebiT moqmedebs kidev Zalebi:    Fk  2m 0  da

    Fc  m  r  .

(22.9) (22.10)

pirvels ewodeba in e rc ii s k o ri oli s i s Z al a , xolo meores _ ine r c i i s c e ntr i d an u l i Z al a. es uCveulo Zalebia, vinaidan isini gamowveulia 63

ara sxeulTa urTierTqmedebiT, aramed sistemis arainerciulobiT. rogorc maTi gamosaxulebebi gviCvenebs, koriolisis Zala damokidebulia sxeulis siCqareze da ar aris damokidebuli mis adgilmdebareobaze. centridaulni  Zala ki piriqiT _ damokidebulia sxeulis adgilmdebareobaze ( r -ze) da ar aris damokidebuli siCqareze. davadginoT centridanuli aCqarebis (da, maSasadame, Zalisac) sidide da

mimarTuleba:

wertili

brunvisas

 r 

Semoswers R - radiusian wrewirs (sur. 22.2).   veqtoruli namravlis wesis Tanaxmad,  r is sidide



↺

wesis gamoyenebiT,   advilad davadgenT, rom  r  veqtori mimarTulia

burRis

naxazis

sibrtyis



 r    r  sin   R. rac Seexeba

mimarTulebas,

 o 



 r

marTobulad

Cvengan.

   r  amrigad,

     R  sin 90    2 R

centridanuli aCqarebis

sur. 22.2

sididi-

saTvis gvaqvs: ac   2 R

(22.11)

2 Sesabamisad, centridanuli Zalis sidide Fc  m R

rac Seexeba mimarTulebas, burRis wesis Tanaxmad, veqtori

  r

mimarTulia radiusis gaswvriv centrisken, TviT centridanuli aCqareba ki (vinaidan win aqvs minusi niSani) mimarTuli iqneba radiusis gaswvriv centridan gareT. swored amitom ewodeba mas centridanuli aCqareba. koriolisis Zalis sididisaTvis (22.9) formulis Tanaxmad gveqneba   FK  2m0 sin  , sadac  aris kuTxe  0 -sa da  -s Soris.    rac Seexeba mimarTulebas, FK marTobia  0 da  veqtorebze gavlebuli sibrtyisa da aqvs im burRis gadataniTi moZraobis mimarTuleba, rom-

  lis saxelursac vabrunebT  0 -dan  –sken (gavixsenoT, rom

0   0  ).

$23. dedamiwa, rogorc arainerciuli sistema Cven aRvniSneT, rom dedamiwa, brunvis gamo, warmoadgens arainerciul sistemas. isic vTqviT, rom dedamiwis dReRamuri brunvis kuTxuri siCqare metismetad mcirea, magram, miuxedavad amisa, igi mainc ganapirobebs mTel rig movlenebs dedamiwaze. kerZod, vnaxoT, Tu ra gavlenas axdens dedamiwis dReRamuri brunva simZimis Zalaze. davuSvaT, rom dedamiwas sferos forma aqvs. vinaidan igi 

brunavs



↺

dasavleTidan

aRmosavleTi-

saken, kuTxuri siCqare mimarTuli iqneba samxreTidan CrdiloeTisaken (sur.

 F1

r  R P0 )

)

(

 P

 F2

 Fc

23.1). vTqvaT, dedamiwis romelime adgilas, romlis ganedia  , Zevs m masis sxeuli. dedamiwa rom ar brunavdes, sxeulze imoqmedebda erTaderTi Zala,  simZimis P0 Zala, romelic mimarTulia radiusis gaswvriv centrisken, magram imis gamo, rom dedamiwa brunavs, sxe-

sur. 23.1

ulze moqmedebs kidev inerciis cen tridanuli Zala Fc *. amitom sxeulze moqmedi

WeSmariti

simZimis

 P

Zala

 warmoadgens am ori Zalis tolqmeds. rogorc suraTidan Cans, P gansxva  vdeba P0 -sagan sididiTac da mimarTulebiTac. swored P Zala gansazRvravs   mocemul ganedze vertikalis mimarTulebas da ara P0 Zala; P ZaliT awve ba sxeuli horizontalur sayrdens da ara P0 ZaliT da a.S.   davamyaroT damokidebuleba P da P0 Zalebis sidideebs Soris. amisa Tvis centridanuli Zala Fc davSaloT or mdgenelad _ dedamiwis radiusis gaswvriv ( F1 mdgeneli) da mis marTobulad ( F2 mdgeneli). suraTidan  Cans, rom F1 Zala iwvevs P0 -is sididis cvlilebas (Semcirebas), xolo F2

* vinaidan sxeuli uZravia dedamiwis mimarT, masze koriolisis Zala ar moqmedebs.

Zala _

 P0 -is mimarTulebis cvlilebas. amitom

 P

Zalis sididisaTvis

gveqneba: P  P0  F1 .

suraTidan Cans, rom F1  Fc  cos   m 2r  cos  ,

amitom P  P0  m 2r  cos  .

P0  mg0 ,

magram

sadac

(23.1)

aris 



simZimis Zalis aCqareba `uZrav~ dedamiwaze, igi

Tanxvdeba

simZimis

Zalis

aCqarebas

polusebze (vinaidan aq centridanuli Zala

Fc  0 ).

garda

amisa,

r  R  cos 

sadac





↺



0

 Fk



dedamiwis radiusia. SevitanoT (23.1)-Si:

P  P0 



0

P0 2  R  cos 2  , g0



0

an   2R  P  P0 1   cos 2   . g0  

gamoTvlebi

gviCvenebs,

amitom

rom



 Fk

(23.2)

 2R g0



0

 0,0034 ,

sur. 23.2



(23.3) P  P0 1  0,0034  cos 2  . (23.3) formula, iseve, rogorc (23.2), gamosaxavs simZimis Zalis cvlilebas geografiuli ganedis mixedviT. es cvlileba maqsimaluria ekvatorze, vinaidan aq   0 da cos 2   1 da minimaluria (nulis tolia) polusze (   90 ,

cos 2   0 da P  P0 ). rogorc (23.3) formula gviCvenebs, gansxvaveba

P -sa da

P0 -s Soris

ekvatorzec ki metismetad mcirea, rac imiT aris gamowveuli, rom inerciis centridanuli

Zala

gacilebiT

mcirea

simZimis

Zalaze.

marTlac,

P0  mg0  9,83m , xolo ekvatorze 2    5 1 Fc  m R   7,27  10  6,4  106 m   m  0,034m .  wm    simZimis Zalis aCqarebis cvlilebas ganedis mixedviT miviRebT, Tu 2

(23.3) formulaSi SevitanT P  mg da P0  mg0 mniSvnelobebs. gveqneba:





g  g0 1  0,0034  cos 2  . 66

(23.4)

sinamdvileSi,

gansxvaveba

gď Ş -s

mniSvnelobebs

Soris

sxvadasxva

ganedze ufro metia, vidre es gamomdinareobs (23.4) formulidan. amis mizezia is, rom dedamiwas ara aqvs idealuri sferuli forma _ centridanuli Zalis moqmedebiT igi gamoberilia ekvatorze. ganvixiloT koriolisis Zalis moqmedebis magaliTi. vTqvaT, sxeuli moZraobs CrdiloeT naxevarsferoSi meridianis gaswvriv CrdiloeTidan samxreTisaken.

maSin,

rogorc

23.2

suraTidan Cans, masze imoqmedebs ď ˛ moZraobisadmi marjvniv mimarTuli Fk koriolisis Zala. imave suraTidan Cans, rom imave naxevarsferoSi samxreTidan CrdiloeTisaken moZraobis SemTxvevaSic

koriolisis

Zala

moZraobisadmi

marjvniv

aris

mimarTuli.

amrigad, CrdiloeT naxevarsferoSi meridianis gaswvriv moZrav sxeulze yovelTvis

moqmedebs

moZraobisadmi

marjvniv

mimarTuli

Zala.

aseve

advilia Cveneba, rom samxreT naxevarsferoSi moZrav sxeulze moqmedebs moZraobisadmi

marcxniv

sxeulze,

mimarTuli

moZraoba

Zala.

warmoebs

ekvatoris

dasavleTidan

gaswvriv

moZrav

aRmosavleTisaken,

sxeulze moqmedebs `zeviT~ mimarTuli koriolisis Zala, rac iwvevs misi wonis Semcirebas. aRmosavleTidan dasavleTisaken moZraobisas, piriqiT, koriolisis Zalis moqmedebiT sxeuli awveba dedamiwas, rac iwvevs misi wonis gazrdas. koriolisis Zalis moqmedebiT aixsneba is faqti, rom mdinareebi, romlebic

miedinebian

meridianis

gaswvriv,

CrdiloeT

naxevarsferoSi

marjvena napirs ufro metad Camorecxaven, samxreT naxevarsferoSi ki _ marcxenas. qvemexidan gasrolili Wurvi CrdiloeT naxevarsferoSi ixreba marjvniv,

samxreT

naxevarsferoSi

marcxniv.

xolo

gasrola

warmoebs ekvatoris gaswvriv, koriolisis Zala Wurvs an aswevvs zeviT (Tu gasrola xdeba dasavleTidan aRmosavleTisaken), an daswevs qveviT. koriolisis

Zalis

moqmedebiT

aixsneba

agreTve

saqanis

rxevis

sibrtyis

mobruneba (fukos cda). koriolisis Zalis moqmedeba haeris nakadze anu qarze ganapirobebs e.w. pasatebis arsebobas. saqme isaa, rom CrdiloeTis polaruli aridan _ arqtikidan ekvatorisaken, e.i. samxreTisaken mimarTuli civi qari koriolisis Zalis moqmedebiT gadaixreba marjvniv anu imoZravebs ara CrdiloeTidan samxreTisaken, aramed Crdilo-aRmosavleTidan samxreT-dasavleTisaken (Crdilo-aRmosavleTis pasati). imave mizeziT samxreT naxevarsferoSi 67

warmoiqmneba samxreT-aRmosavleTis pasati, romelic `uberavs~ antarqtididan ekvatorisaken*. koriolisis Zalis moqmedebiT aixsneba haeris nakadis moZraobis xasiaTi ciklonebsa da anticiklonebSi da sxv. dabolos,

rogorc

aRvniSneT,

marTalia,

dedamiwa

warmoadgens

arainerciul sistemas, magram Tu sxeulze moqmedi Zalebi didia inerciis ZalebTan

SedarebiT,

igi

miaxloebiT

SeiZleba

CavTvaloT

inerciul

sistemad, rogorc Cveulebriv xdeba xolme praqtikaSi.

V Tavi

m u S a ob a d a e n e r g i a $24. muSaoba. cvladi Zalis muSaoba. erTeulebi muSaobis cneba asaxavs m a te ri is m oZ r aob is e rT i fo rm is g ar d a q m n is pr oce s s m e ore f orm a d . magaliTad, Wurvis gasrolisas denTis airi asrulebs muSaobas lulidan Wurvis gamotyorcnaze, misi molekulebis moZraobis intensivoba mcirdeba, Wurvis moZraobis siCqare ki izrdeba, e.i. xdeba materiis moZraobis siTburi formis gardaqmna meqanikur formad. termini `muSaoba~ gamoiyeneba ori azriT: zemoT miTiTebuli procesis azriT da rogorc fizikuri sidide, romelic aris am procesis raodenobrivi maxasiaTebeli, raodenobrivi zoma. meqanikuri muSaoba sruldeba maSin, rodesac sxeulze moqmedebs Zala da igi gadaadgildeba. amitom, bunebrivia, rom m u S aob a iz om e b a Z a lis a d a am Z a lis m oq m e d e b i s m im arT u l e b iT gav li li g zi s n am r av l iT : A F S .

muSaoba

geometriulad

sur.

24.1-ze

(24.1) gamosaxuli

daStrixuli

marTkuTxedis farTis tolia. zogad SemTxvevaSi, roca Zalis moqmedebis mimarTuleba ar emTxveva  gadaadgilebis mimarTulebas (sur. 24.2), F Zalas, paralelogramis wesiT,  Slian or mdgenelad, romelTagan erTi ( FS ) gadaadgilebis Tanxvedrilia,

pasatebis moqmedeba Zlieria okeanis tropikul ganedebze. kontinentur nawilze igi

sustadaa gamoxatuli. 68

 xolo meore ( Fn ) _ gadaadgilebis marTobuli.

 Fn

Zala muSaobas ar

asrulebs, vinaidan misi moqmedebis mimarTulebiT sxeuli ar gadaadgilde ba. amitom mTeli muSaoba FS Zalis muSaobis tolia:

 Fn

 Fn )

 FS

sur. 24.1

sur. 24.2 A  Fs S .

(24.2)

magram naxazidan Cans, rom FS  F cos  , amitom A  FS cos .

(24.3)

e.i. zo g a d Se m T xv e v a S i m u Sa ob a i zo m e b a Z ali s a d a g a d a ad g i l e b is s i d id e e b i s a d a m aT S or is m o T av s e b u l i k u T xi s k os inu s is n am r av l iT . (24.3) formula gviCvevnebs, rom roca  

 2

, muSaoba dadebiTia. igi

maqsimaluria maSin, roca   0 ( A  FS ), xolo roca  



_ muSaoba uar-

yofiTia. igi maqsimaluria maSin, roca    , e.i. roca Zala mimarTulia gadaadgilebis sawinaaRmdegod:

A   F S . aseTi Zalis magaliTia srialis

xaxunis Zala, romelic yovelTvis gadaadgilebis sawinaaRmdegodaa mimarTuli. Tu

gamoviyenebT

skalaruli

namravlis

wess,

(24.3)

formula

Semdegnairad daiwereba:



e.i. m u S aob a

Z al is a



  A  F S . g ad a a dgi le b is

v e q to re b is

(24.4) s k a l aru li

n am r av l is to li a. Cven ganvixileT umartivesi SemTxveva, roca moqmedi Zala iyo mudmivi, xolo moZraoba wrfivi. axla ganvixiloT SemTxveva, roca Zala ar aris mudmivi da gadaadgilebasTan Sedgenili kuTxec cvladia (moZraoba warmoebs mrud wirze). am dros muSaobis gamosaTvlelad mTeli S gza unda dav69

 yoT imdenad mcire S1 , S2, . . . Sn ubnebad, rom TiToeulis farglebSi F Zala da am Zalis mier gadaadgilebasTan Sedgenili  kuTxe SeiZlebodes CaiTvalos mudmivad. maSin TiToeul ubanze Sesrulebuli elementaruli muSaoba (24.4)-is Tanaxmad tolia:   Ai  Fi  S i  Fi S i cos  i  Fis S i





sadac Fis aris Zalis gegmili gadaadgilebis mimarTulebaze. es muSaoba geometriulad gamosaxavs im svetis farTobs, romlis siganea  Si , xolo simaRle F . is

muSaoba mTel S gzaze toli iqneba aseTi elementaruli muSaobebis jamisa:





n n n n   A   Ai   Fi  Si   Fi  Si cos  i   FiS  Si . i 1

i 1

(24.5)

i 1

cxadia, rom rac ufro mcirea gzis TiToeuli ubani Si , miT ufro meti sizustiT SeiZleba CaiTvalos F da  mudmivebad am ubnebze. amitom zusti Sedegis misaRebad unda movnaxoT (24.5) formuliT gamosaxuli jamis zRvari, roca Si  0

A  lim

S  0

F i 1

 Si .

(24.6)

maTematikidan cnobilia, rom aRniSnuli zRvari tolia gansazRvruli S

integralisa

 F dS . S

amrigad, sabolood gvaqvs, rom cvladi Zalis mier

Sesrulebuli muSaoba mrudwirul traeqtoriaze S

A   Fs dS

(24.7)

integrali

geometriulad

tolia

cvladi Zalis ab grafikis qveS moTavsebuli figuris farTisa (sur. 24.3). muSaobis

erTeuls

davadgenT

(24.1)

formulidan. m u Sa ob i s e r T e u l a d m iR e b u -

a Fis Si

sur. 24.3

l i a e r T e u lov an i Z a li s m ie r S e s ru le b u l i m u S aob a e rT e u l ov a n g z aze , r od e s ac Z al is m im a rT u le b a T an x v de b a g a da a d gi le b i s m im arT u le b as . SI sistemaSi muSaobis erTeulia jouli (j). 70

1 j = 1 n

1 m

e.i. jou li a ri s m u S aob a, ro m e ls a c a s ru le b s e rT i niu to ni Z a la e r T m e tr g z aze . teqnikur sistemaSi muSaobis erTeulia kgm. 1 kgm = 1 kgZ . m. vinaidan 1 kgZ=9,81 n-s, amitom 1 kgm=9,81 j.

$25. simZimis Zalis muSaoba. konservatuli da arakonservatuli Zalebi Tu sivrcis yovel wertilSi nawilakze moqmedebs garkveuli Zala, maSin amboben, rom nawilaki imyofeba Zalur velSi. aseTia, magaliTad, gravitaciuli

anu

simZimis

veli,

eleqtromagnituri veli

sxv.

nawilakze moqmedi Zala velis yovel wertilSi sididiT da mimarTulebiT erTnairia,

vels

ewodeba

erTgvarovani.

veli

drois

mixedviT

icvleba, mas ewodeba stacionaruli. simZimis veli dedamiwis zedapirTan axlos erTvagrovanicaa da stacionarulic. gamoviTvaloT simZimis Zalis mier Sesrulebuli muSaoba. vTqvaT, m masis

sxeuli

atanilia

simaRleze

simZimis

Zalis

moqmedebiT

moZraobs a wertilidan b wertilisaken nebismier mrud wirze (sur. 25.1). moZraobis mTel gzaze sxeulze moqmedi P simZimis Zala, cxadia, mudmivia, magram

kuTxe

Zalasa

gadaadgilebas

Soris

ganuwyvetliv

icvleba.

muSaobis gamosaTvlelad ab gza davyoT imdenad mcire Si ubnebad, rom TiToeul ubanze kuTxe  i SeiZlebodes CaiTvalos mudmivad (e.i. Ta-

a hi

vad ubani _ wrfivad),

i

Si

h1 P

gamoviTva-

loT TiToeul ubanze elementaruli muSaobebi da Semdeg SevkriboT. b

Si ubanze P Zalis mier Sesrule-

buli muSaoba (24.3) formulis Tanaxmad tolia:

sur. 25.1

Ai  P  Si cos i . 71

magram

Si cos  i  hi .

amitom

Ai  P  hi .

sruli muSaoba n

i 1

A   Ai   P  hi  P hi  Ph1  h2  . e.i.

A  Ph1  h2  . miviReT

mniSvnelovani

Sedegi:

simZimis

(25.1) Zalis

mier

Sesrulebuli

muSaoba gzis formaze damokidebuli ar aris, damokidebulia mxolod sxeulis sawyis da saboloo mdebareobaze. i s e T v e ls , rom e l Si c s xe u l i s g a d a a dg ile b a ze Se s ru le b u li m u S aob a ar a a d am ok i d e b u li gz is f o r m aze

d a d am ok i de b u l i a m xol o d s xe u lis

s aw y i s

da s a b o -

l oo m de b a re ob az e , e w o de b a k o ns e rv atu li an p ote n c iu r i v e l i, x o lo as e T v e l S i m oq m e d Z ale b s _ k on s e rv a tu l i an po t e n c iu r i Z a le b i. simZimis Zalis garda konservatulia drekadobis Zala, eleqtrostatikuri Zala da, sazogadod, yvela Zala, romelic nawilakebs Soris fundamenturi urTierTqmedebiTaa gamowveuli (ixile petitis teqsti). vinaidan

gadaadgilebis

mimarTulebis

SecvliT

muSaoba

icvlis

niSans, (25.1) formulis Tanaxmad, k o ns e rv atu li Z al is m i e r S e k r u l g z az e Se s ru l e b u l i m u S a ob a nu l is t ol ia . konservatuli Zalebis garda arsebobs e.w. di s i p ac iu r i Zalebi. es iseTi Zalebia, romelTa mier Sesrulebuli muSaoba gzis yovel ubanze uaryofiTia da maSasadame, mTeli muSaoba miT metia, rac metia gavlili gza. amitom cxadia, rom Sekrul gzaze disipaciuri Zalis muSaoba nulis toli ar aris. disipaciuri Zalis magaliTia xaxunis Zala, romlis mier Sesrulebuli muSaoba yovelTvis uaryofiTia. xaxunis Sedegad sxeuli Tbeba.

dros

meqanikuri energia gardaiqmneba siTbur

energiad,

anu

adgili aqvs meqanikuri energiis gabnevas*. meqanikuri muSaobis sididis ganmsazRvreli (24.1) formulidan gamomdinareobs, rom rac ufro met gzas gadis sxeuli Zalis moqmedebiT, miT metia am Zalis mier Sesrulebuli muSaoba. meore mxriv, Cven vnaxeT, rom es yovelTvis ase ar aris: simZimis Zalis muSaoba damokidebulia sxeulis mxolod sawyis da saboloo mdebareobaze, xolo gzis sigrZeze,

* aqedan warmodgeba Zalis dasaxelebac: laTinurad `disipacia~ niSnavs gabnevas.

traeqtoriis formaze damokidebuli ar aris. kidev metic, aRmoCnda, rom es Tviseba sayovelTaoa

damaxasiaTebelia

nawilakebs

Soris

moqmedi

yvela

fundamenturi

ZalisaTvis. rogorc III TavSi gakvriT iyo naxsenebi, bunebaSi gvxvdeba mxolod oTxi tipis fundamenturi urTierTqmedeba da, Sesabamisad, oTxi tipis fundamenturi Zala. esenia gravitaciuli,

eleqtromagnituri,

Zlieri da susti urTierTqmedebis Zalebi. yoveli

maTganis muSaoba damokidebulia mxolod nawilakis sawyis da saboloo mdebareobebze, xolo

gzis

formaze

damokidebuli

aris.

sagulisxmoa,

rom

yoveli

maTgani

damokidebulia manZilze (sxeulis koordinatebze). meqanikaSi ganixileba sami saxis Zala _ gravitaciuli, drekadobis da xaxunis Zalebi. amaTgan pirveli ori Zala konservatulia. maTi sidide damokidebulia manZilze (gravitaciuli Zalisa _ sxeulebs Soris manZilze, xolo drekadobis Zalis _ sxeulis nawilebs Soris manZilze). mesame saxis Zalis _ xaxunis Zalis sidide manZilze damokidebuli ar aris (damokidebulia siCqareze), is ar aris konservatuli Zala, misi muSaoba gavlili manZilis proporciulia. ismis kiTxva: ratom?

sxeulis

Semadgenel

nawilakebs

Soris

moqmedi

yvela

tipis

Zala

xom

konservatulia? saqme isaa, rom xaxunis Zala moqmedebs sxeulebs Soris da ara mis Semadgenel nawilakebs Soris. atomur-molekulur samyaroSi xaxunis movlena ar arsebobs. igi Tavis iCens mxolod makroskopul sxeulTa urTierTqmedebisas.

$26. simZlavre xSirad

saWiroa

vicodeT

ara

marto

muSaobis

sidide,

romlis

Sesrulebac ama Tu im manqanas an meqanizms SeuZlia, aramed muSaobis Sesrulebis tempebic. cxadia, rom ori manqanidan ukeTesia is, romelic mocemul samuSaos ufro swrafad Seasrulebs. fizikaSi, m u Sa ob i s Se s r u l e b is s is w r a fis da s a x as i aT e b l a d S e m oR e b u li a s im Z l av r is c ne b a. m anq a ni s an m e q an izm is s im Z l av re to li a Se s ru le b u li m u S a ob is f a rd ob i s a i m dr os T an , r om l is g a nm av l ob aS ic e s m u S a o b a Se s r u l d a: A . (26.1) t Tu (26.1) formulaSi davuSvebT, rom t  1, maSin N  A . e.i. simZlavre N

ricxobrivad drois erTeulSi Sesrulebuli muSaobis tolia. s im Z l av ri s e rT e u l a d m iR e b u l ia i s e T i m a nq an is s im Z l av r e , r om e li c e rT e u l ov an d ro S i e rT e u lov a n m u S a ob a s as ru le b s . SI sistemaSi simZlavris erTeulia vati (vt).

1vt 

1j . 1wm

e.i. v a ti a ri s is e T i m a nq an is s im Z l av re , ro m e l ic e rT w am Si e r T j ou l

m u S a ob a s

as ru l e b s .

aTasjer

met

erTeuls

kilovati

(kvt)

ewodeba, milionjer mets _ megavati (mgvt). xSirad

gamoiyeneba

simZlavris

sistemgareSe

erTeuli,

romelsac

cxenis Zala (cZ) ewodeba:

1 kgm . 1 wm davamyaroT damokidebuleba cxenis Zalasa da vats Soris: 1 cZ  75

1 cZ 

j 1 kgm  75  9,81  740 vt . 1 wm wm

adamianis saSualo simZlavre daaxloebiT 0,1 cxenis Zalis tolia, e.i. wamSi SeuZlis Seasrulos

 75 jouli muSaoba. magram xanmokle drois ganmavlobaSi, magaliTad,

xtomis an kibeze arbenis dros, mas SeuZlia ganaviTaros 1 c.Z.-is toli da meti simZlavre. sainteresoa aRiniSnos, rom cxens xangrZlivi drois manZilze SeuZlia ganaviTaros 500 vt simZlavre, drois ufro mcire monakveTSi ki _ (700-800) vt simZlavre. swored es monacemebi udevs safuZvlad simZlavris erTeulis `cxenis Zalis~ dadgenas, romelic

 740 vt-is

tolia.

(26.1) erTeulad

formulidan SI

A  N t

aqedan

gamomdinareobs,

rom

m u S aob is

s is te m a Si Se iZ le b a m iv i Cn ioT m u S aob a, ro m e l ic

s r u l de b a e r T i v at i s i m Z l av ris a s 1 w am Si . am e rT e u ls e w o de b a v at -w am i (v t. w m ) : 1 vt.wm =1 j. jouli da, maSasadame, vat-wamic Zalian mcire erTeulebia. amitom praqtikaSi sargebloben ufro didi erTeulebiT _ kilovatsaaTiT (kvt.sT) da megavatsaaTiT (mgvt.sT). 1 kvt.sT = 3,6.106 j; 1 mgvt.sT = 3,6.109 j. avtomanqanebi,

gemebi,

TviTmfrinavebi

xSirad

mudmivi

siCqariT

moZraoben, rac niSnavs, rom Zravas muSaobis Sedegad, maTze moqmedi wevis Zala gawonasworebulia xaxunis ZaliT*. vnaxoT, Tu razea damokidebuli am obieqtebis mier ganviTarebuli maqsimaluri mudmivi siCqare:

* gavixsenoT (§17), rom siTxeSi an airSi moZrav sxeulze moqmedi xaxunis Zala sveli

xaxunis Zalisa da winaaRmdegobis Zalis jamis tolia. 74

N

A F S   F  , t t

(26.2)

saidanac N . (26.3) F (26.3) gviCvenebs, rom Tu xaxunis Zala mudmivia, obieqtis maqsimaluri



siCqare Zravis simZlavris proporciulia. amitom sWirdeba swrafmaval avtomanqanebs da sxva obieqtebs didi simZlavris Zravebi. magram xaxunis Zala xSirad mudmivi ar aris da izrdeba siCqaresTan erTad. kerZod, didi siCqariT moZraobis SemTxvevaSi xaxunis Zalis sidide siCqaris kvadratis proporciulia: F  k   2 Tu am mniSvnelobas SevitanT (26.2)-Si, miviRebT: maSasadame, siCqaris

ara

magaliTad,

didi

pirveli

N  k  3 . siCqaris SemTxvevaSi, xarisxis,

TviTmfrinavis

aramed

siCqaris

simZlavre

kubis.

2-jer

imas

gasazrdelad

proporciulia niSnavs, misi

rom

Zravebis

simZlavre 8-jer unda gaizardos. amitomaa gaZnelebuli siCqaris gazrda. (26.2)-dan F



e.i. mudmivi simZlavris SemTxvevaSi, Zravis mier ganviTarebuli wevis Zala siCqaris ukuproporciulia _ rac naklebia siCqare, miT metia wevis Zala. amitomaa, rom avtomanqanis mZRoli, aRmarTze moZraobisas, roca meti wevis Zalaa saWiro, Zravas mcire siCqareze gadarTavs xolme. (26.1) formula gamosadegia im SemTxvevaSi, roca drois tol SualedebSi erTnairi muSaoba sruldeba. Tu muSaoba araTanabrad sruldeba, e.i. Tu simZlavre drois mixedviT icvleba, unda SemoviRoT saSualo simZlavre: A . t simZlavris myisi mniSvneloba miiReba zRvarze gadasvliT: NsaS 

A dA .  t dt magram (24.2)-is Tanaxmad A  FS  S , amitom N  lim

t  0





  dS  FS    F   . (26.4) dt maSasadame, m y i s i s im Z lav re t ol ia Z al is a d a s i Cq a ris v e q N  FS 

t o r e b is s k al a ru l i n am r av l is .

§28. energia. energiis mudmivobis kanoni viciT, rom moZraoba aris materiis pirveladi, Tandayolili Tviseba _ ar arsebobs materia moZraobis gareSe. materiis moZraobis sxvadasxva formebs (meqanikurs, atomur-molekulurs anu siTburs da a.S.) Sesabamis pirobebSi SeuZliaT urTiergardaqmna. amasTan, materiis moZraobis erTi forma

yovelTvis

formaSi.

aman

m k ac r ad

warmoqmna

g a rk v e u li

SesaZlebloba,

r ao de nob iT materiis

gadadis

moZraobis

meore

sxvadasxva

forma gaizomos erTi sazomiT, rac daedo safuZvlad energiis cnebis SemoRebas. e ne r g ia

e w o de b a

s k al a ru l

s i di de s ,

rom e l ic

ar is

m at e r i is m oZ ra ob i s s xv ad as xv a f o rm i s s ae r T o s a zom i. moZraobis sxvadasxva formebi xasiaTdeba energiis Sesabamisi saxeebiT: meqanikuri, Sinagani, eleqtromagnituri da sxva. sistemis meqanikuri energia icvleba muSaobis Sesrulebis procesSi. amasTan

e ne rg ii s

cv li le b a

Se s ru l e b u l i

m u S a ob is

t ol i a .

davuSvaT, gare F Zalis mier Sesrulebuli A muSaobis Sedegad sistemis meqanikuri

energia

Seicvala

E2 -mde.

E1 -dan

maSin

energiis

cvlileba

E  E2  E1 , zemoTqmulis Tanaxmad, Sesrulebuli A muSaobis tolia: (28.1) E  E2  E1  A . (28.1) gviCvenebs, rom Tu gare Zalebis mier Sesrulebuli muSaoba dadebiTia

A  0 ,

maSin E2  E1 anu sistemis energia izrdeba da piriqiT _

Tu gare Zalebis muSaoba uaryofiTia

A  0 ,

sistemis energia mcirdeba. am

SemTxvevaSi, dadebiTia s is t e m i s m i e r S e s r u le b u li m u S aob a , anu im Zalebis mier Sesrulebuli muSaoba, romliTac sistema moqmedebs gareSe sxeulebze. maSasadame, sistemas dadebiTi muSaobis Sesruleba SeuZlia mxolod sakuTari energiis Semcirebis xarjze. a m i tom , e ne r g i as x S ir a d g a nm ar t av e n r o gor c f iz ik u r s id i de s , rom e li c a x as iaT e b s s xe u l is a n s xe u l T a s is t e m i s m ie r m u S aob is S e s ru le b is u n ar s *. sxeuls aqvs energia niSnavs, rom mas SeuZlia muSaobis Sesruleba**. (28.1) formulidan gamomdinareobs,

rom

energia imave erTeulebiT

izomeba, riTac muSaoba. *

energia berZnulad niSnavs moqmedebas, moRvaweobas. termini `energia~ SemoiRo 1860 wels kelvinma, nacvlad terminisa `bunebis Zala~. ** unda aRiniSnos, rom energiis cnebis fizikuri Sinaarsi da misi sayofacxovrebo gageba erTnairia: cxovrebaSic energiuls uwodeben muSaobis unaris mqone adamians. 76

vTqvaT,

sistema

izolirebuli

anu

F  0  .

Caketilia

maSin

A  0, E  0 da E  const . e.i. i zo li re b u li s is te m i s m e q an ik u ri e n e r g i a m u dm iv i s i di de a . am daskvnas ewodeba meqanikuri energiis mudmivobis kanoni. es kanoni samarTliania im SemTxvevaSi, roca sistemaSi moqmedebs mxolod konservatuli (gravitaciuli da drekadi) Zalebi. sistemis sxeulebs

Soris

arakonservatuli

Zalebis

(magaliTad,

xaxunis

Zalis)

moqmedebis SemTxvevaSi kanoni irRveva, vinaidan meqanikuri energiis nawili gadadis siTbur energiaSi. es SezRudva ar exeba s ru l i e ne r g ii s mudmivobis kanons, romlis kerZo SemTxvevasac warmoadgens meqanikuri energiis mudmivobis kanoni da romelic ase gamoiTqmis: i zo li re b u li s is t e m i s s ru li e n e r gi a m u dm iv i s id id e a . s i s t e m a Si m im di na re pr oc e s e b i s Se de g ad e ne r g ii s e rT i s ax e S e i Z l e b a g ai z ar dos , m e or e Se m ci rd e s , m a gr am i s e , ro m m aT i j am i a nu s is t e m i s s ru l i e ne rg i a d arC e s m u dm iv i. energiis mudmivobis kanoni bunebis fundamenturi, uzogadesi kanonia. igi sruldeba rogorc makro, ise mikrosamyaroSi, yovelgvari gamonaklisis gareSe. yoveli axlad aRmoCenili faqti Tu movlena mxolod adasturebs am kanons. dabolos, SevniSnoT, rom (28.1) toloba gamosaxavs meqanikuri energiis mudmivobis kanons araizolirebuli sistemisaTvis.

§29. kinetikuri da potenciuri energia

arsebobs meqanikuri energiis ori saxe: kinetikuri

W 

da potenciu-

ri U  . sruli meqanikuri energia maTi jamis tolia: E  W U . wina paragrafSi gamoTqmul meqanikuri energiis mudmivobis kanonSi,

cxadia, igulisxmeba sruli meqanikuri energia da kanoni Semdegnairad unda gavigoT: izolirebul sistemaSi kinetikuri energiis zrda iwvevs potenciuri energiis Semcirebas da piriqiT, magram ise, rom maTi jami anu sruli meqanikuri energia rCeba mudmivi. ganvixiloT es energiebi cal-calke.

k ine tik u ri e ne r gi a. k ine tik u ri e n e r gi a e w o de b a e n e r gi as , r o m e l i c s xe u ls g aa C ni a m oZ r a ob is Se d e g a d . Tu sxeuli ar moZraobs misi kinetikuri energia nulis tolia. miviRoT kinetikuri energiis gamosaTvleli formula. ganvixiloT umartivesi SemTxveva, rodesac sxeuli moZraobs wrfeze da masze moqmedebs mudmivi F Zala, romelic am wrfis gaswvrivaa mimarTuli. vTqvaT, am Zalis moqmedebiT m masis sxeulma gaiara S manZili da misi siCqare Seicvala 1 -dan  2 –mde.

am SemTxvevaSi Zalis mier Sesrulebuli

muSaoba A  F S .

magram F  ma , xolo S 

 22

 12 2a

(29.1)

(vinaidan F  const , moZraoba TanabaraCqare-

bulia). SevitanoT (29.1)-Si:

A  F  S  ma  miviReT,

rom

sxeulze

 22  12 2a



moqmedi

m 22 m12  . 2 2 Zalis

(29.2)

muSaoba

m 2 2

sididis

cvlilebis tolia. (28.1) formulis Tanaxmad, es sidide unda iyos energia da vinaidan roca   0 es energiac nulis tolia, bunebrivia CavTvaloT, rom igi kinetikuri energiaa. e.i.

W

m 2

. (29.3) 2 (29.2)-is Tanaxmad, s x e u l ze m oq m e di Z al is m u S a ob a s xe u li s k ine ti kuri

e ne r g ii s

cv li le b i s

t oli a. am debulebas ewodeba T e or e m a

k ine tik u r i e ne r gi is Se s a xe b . Tu sxeulze moqmedi Zalis muSaoba dadebiTia (Zalis mimarTuleba emTxveva moZraobis mimarTulebas), maSin (29.2)-is Tanaxmad

m 22 m12  2 2

anu sxeulis kinetikuri energia izrdeba da piriqiT,

roca A  0 , sxeulis kinetikuri energia mcirdeba. vTqvaT 1  0 da  2   . maSin A 

masis

sxeulis

kinetikuri

m 2 . maSasadame,  siCqariT moZravi 2

energia

muSaobis

tolia,

romelsac

asrulebs sxeulze moqmedi Zala misTvis am siCqaris misaniWeblad. igive muSaoba sruldeba  siCqariT moZravi sxeulis gaCerebisas.

(29.2) formula, romelic Cven miviReT kerZo SemTxvevisaTvis, zogadia da marTebulia im SemTxvevaSic, roca Zala aris cvladi, xolo traeqtoria _ mrudi.

 F cvladi Zalis moqmedebiT sxeulis siCqare Seicvala    1 -dan  2 -mde. mcire dt droSi siCqaris cvlileba iqneba d . Zalis mier Sesrulebuli marTlac, davuSvaT, rom

elementaruli muSaoba (24.4)-is Tanaxmad



magram F  ma  m



  d , xolo dS  dt , dt

amitom





  dA  F  dS .

    d   dA  m  dt   m  d   md .  dt 

(29.4)

 F Zala siCqaris d   sididiT Sesacvlelad. sruli muSaoba, romelic sruldeba siCqaris 1 -dan  2 -mde (29.4) formula gamosaxavs im muSaobas, romelsac asrulebs cvladi

Sesacvlelad miiReba am formulis integrebiT: 2

2

1

m 2 A   dA   md  2 

m 22 m12 .   2 2

rac emTxveva (29.2)-s.

p o te n ciu ri e ne r gi a. po te n ciu ri e ne r gi a e w o de b a e ne r gi as , r o m e l i c s xe u ls g a a Cn i a u rT ie r T q m e de b i s Se de g ad. ig i g an is a z R v r e b a s xe u le b i s an e r T i d a i m av e s xe u l is n aw ile b is u r T i e r T m de b are ob iT . maSasadame, warmoadgens mdebareobis, e.i. koordinatebis funqcias:L U  U x, y, z  . aseTi energia gaaCnia, magaliTad, dedamiwis zedapiridan raime simaRleze atanil sxeuls, deformirebul sxeuls da sxv. miviRoT potenciuri energiis gamosaTvleli formula. unda aRiniSnos, rom am formulis saxe damokidebulia im Zalebis bunebaze, romliTac sxeulebi urTierTqmedeben. gamovTvaloT jer energia sxeulisa, romelzec simZimis

Zala

moqmedebs. viciT (§25), rom simZimis Zalis mier Sesrulebuli muSaoba

masis sxeulis h1 simaRlidan h2 -mde daSvebisas gzis formaze damokidebuli ar aris da gamoiTvleba formuliT

A  Ph1  h2   mgh1  h2  . am formulas SeiZleba mivceT sxva saxec: A  mgh2  mgh1  . 79

(29.5)

miviReT, rom s im Z im is Z a lis m u S aob a mgh s ididis cv lile b is toli a Se b ru ne b u li ni SniT . (28.1) formulis Tanaxmad es sidide unda iyos energia da vinaidan igi ganisazRvreba h -iT anu dedamiwisa da m masis sxeulis urTierTmdebareobiT, unda CavTvaloT, rom mgh aris potenciuri energia Tu davuSvebT, rom h2

U  mgh aris nulovani done

(29.6) h2  0 *, xolo nulovani

donidan aTvlil simaRles aRvniSnavT h -iT, e.i. h1  h , maSin A  mgh . maSasadame, nulovani donidan h simaRleze atanili m masis sxeulis potenciuri energia tolia im muSaobisa, romelsac asrulebs simZimis Zala am sxeulis nulovan donemde daSvebisas. aseTive muSaobas asruleben gare Zalebi m masis sxeulis h simaRleze atanisas. minusi

niSani

potenciuri

energiis

 F

cvlilebis win (29.5) formulaSi imis maCvenebelia, rom simZimis Zalis mier dadebiTi

 F

muSaobis Sesrulebisas es energia mcirdeba da piriqiT, rodesac simZimis Zala asrulebs

uaryofiT

muSaobas

(sxeuli

zeviTaa asrolili), sxeulis potenciuri

energia izrdeba. miviRoT axla drekadad deformirebuli sxeulis potenciuri energiis for-

x sur. 29.1

x 2 x1

mula. davuSvaT, drekadi zambara, romlis sigrZe aradeformirebul mdgomareobaSi

l0 -ia, marcxena boloTi damagrebulia, xolo marjvena boloze

mimagrebulia burTula (sur. 29.1). Tu zambaravs gavWimavT (e.i. burTulas ga davaadgilebT marjvniT) raime x sididiT, masSi aRiZvreba drekadobis F Zala, romelic cdilobs daabrunos zambara pirvandel mdgomareobaSi. am ZaliT zambara imoqmedebs burTulaze. vinaidan x gansazRvravs zambaris deformacias, drekadi Zalis sidide, hukis kanonis Tanaxmad, ganisazRvreba formuliT F  kx , sadac k zambaris sixistea (§16). Tu zambaras gavanTavisuflebT, misi marjvena bolo da, maSasadame, burTulac daiwyebs moZrao* Cveulebriv, nulovan doned iReben dedamiwis zedapirs.

bas marcxniv da drekadobis Zala Seasrulebs muSaobas. gamoviTvaloT es muSaoba. vTqvaT, zambara gaWimuli iyo x1 sididiT da ganTavisuflebis Semdeg raRac droSi igi Semcirda

x2 sididemde. sxeulis gadaadgileba toli

iqneba ( x1  x2 )-is. vinaidan Zalis moqmedebis mimarTuleba emTxveva gadaadgilebis mimarTulebas, elementaruli muSaoba, romelsac asrulebs drekadobis Zala deformaciis dx sididiT Semcirebisas, tolia: dA  F  dx  kxdx .

sruli

muSaoba,

romelsac

asrulebs

(29.6) drekadobis

Zala

zambaris

deformaciis x1 -dan x2 -mde Semcirebisas miiReba (29.6)-is integrebiT: x1

A   dA  k  xdx  k x2

x2 2



 kx2 kx2  kx12 kx22    2  1  . 2 2 2   2 kx2

e.i. Sesrulebuli muSaoba tolia

(29.7)

sididis cvlilebisa Sebrunebu-

li niSniT. simZimis Zalis muSaobis ganxilvisas Catarebuli msjelobis gameorebiT davaskvniT, rom

kx2 2

aris drekadad deformirebuli zambaris

potenciuri energia:

U amrigad,

d re k adob is

kx2

Z alis

m ie r

Se s ru leb u li

m u S aob ac

toli a pote nciu ri e ne rgiis cv lile b is a Se b ru ne b u li niSniT . mocemul

paragrafSi

miRebuli

Sedegebi

saSualebas

gvaZlevs,

martivad

gamoviyvanoT sruli meqanikuri energiis mudmivobis kanoni. davuSvaT, mocemuli gvaqvs Caketili sistema, romelSic moqmedeben mxolod konservatuli Zalebi. konservatuli Zalebis

muSaoba

tolia

potenciuri

energiis

cvlilebisa

Sebrunebuli

niSniT:

A  U 2  U1  . vinaidan sistemaSi sxva Zalebi ar moqmedeben, igive muSaoba tolia kinetikuri

energiis

cvlilebisa:

A  W2  W1 .

formulebis

Sedareba

gvaZlevs:

 U 2  U1   W2  W1 , an W1  U1  W2  U 2 , saidanac W  U  E  const .

bolos, mivaqcioT yuradReba erT garemoebas. mudmivobis Tvisebis gamo energia iZens ZiriTadi fizikuri cnebis mniSvnelobas. amitom, mizanSewonili xdeba energia ganisazRvros ara muSaobis saSualebiT (`energia aris muSaobis Sesrulebis unari~), aramed _ piriqiT. §24-is dasawyisSi 81

naTqvami iyo, rom muSaobis cneba asaxavs moZraobis erTi formis meore formad

gardaqmnis

process.

moZraobis

sxvadasxva

formebi

xasiaTdeba

energiis Sesabamisi saxeebiT. amitom SeiZleba iTqvas, rom muSaoba asaxavs energiis erTi formis meored gardaqmnis process. magram muSaobis Sedegad adgili aqvs energiis ara marto gardaqmnas, aramed gadacemasac. magaliTad, sxeulebis

Sejaxebisas

sruldeba

muSaoba,

romlis

drosac

meqanikuri

energiis nawili gardaiqmneba siTbur energiad, nawili ki gadaecema erTi sxeulidan

meores.

zemoTqmulidan

gamomdinare,

m u S a ob a

Se i Z le b a

g a n im a r t os ro g orc e ne r gi is g ar d aq m nis a d a g a d ace m is for m a.

§30. kavSiri Zalasa da potenciur energias Soris sxeulTa urTierTqmedeba SeiZleba aiweros an Zalis saSualebiT, an potenciuri energiis, rogorc urTierTmoqmed sxeulTa koordinatebis funqciis saSualebiT. klasikur meqanikaSi gamoiyeneba orive xerxi. pirveli xerxi ramdenadme ufro zogadia, vinaidan igi gamoiyeneba iseTi Zalis SemTxvevaSic, romlisTvisac ar SeiZleba potenciuri energiis SemoReba (magaliTad,

xaxunis Zala). meore xerxi gamoiyeneba mxolod konservatuli

ZalebisTvis. kvanturi meqanika Seiswavlis mikrosamyaroSi mimdinare movlenebs, sadac xaxunis Zalebi ar aris. amitom igi nawilakTa urTierTqmedebis aRwerisaTvis iyenebs mxolod da mxolod meore xerxs. kvanturi meqanikis moZraobis gantolebebSi Zala ar Sedis, Sedis mxolod urTierTmoqmedi nawilakebis potenciuri energia.

wina paragrafSi cnobili Zalebis

P  mg,

F  kx saSualebiT gamo-

viTvaleT gravitaciul velSi myofi da drekadad deformirebuli sxeulebis potenciuri energiebi (Sesabamisad, U  mgh da U 

kx2 ). gadavwyvitoT 2

axla Sebrunebuli amocana _ potenciuri energiis saSualebiT gamoviTvaloT Zala. vTqvaT,

konservatuli

 usasrulod mcire dr

 F

Zalis

moqmedebiT sxeulma Seasrula  gadaadgileba. maSin F Zalis mier Sesrulebuli

elementaruli muSaoba

 

  dA  Fdr  Fdr cos   Fr dr ,   sadac Fr  F cos  aris F Zalis gegmili r -is mimarTulebaze.

es muSaoba potenciuri energiis usasrulo mcire

(30.1)

cvlilebis

tolia Sebrunebuli niSniT: dA  dU 82

(30.2)

(30.1) da (30.2) tolobebis Sedareba gvaZlevs:

Fr  dr  dU saidanac dU dr mimarTulebaze Fr  

e.i.

Zalis

gegmili

dr -is

(30.3) udris

potenciuri

energiis

warmoebuls imave mimarTulebiT, Sebrunebuli niSniT. rogorc wina paragrafSi aRvniSneT, potenciuri energia warmoadgens koordinatebis funqcias:

U  U x, y, z  .

s k a la ru li

U x, y, z 

f u nq c i is

w ar m o e b u ls im m im arT u le b i T , s a iT k e n ac e s w a rm oe b u li m aq s im a l u r i a, am fu nq ci is g r ad ie nt i e w ode b a. aRiniSneba grad U -Ti. funqciis gradienti veqtoria, romelic mimarTulia am funqciis uswrafesi zrdis mxares. SeiZleba Cveneba, rom sazogadod, Zala tolia potenciuri energiis gradientisa Sebrunebuli niSniT:  F   grad U

grad U -s warmoebulebi*

mdgenelebi

RerZebze

U U U , , . x y z

amitom

(30.4)

aris (30.4)-is

potenciuri

energiis

dagegmilebiT

kerZo

koordinatTa

RerZebze miviRebT:

U U U , Fy   , Fz   . (30.5) x y z ganvixiloT konkretuli magaliTi _ gamoviTvaloT deformirebul Fx  

zambaraSi aRZruli drekadobis Zala F . viciT rom Tu x RerZi mimarTulia zambaris RerZis gaswvriv (sur. 29.1), maSin drekadad deformirebuli zambaris potenciuri energia gamoiTvleba formuliT U  vaSi U aris mxolod x -is funqcia. amitom

Fx  F  

kx2 . mocemul SemTxve2

U dU  gveqneba: x x

dU d  kx2    kx    dx dx  2 

kerZo warmoebuli SemoRebulia mravali cvladis funqciisaTvis da aRniSnavs am funqciis warmoebuls erTi cvladis mimarT im daSvebiT, rom danarCeni cvladebis mniSvnelobebi fiqsirebulia, anu am gawarmoebis dros isini ar icvleba. kerZo warmoebuli d simbolos nacvlad aRiniSneba  simboloTi. Tu f aris mxolod x -is funqcia, maSin

f df  x dx 83

§31. nawilakebis dajaxeba dajaxeba

aris

nawilakTa

urTierTqmedebis

umniSvnelovanesi

SemTxveva. misi kanonzomierebebis dadgena advildeba imiT, rom amisaTvis ar aris saWiro niutonis moZraobis diferencialuri gantolebebi da sakmarisia CvenTvis cnobili impulsisa da energiis mudmivobis kanonebis gamoyeneba. rogorc viciT, es kanonebi bunebis uzogadesi kanonebia, romlebic upirobod sruldeba rogorc makro, ise mikrosamyaroSi. amitom nawilakebad SeiZleba vigulisxmoT rogorc makrosxeulebi (magaliTad, biliardis an plastelinis birTvebi), ise mikronawilakebi (elementaruli nawilakebi, atomebi, molekulebi). mxedvelobaSi unda viqonioT is garemoeba, rom nawilakTa Sejaxeba yovelTvis ar daiyvaneba mxolod dartymamde, rogorc es biliardis birTvebis SemTxvevaSia. magaliTad, erTi niSniT damuxtuli mikronawilakebi, eleqtruli velis moqmedebiT, uSualo Sexebis gareSe, icvlian siCqaris mimarTulebas, magram aseT processac dajaxeba ewodeba. ganvixiloT

nawilakebis

centraluri

dajaxeba,

e.i.

iseTi,

roca

nawilakebis siCqareebi mimarTulia maTi centrebis SemaerTebeli wrfis gaswvriv. dajaxeba SeiZleba iyos drekadi an aradrekadi. simartivisaTvis ganvixiloT ori ukiduresi SemTxveva _ roca dajaxeba aris absoluturad drekadi, an absoluturad aradrekadi. absoluturad drekadi dajaxeba ewodeba iseT dajaxebas, romlis Semdegac nawilakebis Sinagani mdgomareoba mTlianad aRdgeba da piriqiT, absoluturad aradrekadi dajaxeba ewodeba iseT dajaxebas, romlis Semdegac dajaxebiT gamowveuli nawilakebis Secvlili Sinagani mdgomareobis aRdgena srulebiT ar xdeba. ganvixiloT jer a b s olu tu r a d a r ad re k ad i d a j a xe b a. aseTia, magaliTad, plastelinisagan damzadebuli birTvebis dajaxeba. ganmartebis Tanaxmad, am dros srulebiT ar xdeba dajaxebiT gamowveuli Secvlili Sinagani mdgomareobis aRdgena anu deformaciis aRdgena. amitom, dajaxebis Sedegad, es bir Tvebi erTdeba. gamoviTvaloT SeerTebuli birTvebis siCqare u . amisaTvis gamoviyenoT impulsis mudmivobis kanoni. amasTan, vinaidan siCqareebi mimarTulia erTi wrfis gaswvriv, es kanoni SeiZleba davweroT skalarulad. burTulebis masebi aRvniSnoT m1 -iT da m2 –iT, xolo siCqareebi Sejaxebamde _ Sesabamisad, 1 -iT da  2 -iT (sur. 31.1). maSin impulsis mudmivobis kanonis Tanaxmad: 84

m11  m2 2  m1  m2 u ,

1

saidanac

2

m11  m2 2 (31.1) m1  m2 vTqvaT, birTvebi moZraoben u

sur.

31.1

Semxvedri

m11  m2 2

mimarTulebiT.

maSin

u  0 . e.i. Tu toli impulsis mqone birTvebi moZraoben

Semxvedri mimarTulebiT, dajaxebis Sedegad miRebuli SeerTebuli birTvebis moZraoba wydeba. kerZo SemTxvevaSi, roca birTvebis masebi tolia, es piroba sruldeba, roca siCqareebia toli (marTlac, Tu m1  m2  m maSin u

1   2 2

; roca 1  2 , maSin u  0 ).

vTqvaT m2  m1 , maSin (31.1)-dan (Tu tolobis marjvena nawilis yvela wevrs gavyofT m2 -ze da

m1 -s ugulebelvyofT), gveqneba m2

m1 1   2 m2 u  2 m1 1 m2 aseT SemTxvevas aqvs adgili, magaliTad, moZravi vagonis kedelze plastelinis birTvis dajaxebisas _ birTvi miekvreba kedels da imoZravebs masTan erTad. absoluturad

aradrekadi

Sexlis

dros

meqanikuri

energiis

mudmivobis kanoni ar sruldeba, vinaidan birTvebis kinetikuri energiis nawili ixarjeba birTvebis deformaciaze da gardaiqmneba maT Sinagan energiad. es energia kinetikuri energiebis sxvaobis tolia (birTvebis urTierTqmedebis

potenciuri

energia

nulis

tolia,

vinaidan

arc

dajaxebamde da arc Semdeg isini ar urTierTqmedeben):

E a b s olu tu r a d

m112 m2 22  m1  m2 u 2       2 2 2  

dre k a di

d a j ax e b a .

aseT

Sejaxebas

waagavs,

magaliTad, biliardis birTvebis Sejaxeba. ganmartebis Tanaxmad, am dros xdeba

dajaxebis

Sedegad

Secvlili

Sinagani

mdgomareobis

mTliani

aRdgena: dajaxebis Sedegad birTvebi deformirdebian, magram deformaciis Sedegad

aRZruli

drekadi

Zalebis 85

moqmedebiT

isini

myisve

aRidgenen

pirvandel formas da deformaciaze daxarjuli energia isev uklebliv gardaiqmneba

kinetikur

energiad.

maSasadame,

SemTxvevaSi

meqanikuri

energiis mudmivobis kanoni sruldeba. formis aRdgenisas xdeba birTvebis daSoreba. amasTan (31.1) suraTze gamosaxul SemTxvevaSi

1   2  ,

pirveli

birTvi moqmedebs meoreze moZraobis mimarTulebiT da zrdis mis siCqares, meore ki moqmedebs pirvelze moZraobis sawinaaRmdego mimarTulebiT da amcirebs mis siCqares. pirveli da meore birTvebis Secvlili siCqareebi   aRvniSnoT Sesabamisad u1 -iT da u 2 -iT. am siCqareebis gasagebad davweroT impulsisa da energiis mudmivobis kanonebi.

m11  m2 2  m1u1  m2u 2 .

(31.2)

m112 m2 22 m1u12 m2 u 22 .    2 2 2 2

(31.3)

m1 1  u1   m2 u 2   2  .

(31.4)









m1 12  u12  m2 u 22   22 .

gavyoT

(31.5)

toloba

(31.4)-ze

(31.5) gaviTvaliswinoT,

rom

12  u12  1  u1 1  u1  :

1  u1   2  u 2 . aqedan

(31.6)

u 2  1  u1   2 .

CavsvaT u 2 -s es mniSvneloba (31.2) gantolebaSi:

m11  m2 2  m1u1  m21  m2u1  m2 2 , saidanac

u1 

m1  m2 1  2m2 2 .

(31.7)

m1  m2

Tu (31.6)-dan amovxsniT u1 -s da SevitanT (31.2)-Si, u 2 -sTvis miviRebT:

u2 

m2  m1  2  2m11 m1  m2

(31.8)

miRebuli formulebi gamosadegia im SemTxvevaSic, roca birTvebi moZraoben erTmaneTis Sesaxvedrad, oRond 1

da  2

niSnebi eqnebaT. ganvixiloT kerZo SemTxvevebi: 1. birTvebis masebi tolia: m1  m2  m , maSin 86

siCqareebs sawinaaRmdego

u1   2

u 2  1

e.i. am SemTxvevaSi adgili aqvs siCqareTa gacvlas birTvebs Soris. kerZod, Tu dajaxebamde meore birTvi uZravi iyo, e.i.  2  0 , maSin

u1  0

da u 2  1

anu dajaxebis Semdeg pirveli birTvi gaCerdeba, meore ki amoZravdeba im siCqariT, romelic pirvels hqonda dajaxebamde. 2. meore birTvis masa gacilebiT metia pirvelze: m2  m1 , maSin tolobebis marjvena nawilis yvela wevris m2 -ze gayofiT miviRebT:

u1  2 2  1 da u2  2 e.i. dajaxebis Semdeg didi masis birTvis siCqare ar icvleba. kerZod, Tu igi Tavidan uZravi iyo (vTqvaT, birTvi ejaxeba uZrav kedels), e.i.  2  0 , maSin u 2  0 da u1  1 anu dajaxebis Semdeg didi masis birTvi uZravi darCeba, mcire masis birTvi ki ukuigdeba misgan imave sididis siCqariT. es Sedegi gamoiyeneba airis kinetikuri Teoriis ZiriTadi gantolebis gamoyvanisas. vTqvaT, birTvi absoluturad drekadad ejaxeba uZrav kedels iribad, ise, rom



n

) 

)

 1

 t

sur. 31.2

birTvis



 n

 u

moZraobis

mimarTuleba

kedlis

normalTan adgens  kuTxes (sur. 31.2). dav SaloT birTvis 1 siCqare kedlis marTo  bul  n da mxeb  t mdgenelebad. mxebi mdgeneli ar icvleba, xolo normaluri mdgeneli,

zemoTqmulis Tanaxmad, icvlis    mimarTulebas, e.i. xdeba - n -is toli.  t da -  n veqtorebis SekrebiT  miviRebT birTvis u siCqares dajaxebis Semdeg. naxazidan Cans, rom igi

 sididiT 1 -is tolia da adgens normalTan igive 

kuTxes. e.i. kedelze

iribi dajaxebis dros siCqaris sidide ar icvleba, mimarTuleba ki icvleba ise, rom dacemis kuTxe arekvlis kuTxis tolia.

VI Tavi

m y a r i s xe u l i s d i n a mi k a §32. myari sxeulis gadataniTi moZraobis ZiriTadi gantoleba myaris sxeulis gadataniTi moZraobisas misi yvela wertilis siCqare da aCqareba erTnairia. amitom myari sxeulis gadataniTi moZraobis dasaxasiaTeblad sakmarisia misi erT-erTi wertilis moZraobis Seswavla.

marTlac, davuSvaT, mocemuli gvaqvs myari sxeuli, romlis masa aris  da moZraobs a aCqarebiT. davyoT azrobrivad es sxeuli mcire m

elementebad, TiToeuli elementisTvis davweroT niutonis meore kanoni da Semdeg SevkriboT. i -uri elementisaTvis: n    mi a  Fi   Fki ,

(32.1)

k 1

 sadac Fi aris mi elementze moqmedi gareSe Zala, xolo

Zalebis

tolqmedi,

elementebi

anu

moqmedeben

Zalebis

mocemul

tolqmedi,

elementze.



 Fki

_ Sinagani

romliTac

danarCeni

k 1

elementebis

ricxvia.

avjamoT (32.1) toloba i indeqsiT 1-dan n –mde ( k  i ): n  n   m a  F   i  i  n

i 1

 F  ki . n

(32.2)

k 1

ganvixiloT TiToeuli wevri: n

  n   m a  i  a  mi  ma , i 1

i 1

sadac m   mi aris mTeli sxeulis masa. i 1

 Fi  F

_ aris gareSe Zalebis

i 1

veqtoruli jami da mas ewodeba g a r e S e Z ale b is m T av ar i v e q t or i. n n      Fki  0 niutonis mesame kanonis Tanaxmad ( Fik  Fki ). amitom (32.2) miiRebs i 1

k 1

saxes:

  ma  F .

(32.3)

miRebuli Sedegi gviCvenebs, rom myari sxeulis gadataniTi moZraobis ganxilva marTlac SeiZleba Seicvalos misi erTi wertilis moZraobis ganxilviT, oRond unda davuSvaT, rom am wertilSi Tavmoyrilia mTeli sxeulis masa da masze moqmedebs gareSe Zalebis mTavari veqtori. 88

myari sxeulis ufro rTuli, aragadataniTi moZraobis dros misi sxvadasxva wertilis

siCqare

da aCqareba sxvadasxvaa,

magram

SeiZleba

sxeuli davyoT imdenad mcire mi elementebad, rom TiToeulis farglebSi  es sidideebi CaiTvalos mudmivad. i –uri elementis aCqareba aRvniSnoT ai iT, maSin wina SemTxvevaSi Catarebuli moqmedebebis zusti gameorebiT miviRebT:

   m a  F  i i . n

(32.4)

i 1

(32.4)

gantolebis uSualo dayvana (32.3) tipis gantolebaze ar  SeiZleba, vinaidan ai sxvadasxva elementisaTvis sxvadasxvaa. SevcvaloT am 2 2  d r d r  i a gantolebaSi i misi toli -iT ( ai  2i 2 dt dt

 sadac ri aris mi elementis

radius-veqtori) da gaviTvaliswinoT, rom (vinaidan mi  const ) n

 mi i 1

 d 2 ri dt

 d2 d2    m r  i i 2 dt 2 i 1 dt n





 mi ri ,* i 1

gveqneba:

d2 dt 2



 mi ri  F

(32.5)

i 1

SemovitanoT masaTa centris cneba. nivTier wertilTa sistemis masaTa centri an inerciis centri** ewodeba iseT c wertils, romlis radiusveqtoric gamoiTvleba tolobiT: 

 rc 

 m r

i i

i 1

m n

sadac m mTeli sistemis masaa. (32.6)-dan

(32.6)



i i

 m r  mr i 1

Tu am mniSvnelobas SevitanT (32.5)-Si, miviRebT:   d 2 rc d2   mr  m  F , c dt 2 dt 2 an

  mac  F ,

(32.7)

gamoyenebulia is faqti, rom jamis warmoebuli tolia SesakrebTa warmoebulebis jamisa.

** sistemis masaTa centri Tanxvdeba mis simZimis centrs, Tu davuSvebT, rom simZimis Zalis

g aCqarebis mniSvneloba sistemis yvela wertilSi erTnairia, anu Tu sistema

moTavsebulia erTgvarovan simZimis velSi. aseTi daSveba samarTliania,

Tu sistemis

xazovani zoma gacilebiT naklebia dedamiwis radiusze, rac praqtikulad yovelTvis sruldeba. 89

2 sadac ac  d rc - masaTa centris aCqarebaa. dt 2

(32.7)

gantolebidan

gamomdinareobs,

rom

masaTa

centri

moZraobs

rogorc nivTieri wertili, romlis masa mTeli sistemis masis tolia da romelzec moqmedebs gareSe Zalebis mTavari veqtori. am daskvnas ewodeba Teorema masaTa centris moZraobis Sesaxeb. myari sxeuli SeiZleba ganvixiloT rogorc urTierTmoqmed nivTier  wertilTa sistema. myari sxeulis gadataniTi moZraobis dros sxeulis a  aCqareba, cxadia, misi masaTa centris ac aCqarebis tolia. amitom (32.7) gantolebas

xSirad

uwodeben

myari

sxeulis

gadataniTi

moZraobis

dinamikis ZiriTad gantolebas.

§33. myari sxeulis brunviTi moZraobis dinamikis damaxasiaTebeli sidideebi myari sxeulis brunviTi moZraobis dinamikis ganxilvisas, garda im sidideebisa, romlebic SemoRebuli iyo brunviTi moZraobis kinematikis dasaxasiaTeblad (kuTxuri siCqare, aCqareba), saWiroa damatebiTi sidideebis SemoReba. esenia: Zalis momenti, inerciis momenti, impulsis momenti da sxv. gavecnoT am sidideebs. Z a lis romelime

m om e nt i . davuSvaT, myari sxeulis  wertilze moqmedebs Zala A F

(sur. 33.1).

am Zalis momenti O wertilis mimarT

ewodeba O wertilidan A wertilamde gavlebuli  r radius-veqtoris veqtorul namravls Zalaze.  aRiniSneba M -iT:

 

   M  r F .

(33.1)

veqtoruli

 F

  r

o l

sur. 33.1

namravlis wesis Tanaxmad, Zalis momenti veqtoria,   romelic marTobia r da F veqtorebze gavlebuli sibrtyisa da aqvs im  burRis gadaadgilebis mimarTuleba, romlis saxelursac vabrunebT r  veqtoridan F -sken. Cvens suraTze Zalis momenti mimarTulia suraTis sibrtyis marTobulad Cvensken. sididiT 90

sadac 

aris kuTxe

 r

M  r  F  sin  ,  F veqtorebs Soris. suraTidan Cans, rom

r  sin   l ,

M  F l

amitom

(33.2)

l aris O wertilidan Zalis mimarTulebaze daSvebuli perpendikularis

sigrZe. mas O wertilis mimarT Zalis mxari ewodeba. maSasadame, Z a li s m o m e n t i s i di d iT Z a lis a d a m xri s n am r av l is t ol i a. SI sistemaSi Zalis momentis erTeulia nm. Cven RerZis

ganvixileT

mimarT

Zalis

momenti

ganisazRvreba

wertilis

rogorc

mimarT.

RerZis

Zalis

nebismieri

momenti

wertilis

mimarT Zalis momentis mdgeneli amave RerZze. im kerZo SemTxvevaSi, roca gansaxilveli oz RerZi 0  wertilze da F Zalaze gavlebuli sibrtyis mar-

z  M  F

marT mdebareobs oz RerZze da Tanxvdeba Zalis moments amave RerZis mimarT.

 r

Tobia (sur. 33.2), Zalis momenti 0 wertilis mi-

Zalis momentis msgavsad ganisazRvreba ne-

sur. 33.2

bismieri veqtoris momenti. Z a lT a

w y v i li s

m om e n ti . ZalTa wyvili anu wyvilZala ewodeba

sididiT toli da mimarTulebiT sawinaaRmdego ori Zalis erTobliobas, romlebic erT wrfeze ar mdebareoben. umokles manZils ZalTa qmedebis mimarTulebebs Soris ewodeba wyvilZalis l mxari (sur. 33.3). wyvilZalis momenti raime 0 wertilis mimarT tolia calkeuli Zalebis momentebis    veqtoruli jamisa: M  M1  M 2 sadac        M1  r1  F da M 2  r2  F vinaidan M 1 -ic )   F  da M 2 -ec mimarTulia suraTis sibrtyis r2 l2 marTobulad Cvengan, amitom sididiT



o

 F

 r1



e.i. w y v ilZ a li s d id e da

sur. 33.3



M  M1  M 2  F  l1  F  l 2  F l1  l2   Fl .

(



e r T - e rT i

w y v il Z al is

t ol i a. 91

m om e n ti s

Z a li s m xr is

si-

s i d i d is a n am rav li s

i ne r c iis uZravi

RerZis

m om e nt i . m masis nivTieri wertilis inerciis momenti mimarT

ewodeba

wertilis

masis

namravls

RerZamde

manZilis kvadratze (sur. 33.4):

I  mr 2 .

(33.3)

inerciis momenti skalaruli sididea. SI

sistemaSi misi erTeulia

kg.m2. myari sxeulis inerciis momentis misaRebad igi unda davyoT imdenad mcire mi elementebad, rom TiToeulis farglebSi ri iyos mudmivi. gamoviTvaloT yoveli elementis inerciis momenti da Semdeg SevkriboT. amri2 gad, i -uri elementis inerciis momenti I i  mi ri , xolo mTeli sxeulis:

i 1

I   Li  mi ri 2 .

(33.4)

sxeulSi nivTierebis uwyveti ganawilebisas (e.i. roca n   xolo

mi  0, ),

jami

gadadis

dm  dV ,

magram

I   r 2 dm .

integralSi:



sadac

simkvrivea.

amitom

I   r 2 dV . Tu sxeuli erTgvarovania,   const da

I    r 2 dV .

(33.5)

magaliTis saxiT gamoviTvaloT zogierTi (simetriuli)

sur. 33.4

sxeulis inerciis momenti. 1. d i s k o s inerciis momenti. mocemulia erTgvarovani disko, romlis radiusia R , xolo sisqe _ h . gamoviTvaloT inerciis momenti misi simetriis 00' RerZis mimarT (sur. 33.5). davyoT disko dr siganis wriul fenebad. aseTi fenis moculoba

dV  2rh  dr . visargebloT (33.5) formuliT: R

R4 I    r dV    r 2rhdr  2h  r dr  2h 4 0 0 0 2

magram

R 2 h  m

aris diskos masa. amitom

I

1 mR2 . 2

aseTive formuliT gamoiTvleba m T l i a n i c i l i n d r i s inerciis momenti simetriis RerZis mimarT.

2. T x e l k e d l i a n i c i l i n d r i s inerciis momenti. mocemulia Txelkedliani ci92

sur. 33.5

lindri, romlis radiusia

R , xolo masa _ m gamoviTvaloT inerciis momenti misi

simetriis oo' RerZis mimarT (sur. 33.6). am amocanis gadawyveta advildeba (33.4) formulis gamoyenebiT. marTlac, vinaidan kedlis sisqe Zalian mcirea, cilindris zedapiris yoveli elementi erTi da igive R manZiliT iqneba daSorebuli oo' RerZidan. amitom n

i 1

I   mi ri 2  mi R 2 R 2  mi mR 2 . danarCeni Sedegebi moviyvanoT gamoyvanis gareSe. s q e l k e d l i a n i c i l i n d r i s inerciis momenti simetriis RerZis mimarT:

I





1 m R12  R22 , 2

sadac R1 da R2 cilindris Siga da gare zedapirebis radiusebia, m cilindris masaa. s f e r o s inerciis momenti sferos centris mimarT:

I

3 mR2 . 5

xolo centrSi gamavali RerZis mimarT:

I

2 mR2 , 5

sadac R sferos radiusia, m _ sferos masa. R e r o s inerciis momenti Reros Sua wertilSi gamavali marTobi RerZis mimarT:

I

1 2 ml , 12

(33.6)

xolo Reros boloze gamavali marTobi RerZis mimarT

o sur.33.6

1 I  ml 2 . 3

(33.7)

l _ Reros sigrZea, m _ Reros masa.

ukanaskneli magaliTi cxadyofs, rom myari sxeulis inerciis momenti damokidebulia ara marto

sxeulis zomebsa da masaze,

aramed brunvis RerZis mdebareobazec.

S t e i n e r m a daamtkica Teorema: inerciis momenti nebismieri AB RerZis mimarT

I  I c  md 2 o'

(33.8)

sadac I c aris inerciis momenti sxeulis masaTa centrSi gamavali da

AB RerZis paraleluri 00' RerZis mimarT. d RerZebs Soris manZilia (sur. 33.7). mkiTxvels vTavazobT Seamowmos, ramdenad eTanxmeba (33.6) da

(33.7) formulebiT gamosaxuli Sedegebi Steineris Teoremas. 

o sur.

33.7

i m pu ls is m om e nt i . davuSvaT, mocemulia nivTieri  wertili A , romlis masaa m , xolo siCqare _  . maSin mi  si impulsi P  m . am wertilis impulsis momenti 0 wertilis mimarT ewodeba 0 wertilidan A wertilamde gav93

 lebuli r radius-veqtoris veqtorul namravls impulsze:       L  r P an L  r  m 

 

(33.9) impulsis momentis sidide da mimarTuleba, agreTve misi mniSvneloba

RerZis mimarT ganisazRvreba zustad ise, rogorc es iyo Zalis momentis SemTxvevaSi. kerZod, roca oz RerZi (brunvis RerZi) marTobia 0 wertilze  da m veqtorze gavlebuli sibrtyisa, impulsis momenti 0 wertilis mimarT mdebareobs brunvis RerZze da Tanxvdeba impulsis moments amave Rer  Zis mimarT (am SemTxvevaSi L -is mimarTuleba Tanxvdeba  -s mimarTulebas).    (33.9) formulaSi SevitanoT    r , sadac  wertilis brunvis kuTxuri siCqarea. miviRebT:    L  mr  r .L

(33.10) visargebloT ormagi veqtoruli namravlis gardasaqmneli formuliT:

abc  ba  c   ca  b  . gveqneba:

       L  m r  r   mr r    .

skalaruli namravlis    r     0   90 . amitom





wesis

Tanaxmad

r  r   r 2 ,

xolo

  L  mr 2 .

2 magram mr  I aris A wertilis inerciis momenti 0 wertilis mimarT. e.i.   (33.11) L  I .

myari sxeulis impulsis momenti tolia misi calkeuli elementebis (nawilakebis) impulsis momentebis jamis:



n    L   ri  Pi i 1



  sadac ri da Pi Sesabamisad i -uri elementis (nawilakis) radius-veqtori da

impulsia. (33.11) toloba samarTliania myari sxeulisTvisac, oRond am SemTxvevaSi I aris myari sxeulis inerciis momenti. SI sistemaSi impulsis momentis erTeulia kgm2wm-1.

m b r u n av i

m y a ri

s xe u li s

k i ne t ik u ri

e ne r gi a . mbrunavi myari

sxeulis kinetikuri energiis gamosaTvlelad formula W 

m 2 ar gamod2

geba, vinaidan mbrunavi sxeulis sxvadasxva wertils sxvadasxva wiriTi siCqare aqvs. mivmarToT CvenTvis cnobil xerxs: davyoT azrobrivad myari 94

sxeuli imdenad mcire mi elementebad, rom TiToeulis farglebSi siCqare iyos mudmivi. gamoviTvaloT yoveli maTganisTvis kinetikuri energia da Semdeg SevkriboT. i -uri elementis kinetikuri energia Wi 

mi i2 , xolo 2

mTeli sxeulis kinetikuri energia

mi i2 . 2 i 1 i 1 Tu brunva warmoebs  kuTxuri siCqariT, maSin  i   ri da n

W   Wi  

2

W n

magram

 m r i 1

i i

 mi ri2 . i 1

 I aris sxeulis inerciis momenti. amitom

I 2 . (33.12) 2 aseTia brunavi sxeulis kinetikuri energiis formula. igi savsebiT W

analogiuria gadataniTi moZraobis kinetikuri energiis formulisa, oRond masa Secvlilia inerciis momentiT, xolo wiriTi siCqare _ kuTxuri siCqariT. Tu sxeuli gadataniT moZraobasTan erTad brunviT moZraobasac asrulebs,

maSin

misi

sruli

kinetikuri

m c2 I 2 , W  2 2

energia

sadac

c

sxeulis masaTa centris siCqarea.

§34. brunviTi moZraobis dinamikis ZiriTadi gantoleba davuSvaT, myari sxeuli brunavs 0 wertilSi gamavali uZravi RerZis  irgvliv mudmivi  kuTxu-ri siCqariT (sur. 34.1). am moZraobis gantolebis misaRebad davyoT azrobrivad sxeuli n raodenobis 



mcire elementebad da davweroT impulsis momentis

↺

gamosaxuleba mi

 ri

i -uri

erT-erTi,

elementisaTvis.

ganmartebis Tanaxmad





  Li  ri  Pi ,

(34.1)    sadac Pi  mi i aris i -uri elementis impulsi, xolo ri _ am elementis radius-veqtori.

sur. 34.1

gavawarmooT (34.1) gantoleba droiT:

miRebuli

gantolebis

   dLi  dri     dPi    Pi   ri  . dt  dt   dt  marjvena nawilis pirveli

(34.2) Sesakrebi veqtoruli

namravlis wesis Tanaxmad nulis tolia:     dri      dt  Pi    i mi i   mi  i   i   0 .   meore Sesakrebi gardavqmnaT niutonis meore kanonis saSualebiT:  n   dPi  Fi   Fik dt K 1 n   sadac Fi aris i -ur elementze moqmedi gareSe Zala, xolo  Fik _ Sinagani K 1

Zalebis tolqmedi. gveqneba:

 n dLi      ri  Fi   ri  Fik (34.3) dt K 1 (34.3) toloba gamosaxavs myari sxeulis i -uri elementis impulsis









momentis cvlilebis kanons. imisaTvis, rom miviRoT impulsis momentis cvlilebis kanoni mTeli sxeulisaTvis, saWiroa (34.3) gantoleba davweroT yvela n elementisaTvis da wevr-wevrad SevkriboT, anu, rogorc vambobT xolme, movaxdinoT (34.3) gantolebis ajamva i indeqsiT 1-dan n -mde:  n n n n dLi      r  F  r     i  Fik i i i 1 dt i 1 i 1 K 1









(34.4)

ganvixiloT (34.4) gantolebis TiToeuli wevri:   n  n  dLi d n  dL , sadac L   Li aris mTeli sxeulis impulsis  L   dt dt  i dt i 1 i 1 i 1

 r  F    M n

momenti.

i 1

  i

i 1



  M. M

aris gareSe Zalebis momentebis veqtoruli

jami da ewodeba g a re Se Z a le b is m T a v ar i m om e n ti . niutonis mesame kanonis gamoyenebiT mtkicdeba, rom tolobis mesame

 ri  Fik   0 . n

wevri

 

i 1 k 1

miRebuli

mniSvnelobebis

CasmiT

(34.4)

gantolebaSi

sabolood

miviRebT:

 dL  M. dt

(34.5)

(34.5) gamosaxavs myari sxeulis brunviTi moZraobis ZiriTad gantolebas. am gantolebis Tanaxmad, sxeulis impulsis momentis cvlilebis siCqare tolia gareSe Zalebis mTavari momentisa da igive mimarTuleba aqvs.  (34.5) formulas SeiZleba mivceT sxva saxe, Tu masSi SevitanT L -is mniSvnelobas (33.11)-dan:  d I    M dt Tu myari sxeulis forma ar icvleba, misi inerciis momenti I  const   d  maSin I M dt

  I   M

(34.6)

e.i. myari sxeulis inerciis momentisa da kuTxuri aCqarebis namravli udris masze moqmedi gareSe Zalebis mTavar moments. (34.5)

(34.6)

analogiuria niutonis    dP   F da ma  F . kanonis formulebisa gadataniTi moZraobisaTvis: dt maSasadame,

formulebi

rolsac

savsebiT

impulsi,

Zala,

masa

wiriTi

meore

aCqareba

asrulebs gadataniTi moZraobisas, igive rols asrulebs impulsis momenti, Zalis momenti, inerciis momenti da kuTxuri aCqareba brunviTi moZraobis dros. iseve, rogorc gadataniTi moZraobis dros sxeulis inerciis zoma aris misi masa, brunviTi moZraobis dros sxeulis inerciis zoma misi inerciis momentia. marTlac, (34.6) formulis Tanaxmad, rac metia sxeulis I  inerciis momenti, miT nakleb kuTxur aCqarebas iRebs sxeuli M Zalis momentis moqmedebiT.

§35. impulsis momentis mudmivobis kanoni Cven gavecaniT bunebis or fundamentur kanons _ energiis mudmivobisa da impulsis mudmivobis kanons. gavecnoT axla mesame fundamentur kanons _ impulsis momentis mudmivobis kanons. davweroT myari sxeulis brunviTi moZraobis dinamikis ZiriTadi gantoleba:   dL M. dt

davuSvaT,

sxeuli

izolirebulia.

  dL M  0, 0 dt

maSin

 L  const

 I  const (35.1) e.i. izolire b u li s xe u lis im pu ls is m om e nti m u dm iv i s idide a. am

daskvnas ewodeba impulsis momentis mudmivobis kanoni. es kanoni izolirebul sistemasTan mimarTebaSi gulisxmobs, rom sistemaSi mimdinare procesebis Sedegad calkeuli sxeulebis impulsis momentebi SeiZleba Seicvalos, magram ise, rom maTi jami anu sistemis impulsis momenti darCes mudmivi. (35.1) toloba veqtoruli tolobaa, rac niSnavs, rom inaxeba impulsis momentis rogorc sidide, ise mimarTuleba.  I namravlis mudmivoba (Senaxva) niSnavs, rom inerciis momentis Secvla, vTqvaT, gazrda, aucileblad gamoiwvevs kuTxuri siCqaris imave sididiT Semcirebas da piriqiT. impulsis momentis mudmivobis kanonis es Sedegi SeiZleba Semowmdes Semdegi cdiT: advilad mbrunav magidaze (e.w. Jukovskis merxi) dgas adamiani, romelsac orive xelSi ukavia tvirTebi inerciis

momentis

SesamCnevad

gadidebis

Semcirebis

mizniT.

magida

moyavT brunviT moZraobaSi. roca adamiani gaSlis xelebs, xelebSi da tvirTebSi Tavmoyrili masa brunvis RerZs Sordeba ( r izrdeba), amitom misi inerciis momenti izrdeba, rac iwvevs brunvis kuTxuri siCqaris Semcirebas. xelebis

daSvebisas

piriqiT

inerciis

momenti

mcirdeba,

ris

gamoc

izrdeba brunvis kuTxuri siCqare da magida ufro swrafad iwyebs brunvas.  I -s mimarTulebis mudmivoba niSnavs kuTxuri siCqaris veqtoris da, maSasadame, sxeulis brunvis RerZis sivrculi mimarTulebis ucvlelobas. amitomaa, rom Sorsmsroleli qvemexis (agreTve SaSxanis) lulis SigniTa zedapirs akeTeben xraxniseburs*, riTac Wurvs gadataniT moZraobasTan erTad aniWeben brunviT moZraobasac. amis Sedegad Wurvi frenisas ar icvlis

orientacias,

ris

gamoc

izrdeba rogorc

frenis

siSore,

ise

damiznebis sizuste. brunvis RerZis mimarTulebis ucvleloba ganapirobebs moZravi

velosipedis

vertikalur

mdgradobas,

mdgradobas da sxv.

amas asaxavs, kerZod, SaSxanis rusuli dasaxeleba Винтовка. 98

bzrialas

vertikalur

§36. myari sxeulis wonasworobis pirobebi myari sxeulis dinamikisadmi miZRvnili Tavi davamTavroT paragrafiT, romelSic ganxilulia myari sxeulis s tatik is ZiriTadi sakiTxi – wonasworobis pirobebi. pirvel rigSi gavarkvioT, Tu ra igulisxmeba sxeulis (iseve, rogorc nivTieri wertilis) wonasworobaSi: s xe u li w on a s w orob aSia, T u i gi, m oy v anili u Z rav m dgo m areo b aSi, u Z rav i rCe b a Se m de g Sic. amrigad, mxolod u Z r av sxeulze SeiZleba iTqvas, rom wonasworobaSia, miuxedavad imisa, rom rogorc qvemoT vnaxavT, wonasworobis maTematikur pirobas akmayofilebs wrfivad da Tanabrad moZravi sxeulic. myari sxeulis wonasworobis pirobis dadgena martivad xdeba Tavis wina paragrafebSi gamoyvanili gadataniTi da brunviTi moZraobis dinamikis ZiriTadi gantolebebis safuZvelze. am gantolebebs aqvs saxe: 



ma  F 



I M

(36.1)



sadac  da  Sesabamisad sxeulis wiruli (xazovani) da kuTxuri aCqa



rebebia, xolo F da M - gareSe Zalebis da gareSe momentebis veqtoruli jami. uZraobis mdgomareobis SenarCuneba gulisxmobs, rom sxeulma ar unda daiwyos arc gadataniTi da arc brunviTi moZraoba, anu ar unda SeiZinos arc wiruli da arc kuTxuri siCqare. es imas niSnavs, rom nulis toli unda iyos 



  0 . (36.1) tolobebidan

wriuli aCqarebac da kuTxuri aCqarebac:   0 , 



gamomdinareobs, rom amisTvis saWiroa F -isa da M -is nulTan toloba: 

(36.2)

tolobebi



F  0, M  0. gamosaxavs myari sxeulis

wonasworobis

(36.2) pirobebs,

romelic ase gamoiTqmis: my ari sxeulis wonasworo b isaTvis s aWiroa, rom sxeulze m oqme di gareSe Z alebis veqtoru li jami (gare Se Z a lebis mT avari veqtori) da gareSe Z alebis mome ntebis veqto ruli jami (gare Se Zalebis mTavari momenti) iyos nulis toli. 

igulisxmeba absoluturad myari, anu aradeformadi sxeuli. wonasworobis pirobebis dadgena sxeulebisa, romlebic Zalis moqmedebiT mniSvnelovnad deformirdebian, rTuli amocanaa, vinaidan Zalis modebis Semdeg faqtiurad miiReba axali geometriuli zomebisa da formis sxeuli. magram praqtikaSi, sxeulebis umravlesobis deformacia imdenad umniSvneloa, rom SeiZleba misi ugulebelyofa da absoluturad myari sxeulis wonasworobis pirobebis maTze gavrceleba.  es arcaa gasakviri – statika xom dinamikis kerZo SemTxvevaa, vinaidan sxeulis uZraoba, roca masze Zala moqmedebs, aris moZraobis kerZo SemTxveva   0  



VII Tavi

r x e v e b i d a t al R e b i §37. harmoniuli rxeva rxeviTi moZraoba anu rxeva farTod aris gavrcelebuli bunebasa da teqnikaSi. r xe v a e w ode b a y ov e l gv a r pr oce s s , rom e li c zu s t a d a n m ia xl oe b iT

m e or de b a

dr oi s

to li

S u a le de b i s

Se m de g.

procesis

ganmeorebadoba anu perioduloba rxevis ZiriTadi Tvisebaa. rxeviTi procesis fizikuri bunebis mixedviT ansxvaveben: m e q an ik u r r xe v e b s (saqanelas rxeva, simis rxeva, Sigawvis Zravis cilindrSi dguSis moZraoba da sxv. am rxevebis dros meordeba, magaliTad, sxeulis koordinatebisa da siCqaris mniSvnelobebi), e l e q t rom a gn itu r r xe v e b s (denis   Zalisa da Zabvis rxevebi cvladi denis wredSi, E da B veqtorebis rxevebi eleqtromagnitur velSi da sxv. am dros meordeba denis Zalisa da Zabvis  mniSvnelobebi, eleqtruli daZabulobis E veqtorisa da magnituri induqciis  B veqtoris mniSvnelobebi), e le q t rom e q a nik u r r xe v e b s da sxv. eleqtromeqanikuri rxevebi `Sereuli~ rxevebia. aseTi rxevebi warmoiSvebian eleqtroakustikur gardamqmnelebSi, romlebSic xdeba eleqtruli rxevebis gardaqmna bgeriT rxevebad (magaliTad, telefonSi, xmamaRlamolaparakeSi) da piriqiT (magaliTad, mikrofonSi). aRmoCnda,

rom

sxvadasxva

fizikuri

bunebis

rxevebi

erTnairad

mimdinareobs da emorCileba sruliad erTnair raodenobriv kanonebs. amitom, maTi kvlevis meTodebic erTnairia. rxeviTi moZraobis kanonzomierebebis Seswavlas didi mniSvneloba aqvs, vinaidan es kanonebi udevs safuZvlad xidebis, sxvadasxva manqanebisa da meqanizmebis, safreni aparatebisa da a.S. simtkicis gamoTvlas. am kanonebzea dafuZnebuli mecnierebis iseTi dargebi, rogorebicaa akustika, radioteqnika, cvladi denebis teqnika da sxva.

100

rxeviTi

moZraobis

Seswavla

daviwyoT

misi

umartivesi

saxiT

harmoniuli rxeviT*. ganvixiloT zambariani qanqaras rxeva. zambariani

masiur

mocmulia horizontalur RerZze. RerZze Ca-

mocmulia agreTve drekadi zambara, romlis erTi

 F

warmoadgens

birTvs, romelic gaburRulia diametrze da Ca-

 F

qanqara

bolo

damagrebulia

RerZis

boloze,

meore ki mimagrebulia burTulaze (sur. 37.1). O mdebareobaSi burTula wonasworobaSia, vi-

naidan zambara deformirebuli ar aris, bur-

sur. 37.1

Tulaze moqmedi simZimis Zala ki kompensirdeba RerZis

reaqciis

ZaliT.

burTulas

gadavxriT marjvniv B mdebareobaSi da gavuSvebT xels, igi daiwyebs rxevas wonasworobis mdebareobis mimarT. gavarkvioT, ra aris am rxevis mizezi. rodesac burTulas gadavxriT marjvniv, zambara daiWimeba da masSi aRiZvreba  drekadobis F Zala, romelic cdilobs daabrunos zambara pirvandel (aradeformirebul) mdgomareobaSi (§16). es Zala modebulia burTulaze. amitom, rodesac burTulas gavuSvebT xels, am Zalis moqmedebiT igi daiwyebs aCqarebul moZraobas wonasworobis O mdebareobisaken. am moZraobisas zambaras deformacia

mcirdeba.

amitom

mcirdeba

burTulaze

moqmedi

drekadi

Zala

burTulas aCqareba, magram izrdeba misi siCqare. wonasworobis mdebareobaSi misvlisas zambaras deformacia nulis tolia. e.i. nulis tolia burTulaze moqmedi drekadobis Zalac da burTulas aCqarebac, magram maqsimaluria misi siCqare. amitom burTula ar gaCerdeba, igi inerciiT ganagrZobs moZraobas wonasworobis mdebareobidan marcxniv da iwyebs zambaris SekumSvas. SekumSul zambaraSi kvlav aRiZvreba drekadobis Zala, romelic axla ukve moZraobis sawinaaRmdegodaa mimarTuli, ris gamoc burTulas moZraoba gaxdeba Senelebuli da romeliRac C wertilSi igi gaCerdeba. es gaCereba aris myisieri.

harmoniuli rxevis roli gansakuTrebulia fizikasa da teqnikaSi, vinaidan nebismieri

rTuli perioduli rxeva, romlis sixSirea rxevebis jamis saxiT, romelTa sixSireebia



SeiZleba warmodgenili iqnes harmoniuli

 , 2 , 3 101

da a.S. (furies Teorema).

SekumSul zambaraSi aRZruli drekadobis Zalis moqmedebiT burTula daiwyebs aCqarebul moZraobas marjvniv, inerciiT gairbens wonasworobis O mdebareobas da aRmoCndeba B mdebareobaSi, saidanac daiwyo man rxeva. e.i. burTula Seasrulebs erT srul rxevas. amis Semdeg yvelaferi gameordeba Tavidan. d r os , r om li s ga nm av l ob aS ic s xe u li a s ru le b s e rT s ru l r xe v as , p e r io d i e w o de b a, aR in i Sne b a T a s oT i , i zom e b a w am e b iT . rogorc Catarebuli msjelobidan Cans, burTulas rxevis mizezia gaWimul da SekumSul zambaraSi aRZruli drekadobis Zala da burTulas inercia. wonasworobis mdebareobidan burTulas gadaxra aRvniSnoT x –iT. cxadia, igi gansazRvravs zambaris deformacias. hukis kanonis Tanaxmad, deformirebul zambaraSi aRZruli drekadi Zala deformaciis proporciulia: F  kx ,

(37.1)

proporciulobis k koeficients ewodeba zambaris sixiste. igi ricxobrivad im Zalis tolia, romelic aRiZvreba zambaraSi misi sigrZis erTeuliT deformaciisas. k –s mniSvneloba damokidebulia zambaris zomebze da masalis gvarobaze. Tu gaviTvaliswinebT im faqts (sur. 37.1), rom F Zala yovelTvis mimarTulia wonasworobis mdebareobisaken, x gadaxra ki aiTvleba wonasworobis mdebareobidan, (37.1) formula ase daiwereba: F  kx *.

(37.2)

aseTi ZaliT warmoebs zambariani qanqaras rxeva. F  kx Z a liT g am ow v e u l r xe v as h ar m on iu li r xe v a e w od e b a . bunebaSi

xSirad

gvxvdeba

iseTi

Zalebi,

romlebic

Tavisi

bunebiT

drekadni ar arian, magram maTi sididis damokidebuleba gadaxraze (37.2) formuliT gamoisaxeba. maT k v az i dre k ad i (drekadis magvari) Zalebi ewodeba. cxadia, kvazidrekadi ZaliT gamowveuli rxevac harmoniuli iqneba. amis magaliTia maTematikuri qanqaras rxeva, romelic warmoebs simZimis Zalis mdgenelis moqmedebiT. miviRoT harmoniuli rxevis gantoleba, anu davadginoT, Tu rogor aris damokidebuli harmoniulad merxevi sxeulis x gadaxra t droze. amisaTvis

faqtobrivad, formulaSi niSani minusi gamosaxavs im faqts, rom deformaciis Sedegad

aRZruli drekadi Zala deformaciis Semcirebisken aris mimarTuli.

102

davweroT moZraobis gantoleba anu niutonis meore kanoni: F  kx , xolo a 

d 2x . CavsvaT: dt 2

ma  F . magram

d 2x  kx . dt 2

tolobis orive mxare gavyoT m -ze:

d 2x k   x. m dt 2

(37.3)

k   02 . m

(37.4)

SemoviRoT aRniSvna

 0 -s ewodeba wriuli an cikluri sixSire. maSin (37.3) gantoleba miiRebs saxes:

d 2x   02 x . 2 dt

(37.5)

miviReT meore rigis umartivesi diferencialuri gantoleba. CasmiT advilad davrwmundebiT, rom am gantolebis amoxsnas aqvs saxe: x  A cos0 t    .

A aris rxevis amplituda,

0t    -s

(37.6)

ewodeba rxevis faza, xolo  -s _

sawyisi faza. vinaidan cos -is maqsimaluri mniSvneloba erTia, amitom, roca

cos0 t     1 , maSin x  x maqs.  A , maSasadame, a m pl itu d a a r is w on a s w o r o b is m de b ar e ob id a n m e r xe v i w e rti li s u di de s i g ad a xr a . (37.6)-dan Cans, rxevis faza gansazRvravs sxeulis mdebareobas ( x gadaxras) drois nebismier

t momentSi, xolo sawyisi faza momentSi. marTlac,

_ sxeulis mdebareobas drois sawyis t  0

roca t  0 , x  x0  A cos  . Tu sawyisi faza   0 , maSin

x  A cos 0 t , Tu t  0 , maSin x  A e.i. rxeva iwyeba udidesi gadaxris mdebareobi-

dan (rogorc es Cven gvqonda zemoT ganxilul magaliTSi). (37.5) diferencialur gantolebas, rogorc meore rigis gantolebas, aqvs meore amoxsnac: x  A sin 0 t   

(37.7)

oRond, am SemTxvevaSi, sawyisi fazis mniSvneloba iqneba sxva. (37.6) gantoleba, iseve rogorc (37.7) gantoleba, gamosaxavs harmoniulad merxevi sxeulis

gadaxris damokidebulebas

droze. maT harmoniuli

rxevis gantolebebi ewodeba. rogorc vxedavT, es gantolebebi sin -is da cos -is 103

funqciebiT gamoisaxeba. amitom, harmoniul rxevas xSirad ganmartaven rogorc iseT rxevas, romelic warmoebs sinusis an kosinusis kanoniT. periodis ganmartebis Tanaxmad, T drois gasvlis Semdeg sxeuli ubrundeba im mdebareobas, saidanac man daiwyo rxeva. maSasadame, rodesac harmoniuli rxevis gantolebaSi t -s SevcvliT t  T -Ti, x ar unda Seicvalos, magram sinusi da kosinusi perioduli funqciebia 2 periodiT. amitom zemoT xsenebuli Secvlisas

rom ar Seicvalos, argumenti (e.i. rxevis faza) unda

Seicvalos (gaizardos) 2 -T. e.i.

0 t  T     0 t    2 , saidanac

T

2

0

(37.8)

Tu am formulaSi SevitanT  0 -is mniSvnelobas (37.4)-dan, zambariani qanqaras rxevis periodisaTvis gveqneba:

T  2

m , k

(37.9)

e.i. rxevis periodi damokidebulia burTulas masaze da zambaris sixisteze. ganmartebis Tanaxmad, rxevis sixSire  aris rxevaTa ricxvi erT wamSi. igi periodis Sebrunebuli sididea (§8):  

1 T

(37.8) formulidan wriuli

sixSire 2  2 . T maSasadame, wriuli sixSirec rxevaTa ricxvia, magram ara erT, aramed 2 wamSi.

0 

zemoT ganxilul magaliTSi zambariani qanqaras rxeva aRiZra erTxel miniWebuli energiis xarjze _ burTulas gadaxrisas sistemas mivaniWeT potenciuri

energia

kx 2 . 2

aseT

rxevas,

romelic

warmoebs

erTxel

miniWebuli

energiis xarjze, T av i s u f al i an s ak u T a ri rxeva ewodeba, xolo Sesabamis sixSires _ rxevis sakuTari sixSire. sakuTari imitom, rom igi damokidebulia merxevi sistemis sakuTar Tvisebebze (zambariani qanqaras SemTxvevaSi _ burTulis masaze da zambaris sixisteze). maSasadame, harmoniuli rxeva Tavisufali anu sakuTari rxevaa, xolo  0 sixSire _ rxevis sakuTari sixSire. 104

harmoniuli rxevis ganxilvisas ugulebelyofili iyo xaxunisa da haeris winaaRmdegobis Zalebi, ris gamoc adgili ar hqonda erTxel miniWebuli energiis gabnevas da rxevebi zu s ta d Zalebi

yovelTvis

moqmedeben,

ris

meordeboda. realur pirobebSi es

gamoc

energia

gaibneva.

iwvevs

amplitudisa da rxevis maxasiaTebeli sxva sidideebis TandaTan Semcirebas. amitom rxevis gameoreba xdeba m i a xlo e b iT . Cven aRvniSneT, rom maTematikuri qanqaras rxeva aris

harmoniuli

Zalis

rxeva,

romelic

mdgenelis moqmedebiT

Cveneba (kuTxe



warmoebs

(sur. 37.2). advilia

mcirea, amitom sin    ), rom

F  kx ,

sadac

 

simZimis

 N l

mg . k l

 N x

Catarebuli msjelobis zustad gameorebiT (ris gakeTebasac vTavazobT mkiTxvels) maTematikuri qanqaras rxevis periodisTvis miviRebT formulas

T  2

 F

 P

l , g

 F1

 

 P

sadac l qanqaras sigrZea, g _ simZimis Zalis aCqareba.

sur. 37.2

§38. harmoniulad merxevi sxeulis siCqare, aCqareba da energia harmoniulad

merxevi

sxeulis

siCqares

aCqarebas

davadgenT

harmoniuli rxevis gantolebidan: x  A cos0 t    .

siCqare



dx     A0 sin 0 t     A0 cos 0 t     , dt 2 

(38.1) (38.2)

xolo aCqareba

d   A02 cos0 t     02 x  A02 cos0 t      (38.3) dt miRebuli formulebidan Cans, rom siCqare wanacvlebulia x gadaxris mimarT a



faziT anu periodis meoTxediT, xolo aCqareba _  faziT anu naxevar-

periodiT.

105

garda amisa, (38.1) da (38.2) formulebi gviCvenebs, rom roca gadaxra maqsimaluria ( B da C

mdebareobebi 37.1 suraTze; 0t   Sesabamisad tolia 0-is

da  -s), siCqare nulis tolia da piriqiT _ wonasworobis mdebareobaSi gav-





lisas, roca x =0, siCqare maqsimaluria. xolo (38.3) formula a  02 x gviCvenebs, rom aCqareba da gadaxra erTdroulad aRweven Tavis maqsimalur da minimalur mniSvnelobebs, oRond urTierTsawinaaRmdego mimarTulebebi aqvT (swored amas asaxavs wanacvleba 

faziT). marTlac, a yovelTvis mimarTulia

wonasworobis mdebareobisaken, x ki aiTvleba wonasworobis mdebareobidan. merxev sxeuls gaaCnia rogorc kinetikuri, ise potenciuri energia. kinetikuri energia

m 2 mA202 sin 2 0 t    . (38.4)  2 2 potenciuri energiis gamosaTvlelad gamoviyenoT kavSiri Zalasa da W

potenciur

energias

Soris

(30.5).

vinaidan

moZraoba

warmoebs

RerZis

gaswvriv, amitom x

dU , saidanac U    Fdx Fx  F   dx 0 magram harmoniulad merxev sxeulze moqmedi drekadi Zala F  kx , amitom x

U   kxdx  0

kx2 . 2





Tu am formulaSi SevitanT k -s da x -is mniSvnelobebs, gveqneba k  m02 :

U

m02 A 2 cos 2 0 t    . 2

(38.5)

(38.4) da (38.5) formulebi gviCvenebs, rom merxevi sxeulis kinetikuri da potenciuri energiebi drois mixedviT icvleba, amasTan, rodesac kinetikuri energia maqsimaluria, potenciuri energia nulis tolia da piriqiT. sruli meqanikuri energia rxevis nebismieri momentisaTvis

m02 A 2 m02 A 2 m02 A 2 2 2 E  W U  sin 0 t     cos 0 t     . (38.6) 2 2 2 rogorc vxedavT, sruli energiis gamosaxuleba dros ar Seicavs. maSasadame, igi mudmivi sididea. amrigad, harmoniuli rxevis dros sxeulis kinetikuri energia gadadis potenciurSi da piriqiT, magram ise, rom maTi jami anu sruli meqanikuri energia rCeba ucvleli. 106

§39. erTi wrfis gaswvris mimarTuli ori harmoniuli rxevis Sekreba mTeli

rigi

amocanebis

gadawyveta,

 A

maT Soris, erTi wrfis gaswvriv mimarTuli

ori

ioldeba

harmoniuli

rxevis

Sekreba,

 A

veqtoruli

 0t

diagramis gamoyenebiT, romelic Semdegnairad igeba: x RerZze, romlis gaswvrivac irxeva sxeuli, 0 wertilidan gadavzomoT  A amplitudis toli veqtori A (mas amplitud-veqtori

ewodeba),

RerZTan Seadgens 

romelic



x  A cos0 t   

x  A cos 

sawyisi fazis tol

sur. 39.1

kuTxes (sur. 39.1) am veqtoris gegmili x RerZze x  A cos  . vabrunoT es veqtori 0 wertilis irgvliv saaTis isris sawinaaRmdego mimarTulebiT  0 kuTxuri siCqariT. maSin raRac t droSi igi mobrundeba

  0 t kuTxiT da am momentisTvis RerZTan Seadgens 0t   kuTxes. amplitudveqtoris gegmili x RerZze toli gaxdeba: x  A cos0 t    .

amrigad, amplitud-veqtoris brunviT 0 wertilis irgvliv mudmivi  0 kuTxuri

siCqariT,

misi

bolo

wertilis

gegmili

asrulebs

RerZze

harmoniul rxevas imave  0 sixSiriT. amitomac ewodeba  0 -s wriuli sixSire. naTqvamidan gamomdinareobs, rom harmoniuli rxeva SeiZleba mocemul iqnes amplitud-veqtoris saSualebiT, romelic x RerZTan adgens  sawyisi fazis tol kuTxes. vTqvaT, mimarTul

sxeuli

erTdroulad

rxevaSi

resorebiani

vagonis

CamovkidebT

raime

(aseTi

Werze

tvirTs:

erTi

monawileobs

SemTxveva boloTi

tvirTi

irxeva

erTi

realizdeba, damagrebul damagrebis

wrfis

gaswvriv

magaliTad, drekad

zambaraze

wertilis

mimarT,

romelic Tavis mxriv asrulebs rxevas vagonis resorebze. maSasadame, tvirTi 107

erTdroulad

monawileobs

erTi

wrfis

gaswvriv

mimarTul

rxevaSi).

davuSvaT, es rxevebi harmoniulia da warmoebs erTi da imave  0 sixSiriT. e.i. x1  A1 cos0t  1  , x2  A2 cos0 t   2  .

davadginoT SedegiTi moZraobis xasiaTi. radgan rxevebi erTi wrfis gaswvriv

warmoebs,

SedegiTi

gadaadgileba

iqneba

Semadgeneli

rxevebis

gadaadgilebaTa algebruli jami, anu

x  x1  x2 . Tu am formulaSi SevitanT

x1 -sa da

(39.1)

x 2 -is mniSvnelobebs da movaxdenT

Sesabamis trigonometriul gardaqmnebs, davadgenT x -is mniSvnelobas, magram es gza sakmaod grZeli da Sromatevadia. amocana gacilebiT martivdeba da TvalsaCino xdeba, Tu gamoviyenebT veqtorul diagramas. avagoT es diagrama (simartivisaTvis, diagramas vagebT t  0 momentisaTvis). vinaidan

A1 da

amplitud-veqtorebi brunavs erTi da igive  0 kuTxuri siCqariT, SedegiTi  rxevis A amplitud-veqtoric (romelic   A C miiReba A1 -sa da A2 -is veqtoruli SekA2 rebiT) ibrunebs imave siCqariT. garda

  amisa, kuTxe A1 da A2 veqtorebs Soris

 2  1 2 )1 0

iqneba



 2  1 -is

toli

(sur. 39.2).

ucvleli

cxadia, rom

x  x1  x2  A cos 

x an,

drois

roca

sur. 39.2

nebismieri

momentisaTvis,

amplitud-veqtorebi

Semobrundeba

 0 t kuTxiT, x  A cos0 t   

(39.2) romlis

gviCvenebs, A

rom

amplituda

SedegiTi



moZraoba

sawyisi

faza

(39.2) kvlav

harmoniuli

ganisazRvreba

rxevaa,

veqtoruli

diagramidan (sur. 39.2): A2  A12  A22  2 A1 A2 cos   2  1   A12  A22  2 A1 A2 cos 2  1  ,

108

(39.3)

tg  rogorc

(39.3)

A sin 1  A2 sin  2 BC .  1 OB A1 cos 1  A2 cos  2

toloba

gviCvenebs,

(39.4)

SedegiTi

rxevis

amplituda

damokidebulia Sesakreb rxevaTa fazaTa sxvaobaze. ganvixiloT ori kerZo SemTxveva: 1.  2  1  2n , sadac n =0, 1, 2, . , maSin cos 2  1   1, A2  A12  A22  2 A1 A2 da A  A1  A2 e.i.

SedegiTi

rxevis

amplituda

tolia

Sesakrebi

rxevebis

amplitudebis

jamisa. am SemTxvevaSi orive rxeva warmoebs erT fazaSi da isini erTmaneTs aZliereben. rxevebis Sekrebis es SemTxveva mocemulia 39.3 a suraTze. 2.  2  1  2n  1 , maSin cos 2  1   1, A2  A12  A22  2 A1 A2 da A  A1  A2

A A2 A1

sur. 39.3

e.i. SedegiTi rxevis amplituda Sesakrebi rxevebis amplitudebis sxvaobis tolia. am SemTxvevaSi Sesakrebi rxevebi warmoebs sawinaaRmdego fazebSi da isini erTmaneTs asusteben (sur. 39.3 b). gansakuTrebiT sainteresoa SemTxveva, roca Sesakrebi rxevebi warmoebs erTi

wrfis

gaswvriv,

maTi

amplitudebi

erTnairia,

xolo

sixSireebi

umniSvnelod gansxvavdebian erTmaneTisagan. aseTi rxevebis Sekrebis Sedegad miiReba rxeva, romlis amplituda periodulad icvleba. r xe v as ,

pe r io -

d u l a d c v l a di a m p li tu diT , Z ge r a e w o de b a. *

aRebulia uaryofiTi.

absoluturi

sidide,

vinaidan

amplituda

109

Tavisi

arsiT

SeiZleba

iyos

Zgeris warmoSobis axsna advilia. vTqvaT, sawyis momentSi orive rxevis faza Tanxvdeba erTmaneTs. maSin, rogorc viciT, rxevebi aZliereben erTmaneTs da SedegiTi rxevis amplituda tolia Sesakreb rxevaTa amplitudaTa jamisa. Semdeg (vinaidan sixSireebi gansxvavdeba) meore rxeva iwyebs faziT TandaTan CamorCenas pirvelisagan, e.i. fazaTa sxvaoba TandaTan izrdeba, ris gamoc SedegiTi rxevis amplituda TandaTan mcirdeba. roca es CamorCena (anu fazaTa sxvaoba) gaxdeba  -s toli, rxevebi iqnebian sawinaaRmdego fazebSi da isini erTmaneTs Caaqroben, anu SedegiTi rxevis amplituda iqneba nulis toli

A 

A1  A2 , A1  A2  .

fazaTa

sxvaobis

Semdegi

zrdisas

SedegiTi

rxevis

amplituda TandaTan izrdeba da roca fazaTa sxvaoba gaxdeba 2 -s toli, igi miaRwevs maqsimalur mniSvnelobas

A  A1  A2 , vinaidan orive fazis rxeva

Tanxvdeba erTmaneTs da a.S. Sedegad miiReba rxeva perioduli amplitudiT, romlis grafikic mocemulia 39.4 suraTze. zeda ori mrudi Sesakreb rxevaTa grafikebia. SeiZleba Cveneba,

rom

Zgeris

sixSire

tolia Sesakreb rxevaTa sixSireTa sxvaobis:

  2  1 . vinaidan 1 umniSvnelod gansxvavdeba

 2 -sagan,

(Sesakreb

rxevaTa

Zgeris

sixSire

sixSireebTan

mcirea

SedarebiT),

periodi ki _ didi, rac kargad Cans (39.4) suraTze. didi periodi niSnavs, rom amplisur. 39.4

tuda drois mixedviT nela icvleba. amitom Zgera SeiZleba miaxloebiT CaiTvalos harmo-

niul rxevad. amboben, rom Zgera kvaziharmoniuli rxevaa. Zgeris dakvirveba SeiZleba, Tu erTdroulad avaJRerebT or kamertons, romelTa Tavisufali rxevis sixSireebi mcired gansxvavdebian erTmaneTisagan. Cven movismenT bgeris periodulad gaZlierebas da Sesustebas.

110

§40. urTierTmarTobuli harmoniuli rxevebis Sekreba ganvixiloT axla SemTxvea, roca sxeuli erTdroulad monawileobs toli sixSiris or urTierTmarTob harmoniul rxevaSi, romlebic warmoebs x da y RerZebis gaswvriv. rxevis gantolebebs eqnebaT saxe: x  A1 cos0 t  1 ,

SedegiTi

moZraobis

traeqtoriis

y  A2 cos0 t   2 

misaRebad

(40.1)

(40.1) gantolebebidan

unda

gamovricxoT dro. amisaTvis movaxdinoT cos0 t    –s gaSla da gantolebebi gadavweroT Semdegnairad:

x  cos 0 t  cos 1  sin 0 t  sin 1 , A1 y  cos 0t  cos 2  sin 0t  sin 2 . A1 gavamravloT

(40.2)

cos  2 -ze,

(40.3)

cos 1 -ze

(40.2) (40.3)

gamovakloT

erTmaneTs.

miviRebT:

x y (40.4) cos  2  cos 1  sin  0 t  sin  2  1  . A2 A2 axla gavamravloT (40.2) sin 2 -ze, xolo (40.3) ki sin 1 -ze da kvlav gamovakloT erTmaneTs. miviRebT:

x y sin  2  sin 1  cos  0 t  sin  2  1  . A1 A2 Tu

(40.4)

(40.5)

gantolebebs

aviyvanT

kvadratSi

(40.5) da

wevr-wevrad

SevkrebT, miviRebT SedegiTi moZraobis traeqtorias: x2 y2 2 xy  2  cos 2  1   sin 2  2  1  . 2 A1 A2 A1 A2

miRebuli

gantoleba

aris

elifsis

gantoleba,

romelic

dayvanili x da y RerZebis mimarT. ganvixiloT kerZo SemTxvevebi: 1.  2  1  0 . maSin (40.6)-dan: 2

 x A y  x2 y2 2 xy   0 , saidanac y  2 x .  2   0 , an   2 A1 A1 A2 A1 A2  A1 A2 

111

(40.6) ar

aris

miviReT wrfis gantoleba. e.i. am SemTxvevaSi wertili irxeva wrfeze, romelic gadis dekartis koordinatTa sistemis I da III meoTxedebSi. rxevis amplituda

A  A12  A22 (sur. 40.1 a). y

A1 A2

A2 x

A1 a

sur 40.1 2.  2  1   . A maSin y   2 x , A1 e.i.

wertili

irxeva

meoTxedebSi

gamaval

wrfeze

A  A12  A22

amplitudiT (sur. 40.1 b). y

 2  1 

3. wertili

maSin

x2 y2   1, A12 A22

RerZebis

e.i.

mimarT

dayvanil elifsze (sur. 40.2), amasTan, brunva

brunavs



warmoebs saaTis isris moZraobis mimarTulebiT. marTlac, am SemTxvevaSi (40.1) gantolebebs aqvs

  y  A2 cos 0 t     A2 sin 0 t . 2 

saxe:

x  A1 cos 0 t ,

roca

t  0 , maSin x  A1 da y  0 , e.i. mbrunavi

wertili imyofeba c1 mdebareobaSi.

T  T drois gasvlis Semdeg  t   , x  0 da 4 4 

C2 sur. 40.2

y   A2 _ wertili aRmoCnda c2 mdebareobaSi.

112

Tu, garda amisa, rxevebis amplitudebi tolia

 A1  A2  R ,

wertili

imoZravebs wrewirze: x 2  y 2  R 2 . 4.  2  1 

3 2

maSin

x2 y2   1 , e.i. am SemTxvevaSic wertili brunavs A12 A22

elifsze, oRond saaTis isris moZraobis sawinaaRmdegod (ris damtkicebasac vTavazobT mkiTxvels).

1  2  0  ,

Tu Sesakreb rxevaTa sixSireebi toli ar aris

miiReba

rTuli wirebi, romelTa saxe damokidebulia sixSireTa Tanafardobaze. maT l is aJ u s fi gu re b i ewodeba.

§41. milevadi rxeva Cven ganvixileT harmoniuli rxeva, anu

rxeva, romelic warmoebs F  kx Zalis moqmedebiT. miviReT diferencialuri gantoleba

d 2x   02 x 2 dt

(41.1)

da davwereT misi amoxsna:

x  A cos0 t    .

(41.2)

(41.2) formulis grafiks aqvs sur. 41.1-ze mo-

sur. 40.1

cemuli saxe (simartivisTvis davuSvaT, rom

  0 ). grafikidan

Cans,

rom

rxevis

dros

amplituda

mcirdeba,

e.i.

harmoniuli rxeva aramilevadi rxevaa. cxadia, rom bunebaSi harmoniuli rxeva ar

arsebobs,

vinaidan

realur

pirobebSi

sxeulze

yovelTvis

moqmedebs

moZraobis Semaferxebeli Zala. vuwodoT mas, warmoSobis mizezis miuxedavad, garemos winaaRmdegobis Zala. am Zalis dasaZlevad sxeuls uxdeba erTxel miniWebuli energiis ganuwyveteli xarjva, misi Sevseba ki ar xdeba. amis Sedegad, merxevi sxeulis sruli meqanikuri energia da, maSasadame, amplituda, ganuwyvetliv mcirdeba (formula 38.6). r xe v a s ,

k le b a di

am p li tu diT ,

m ile v a di r xe v a e w o de b a. amrigad, Tavisufali rxeva yovelTvis milevadia. 113

gamoviyvanoT milevadi rxevis gantoleba. amisaTvis unda davuSvaT, rom sxeulze, garda F  kx Zalisa, moqmedebs garemos winaaRmdegobis Zala F1 . mcire siCqaris dros winaaRmdegobis Zala siCqaris proporciulia da mimarTulis mis sawinaaRmdegod, e.i. F1  r  r

dx , sadac r - winaRobis koeficientia. dt

niutonis meore kanons am SemTxvevaSi eqneba saxe:

d 2x dx  kx  r , 2 dt dt

(41.3)

d 2x k r dx .  x 2 m m dt dt k r SemoviRoT aRniSvnebi:  02 ,  2 maSin gveqneba: m m d 2x dx .   02 x  2 2 dt dt aseT saxes iRebs rxevis diRerencialuri

(41.4) gantoleba

ganxiluli

SemTxvevisaTvis. imisaTvis, rom es gantoleba daviyvanoT CvenTvis cnobil harmoniuli rxevis (41.1) gantolebaze, SemoviRoT axali cvladi z , romelic

x -Tan Semdegnairadaa dakavSirebuli

x  ze   t .

(41.5)

ganvsazRvroT x -is pirveli da meore rigis warmoebulebi:

d 2x d 2z dz  2  2   2 z e  t . 2 dt dt dt

dx dz  t  e   z e  t , dt dt Tu

x -is,

dx -s da dt

d 2x -is mniSvnelobebs CavsvamT (41.4) gantolebaSi, dt 2

martivi moqmedebebis Semdeg miviRebT:





d 2z   02   2 z . 2 dt davuSvaT, garemos winaaRmdegoba imdenad mcirea, rom

02   2 . maSin

02   2 dadebiTia da SeiZleba aRvniSnoT  2 -iT. e.i. 02   2   2 ,

(41.6)

d 2z   2 z . 2 dt

(41.7)

da sabolood gveqneba:

114

miviReT

harmoniuli

rxevis

diferencialuri

gantoleba.

viciT,

rom

mis

amoxsnas aqvs saxe: z  A0 cost    , sadac A0 mudmivi sididea. z –is am mniSvnelobis Casma (41.5)-Si mogvcems:

(41.8) x  A0 e   t cos t    . (41.8) gamosaxavs damokidebulebas x gadaxrasa da t dros Soris im SemTxvevaSi, roca sxeulze garda drekadi Zalisa, moqmedebs garemos winaaRmdegobis Zala. mas ewodeba m ile v a di

r xe v is

g an to le b a . igi harmoniuli rxevis

(41.2) gantolebisgan imiT gansxvavdeba, rom rxevis amplituda (41.9) A  A0 e   t drois mixedviT eqsponencialurad mcirde-

ba. amplitudis es Semcireba anu rxevis

mileva xdeba miT ufro swrafad, rac metia  

r –is sidide. amitom  -s uwode2m

ben milevis koeficients. (41.9) formulis

Tanaxmad, A  0 , roca t   . sinamdvileSi rxevis mileva xdeba sasrul droSi, romelic miT ufro mcirea, rac metia r

sur. 41.2 40.2

naklebia m .

milevadi rxevis grafiki (e.i. 41.8 gantolebis grafiki) mocemulia 41.2 suraTze   0 . rxevis periodi T  odze T 

2

0

2





2

 02   2

metia sakuTari rxevis peri-

da miT ufro, rac ufro axloa winaRobis  koeficienti rxevis

0 sakuTar sixSiresTan. roca   0 , maSin T   , rac imis maCvenebelia, rom didi winaRobis gamo, rxeva ar warmoebs. cxadia, rxeva ar warmoebs maSinac, roca   0 . marTlac, am dros fesvqveSa gamosaxuleba uaryofiTia da  warmosaxviTia. rxevis milevas axasiaTeben kidev meore sididiT, romelsac milevis logariTmuli dekrementi ewodeba   . igi ganisazRvreba rogorc erTi periodiT daSorebuli ori momdevno amplitudaTa fardobis naturaluri logariTmi: 115

A0 e   t A1   ln  ln  ln e  T  T .   t T  A2 A0 e Tu (41.8)-Si

(41.10)

 -s SevcvliT  -Ti, milevadi rxevis gantoleba miiRebs

saxes:

x  A0 e



t T

cos t    .

 SeiZleba uSualod cdiT ganisazRvros, riTvisac unda gaizomos A1 da A2 . maSin

(41.10)-is

r  2m  2m

 T

saSualebiT

ganisazRvreba

winaRobis

koeficienti:

§42. iZulebiTi rxeva. rezonansi ganvixiloT yvelaze ufro zogadi SemTxveva, rodesac sxeulze, garda F  kx drekadi Zalisa da F1  r

periodulad

cvladi

amplituda,  Tavdapirvelad sixSiriT)

gareSe

dx garemos winaaRmdegobis Zalisa, moqmedebs dt

F2  H cos t.

Zala

aris

cvladi

Zalis

_ misi cvlilebis wriuli sixSire. vinaidan   0 , sxeuli or

rxevaSi

miiRebs

monawileobas:

sakuTar

rxevaSi

( 0

da gareSe ZaliT gamowveul rxevaSi (  sixSiriT). magram sakuTari

rxeva milevadia. amitom garkveuli drois Semdeg (romelsac rxevis damyarebis dro ewodeba) igi Caqreba da darCeba mxolod gareSe ZaliT gamowveuli rxeva



sixSiriT.

aseT

rxevas

iZ u le b iT i

rx e v a

e w o de b a ,

Zalas

maiZulebeli Zala, xolo  -s _ maiZulebeli Zalis sixSire. gamoviyvanoT iZulebiTi rxevis gantoleba. davweroT niutonis meore kanoni:

d 2x dx m 2  kx  r  H cos  t . dt dt tolobis yvela wevri gavyoT m -ze:

d 2x k r dx H   x   cos  t , m m dt m dt 2

116

da SemoviRoT aRniSvnebi:

k r H  02 ;  2 ;  h. m m m

maSin gveqneba:

d 2x dx  02 x  2  h cos  t . 2 dt dt miviReT

iZulebiTi

rxevis

diferencialuri

(42.1) gantoleba.

davuSvaT,

gantolebis amoxsnas aqvs saxe:

x  A cos t    *, sadac

(42.2)

A aris iZulebiTi rxevis amplituda, xolo 

sidideebs

vipoviT

moTxovnidan,

rom

roca

(42.2)

_ sawyisi faza. am

funqcias

CavsvamT

(42.1)

gantolebaSi, igi unda gadaiqces igiveobad. (42.2)-dan dx   A sin  t    dt

x -is,

dx -s dt

d 2x -is dt 2

d 2x   A 2 cos t    . 2 dt

mniSvnelobebi

SevitanoT

(42.1)

gantolebaSi

movaxdinoT cos t    -s da sin  t    -s gaSla**. gveqneba:

 A 2 cos  t  cos   A 2 sin  t  sin    A02 cos  t  cos   A02 sin  t  sin    2A sin  t  cos   2A cos  t  sin   h cos t .

(42.3)

(42.3) gantoleba rom igiveobas warmoadgendes, saWiroa tolobis orive mxares cos  t -sTan da sin  t -sTan mdgomi koeficientebi iyos erTmaneTis toli. es

moTxovna mogvcems or gantolebas:  A 2 cos    A02 cos   2A sin   h an





A cos   02   2  2A sin   h .

(42.4)

da A 2 sin   A02 sin   2A cos 





A sin   02   2  2A cos   0 .

(42.5)

(42.5)-dan

tg  

* **

2 .  02   2

es niSnavs, rom sxeuli asrulebs harmoniul rxevas gareSe Zalis cvlilebis sixSiriT.

cos t     cos  t  cos   sin  t  sin  ; sin  t     sin  t  cos   cos  t  sin  ; 117

(42.6)

amplitudis gasagebad (42.4) da (42.5) gantolebebi aviyvanoT kvadratSi da wevrwevrad SevkriboT. miviRebT



A2 02   2



 A 2 4  2 2  h 2 ,

saidanac

A



02

(42.7)



2 2

  4  Tu sawyisi fazisa da amplitudis miRebul mniSvnelobebs SevitanT 2

(42.2)-Si, miviRebT iZulebiTi rxevis gantolebas:

x



h 2 0

2



 2  . cos  t  arctg 2 2     2 2 0    4 

(42.7) gviCvenebs, rom mocemuli rxeviTi sistemisaTvis ( 0,  fiqsirebulia) iZulebiTi rxevis amplituda damokidebulia maiZulebeli Zalis  sixSireze. gansakuTrebiT sainteresoa

 -s is mniSvneloba, romlis drosac

iZulebiTi rxevis amplituda mkveTrad izrdeba. am sixSires r e z o n a n s u l i s i xS ir e ewodeba, xolo iZ u le b iT i r xe v is a m p li tu dis m k v e T r zr d as , r oc a m aiZ u le b e li Z a lis s ix S ire T an xv de b a re z o n a ns u l s i x Sir e s _ r e z on a ns i. rezonansuli sixSiris gansazRvrisaTvis unda vipovoT (42.7) funqciis maqsimumi, an, rac igivea, am funqciis mniSvnelis fesvqveSa gamosaxulebis minimumi. amisaTvis gamoviyenoT eqstremumis piroba _ gavawarmooT fesvqveSa gamosaxuleba  -Ti da gavutoloT nuls. miviRebT





2 2  4 02  rez .  rez.  8 rez.  0 ,

saidanac

rez.  02  2 2 .

(42.8)

rogorc vxedavT, rezonansuli sixSire ramdenadme naklebia sistemis rxevis

 0 sakuTar sixSireze. roca   0 , maSin rez.  0 . Tu

rezonansuli

sixSiris

miRebul

mniSvnelobas

SevitanT

(42.7)-Si,

miviRebT rezonansuli amplitudis mniSvnelobas: Arez. 

h 2  02   2

118

(42.9)

(42.9) gviCvenebs, rom rac naklebia  , miT metia rezonansuli amplitudis sidide, an rogorc amboben xolme, miT mkveTria rezonansi. 42.1 suraTze mocemulia rezonansuli mrudebi  -s sxvadasxva mniSvnelobebisaTvis. bolos, SevniSnoT, rom bevr SemTxvevaSi, roca xaxuni mcirea, rez. umniSvnelod

gansxvavdeba

imdenad

0 -sagan,

1  0

2

rom

3

rezonansul sixSired SeiZleba TviT 0

4

CavTvaloT. amitom xSirad amboben, rom

1   2   3   4

rezonansisaTvis saWiroa maiZulebeli Zalis

sixSire

Tanxvdes

sistemis

rxevis

sakuTar sixSires. rezonanss

mravalmxrivi

gamoyeneba

aqvs mecnierebaSi da teqnikaSi. magaliTad, eleqtruli

rezonansi

gamoiyeneba

radiomimRebis

rez.



sur. 42.1 sasurvel

sadgurze

`asawyobad~, akustikuri rezonansi _ bgeris analizisa da gaZlierebisaTvis da a.S. rezonass didi zianis motanac SeuZlia. magaliTad, Tu propeleris brunviT gamowveuli rxevis sixSire daemTxva TviTmfrinavis frTebis sakuTar sixSires, aRiZvreba Zlieri vibracia, rasac SeuZlia katastrofis gamowveva.

§43. talRuri moZraoba. talRis gantoleba. talRis energia vTqvaT, drekadi garemos romelime nawilakma daiwyo rxeviTi moZraoba. maSin, vinaidan aseTi garemos nawilakebi erTmaneTTan dakavSirebulni arian drekadi ZalebiT, es rxeva TandaTan gadaecema danarCen nawilakebsac. miviRebT rxevis gavrcelebas drekad garemoSi. rx e v is g av rce le b as dr e k ad g a r e m o Si t alR u ri m o Z r aob a a n t a lR a e w od e b a . ganxilul magaliTSi laparakia drekad talRaze. sazogadod, t a lR a e w od e b a y ov e l i pe r io d u l i pr o c e s i s g am e o re b as s iv rc e S i. talRuri moZraobis dros yoveli nawilaki imeorebs wina nawilakis rxevas, oRond garkveuli dagvianebiT. e s d a gv i a ne b a a nu C am o r Ce n a f az iT gamowveulia imiT, rom erTi nawilaki119

dan meoreze rxevis gadacema xdeba ara meyseulad, aramed garkveuli siCqariT, romelsac talRis gavrcelebis siCqare ewodeba. igi damokidebulia garemos drekad da inerciul Tvisebebze. cxadia, garemos mocemul wertilSi aRZruli rxeva vrceldeba yvela mimarTulebiT, moicavs ra sivrcis axal-axal areebs. im

wertilTa

aRweven,

an,

geometriul rac

igivea,

adgils,

romlebamdisac

romlebic

irxevian

rxevebi

erTnair

erTdroulad

fazebSi,

ta lR i s

z e d a pi r i ewodeba. talRis yvelaze winamdebare zedapirs t a lR i s f r o nt i ewodeba. talRis zedapirebis raodenoba usasrulod bevria, talRis fronti ki _ erTia. talRis zedapiris formis mixedviT asxvaveben brtyel talRas, sferul talRas, elifsur talRas da a.S. kerZod, advili misaxvedria, rom izotropul garemoSi, anu iseTSi, romlis Tvisebebi yvela mimarTulebiT erTnairia, talRis zedapirebi sferuli iqneba. brtyeli talRa, SeiZleba ganvixiloT rogorc sferuli talRa usasrulod didi radiusiT, e.i. iseTi talRa, romlis wyaro usasrulod Sors aris. talRis gavrcelebis mimarTulebas s x iv i ewodeba. sferuli talRis SemTxvevaSi, sxivebs radiusebis mimarTuleba aqvT, brtyeli talRis SemTxvevaSi sxivebi erTmaneTis paraleluri da talRis zedapiris marTobulia. bunebaSi yvelaze ufro xSirad gvxvdeba drekadi talRebi (kerZod, bgera), eleqtromagnituri talRebi (kerZod, sinaTle) da talRebi siTxis zedapirze. yvela talRis ZiriTadi maxasiaTebeli Tviseba, maTi bunebisagan damoukideblad, mdgomareobs imaSi, rom t a lR i s m i e r xd e b a e ne r gi is g a d a t an a

niv T ie re b i s

ga d at a nis

g a re Se .

talRuri

moZraobis

dros

nawilakebi ar gadaadgildebian talRasTan erTad. isini mxolod irxevian wonasworobis

mdebareobis

maxloblad.

amaSi

advilad

davrwmundebiT,

wyalSi, romlis zedapirzec vrceldeba talRa, CavagdebT msubuq sxeuls, vTqvaT, asanTis cariel kolofs. igi ki ar waritaceba talRis mier, aramed daiwyebs rxevas zeviT da qveviT, darCeba ra TiTqmis erTsa da imave adgilze. talRas, romelsac gadaaqvs energia sivrceSi, m s rb ol i t al R a ewodeba, gansxvavebiT m d ga ri talRisagan (ยง45), romlis mier energiis gadatana ar xdeba.

120

drekadi talRa SeiZleba iyos g an iv i an gaswvrivi (grZivi). g a niv i t a lR a e w o de b a is e T t alR a s , ro m e l S ic naw il ak e b is rx e v a w a r m o e b s t a lR i s g av rc e le b is m a rT ob u la d. ganivi talRis magaliTia talRa, romelic vrceldeba simis gaswvriv. drekadi ganivi talRebi aRiZvrebian da vrceldebian mxolod iseT sxeulebSi, romelTa formis Secvla iwvevs maTSi drekadi Zalebis aRZvras (anu gaaCniaT formis drekadoba). aseTebia mxolod myari sxeulebi*. ga s w v r iv i a nu g rZ iv i t a lR a e w o de b a is e T t al R as , r om e l Si a c n aw il ak e b is r xe v a w arm oe b s t alR is g av r ce le b i s g a s w v r iv . gaswvrivi talRis magaliTia bgeriTi talRa. gaswvrivi talRebi aRiZvrebian da vrceldebian iseT sxeulebSi, romelTa moculobis Secvla iwvevs maTSi drekadi Zalebis aRZvras. aseTebia rogorc myari, aseve Txevadi da airadi sxeulebi. drekad talRas ewodeba harmoni-



uli, Tu masSi nawilakebis rxeva harmoniulia

anu

mimdinareobs

sinusis

an c

kosinusis kanoniT. (43.1) suraTze mocemulia harmoniuli ganivi talRa, romelic vrceldeba x RerZis gaswvriv 



 



siCqariT. nawilakebis rxeva warmoebs S RerZis gaswvriv. Tumca warmodgenili

grafiki ( S x, t  funqciisa) analogiuria harmoniuli

rxevis

grafikisa,

maT Soris a r s e b i T i y v e la

sur. 43.1

magram

g a ns xv av e b aa . talRis grafiki gvaZlevs g a r e m o s

n aw i lak is gadaxris damokidebulebas rxevis 0 wyaromde manZilze

drois m oce m u l m om e nt S i , rxevis grafiki ki gvaZlevs e rT i n aw il ak i s gadaxris damokidebulebas droze.

wyalSi qvis Cagdebis Sedegad mis zedapirze warmoSobili talRac ganivi talRaa, magram igi

drekadi

talRa

aris,

vinaidan

ganpirobebulia

zedapiruli daWimulobis ZalebiT.

121

ara

drekadi,

aramed

simZimisa

manZils

uaxloes

nawilaks

  .

fazebSi, talRis sigrZe ewodeba

Soris,

romlebic

irxevian

erTnair

talRis sigrZe tolia im manZilisa,

romelzedac vrceldeba rxevis garkveuli faza T periodis ganmavlobaSi, e.i.

    T , an, vinaidan T  gamovsaxoT



, amitom  

talRuri

 da    , sadac  rxevis sixSirea. 

procesi

analizurad,

anu

miviRoT

talRis

gantoleba. vTqvaT, 0 wertilSi aRiZvra rxeva, romelic warmoebs S RerZis gaswvriv harmoniuli kanoniT S 0  A cos  t . talRis gavrceleba xdeba x RerZis gaswvriv. talRuri procesi daxasiaTebuli iqneba, Tu gvecodineba S gadaxris mniSvneloba

RerZis

nebismieri

wertilisa

drois

nebismieri

momentisaTvis. davuSvaT, 0 wertilSi aRZruli rxeva c wertilSi movida t ' drois Semdeg. maSin Tu 0 wertilis rxevis dawyebidan garkveul momentamde gavida t dro, c wertilis rxevis dawyebidan imave momentamde gasuli iqneba ufro naklebi, kerZod ( t  t ' ) dro. amitom, c

wertilis rxevis gantolebas

eqneba saxe:

S  A cos t  t ' . magram t ' 



, sadac  talRis gavrcelebis siCqarea. amitom

x  S  A cos   t   .   (43.1) gantoleba gvaZlevs wertilisa

drois

(43.1)

gadaxris mniSvnelobas

nebismieri

momentisaTvis.

mas

RerZis nebismieri brtyeli

talRis

gantoleba ewodeba. talRis gantolebas SeiZleba mivceT sxva saxec:

   S  A cos  t  x     sadac

k

 2 2   .  T 

raodenobisa

2

an S  A cos t  kx ,

(43.2)

k -s ewodeba talRuri ricxvi. igi tolia talRaTa

sigrZeze.

talRa

RerZis

sawinaaRmdego

mxares

vrceldeba, (43.2)-Si x Seicvleba  x  -iT: S  A cos t  kx . SeiZleba Cveneba, rom sferuli talRis gantolebas aqvs saxe: S

A cos t  k r  , r

122

(43.3)

saidanac Cans, rom sferuli taRlis amplituda mcirdeba r

radiusis anu

wyarodan manZilis ukuproporciulad. sazogadod, talRis zedapiris gantolebis misaRebad faza (e.i. cos –is argumenti) mudmivad unda CavTvaloT. drois mocemuli momentisaTvis faza x  const , rac

mudmivi iqneba, Tu (43.1) gantolebaSi

RerZis marTobuli

sibrtyis gantolebas warmoadgens. amitom ewodeba (43.1) gantolebas brtyeli talRis

gantoleba.

(43.3)

gantolebis

fazis

mudmivoba

moiTxovs

r -is

mudmivobas: r  const , rac sferos gantolebas warmoadgens. paragrafis dasawyisSi aRniSnuli iyo, rom yoveli talRis ZiriTadi Tvisebaa energiis gadatana nivTierebis gadatanis gareSe _ talRa TiTqosda Tan atarebs energias. drekadi talRis SemTxvevaSi es energia Sedgeba merxevi nawilakebis

kinetikuri

energiisa

drekadad

deformirebuli

garemos

potenciuri energiis jamisagan. mtkicdeba, rom garemos erTeuli moculobis sruli meqanikuri energia, anu energiis simkvrive

  A2 2 sin 2  t  kx ,

(43.4)

sadac  - garemos simkvrivea, A _ rxevis amplituda,  _ wriuli sixSire. rogorc (43.4) formula gviCvenebs, energiis simkvrive drois mixedviT icvleba. amitom SemoaqvT energiis saSualo simkvrivis cneba. vinaidan sin 2  t  kx -is saSualo

mniSvneloba

1 -is 2

tolia,

drekadi

talRis

energiis

saSualo

simkvrivisaTvis gveqneba:

saS.  talRis intensivoba talRis

gavrcelebis

I 

1 2 2 * A  . 2

(43.5)

ewodeba energiis im raodenobas, romelic gadis

mimarTulebis

marTobulad

moTavsebul

farTis

erTeulSi erTi wamis ganmavlobaSi. cxadia, rom farTis erTeulSi drois erTeulSi *

gadaitaneba

energia,

romelsac

Seicavs

marTkuTxa

(43.5) formulis miReba gamartivebuli gziTac SeiZleba, viciT (38.5), rom harmoniulad

merxevi nawilakis sruli meqanikuri energia gamoiTvleba formuliT

E

1 mA2 2 . Tu am 2

gamosaxulebas gavamravlebT molekulebis n raodenobaze moculobis erTeulSi, miviRebT moculobis erTeulis energias anu energiis simkvrives:

  n m

- garemos simkvrivea.

123

1 2

  nE  nmA2 2 

1 2 2 A  , sadac 2

paralelepipedis

moculoba,

romlis

fuZis

farTobi

udris

erTs,

xolo

simaRle talRis gavrcelebis  siCqaris tolia. vinaidan am paralelepipedis moculoba ricxobrivad  -s tolia, amitom I  saS   

1 A 2 2 2

(43.6)

§44. hiugensis principi talRis gavrcelebasTan dakavSirebuli sakiTxebis gadasawyvetad xSirad saWiroa vicodeT talRis frontis agebis meTodi drois raRac momentisaTvis, Tu igi cnobili iyo wina momentisaTvis. es meTodi mocemulia h iu g e ns i s p r in c i p i T , romelic SemdegSi mdgomareobs: s iv r ci s y ov e li w e r ti li , r om e l a m d is ac

aR w e v s

t alR is

s fe r u l i

t al R e b i s

w y a ro.

m om v le b i

w arm o ad ge ns

m om e n t S i.

amasTan,

f r on ti, am

ta lR is

igulisxmeba,

T v iT o n

m e ore u li axal

rom

x de b a

m e or e u l i

s f e ru li

fr on ts

erTgvarovan

t a lR e b i s

dr oi s

m om de v n o

garemoSi

meoreuli

talRebi vrceldeba mxolod win. vTqvaT,

aris talRis fronti drois

momentisaTvis (sur.44.1).

imisaTvis, rom avagoT talRis fronti drois momdevno

t  t 

momentisaTvis,

talRis frontis yoveli wertilidan, rogorc centridan, unda SemovxazoT r    t

radiusis

naxevarsfero

(vinaidan talRebi win vrceldeba,

meore

naxevris Semoxazva ar aris saWiro), sadac  talRis gavrcelebis siCqarea. am naxevarsferuli zedapirebis momvlebi iqneba talRis axali

A' B'

fronti

drois t  t  momentisaTvis. hiugensis

principis

naTelsayofad

SeiZleba

Semdegi

cdis

Catareba.

oTxkuTxa WurWelSi CavasxaT wyali da mis romelime 0 wertilSi dartymis saSualebiT aRvZraT talRa. maSin 0 wertilidan gavrceldeba koncentruli wriuli talRebi (sur. 44.2, mocemulia zedxedi). gadavtixroT WurWeli vertikaluri

ekraniT, romelsac aqvs viwro

xvreli da cda gavimeoroT.

vnaxavT, rom talRebi gavrceldeba B ekranis meore mxaresac, amasTan, maTi centri iqneba ara 0 wertili, aramed c wertili. amrigad, cda adasturebs,

124

rom c xvrelSi myofi wylis nawilakebi, movidnen ra rxevaSi, TviTon gaxdnen talRaTa axali sistemis centri. hiugensis principi zogadi principia. misi saSualebiT SeiZleba talRaTa B

B' t  t 

Bt 

o

sur. 44.2

sur. 44.1

arekvlisa da gardatexis kanonebis miReba, ormagi sxivTtexis axsna da sxva. miviRoT, magaliTad, arekvlis kanoni. davuSvaT, MN

amreklav

zedapirs ecema brtyeli talRa

AB . mas Seesabameba para-

leluri

sxivebi,

romlebic

kuTxes (sur. 44.3), romel-

sac

dacemis

Cvens

kuTxe

suraTze

talRis

i1 

i2 N

ewodeba.

sur. 44.3

gamoyofilia

sami sxivi. hiugensis principis Tanaxmad, masze

i1 

zedapiris normalTan adgenen

frontis

miRwevis

Semdeg,

zedapiris yoveli wertili,

TviTon

gadaiqceva

meoreuli

naxevarsferuli talRis centrad. cxadia, rom vidre AB frontis B wertili miaRwevs MN zedapiris c wertils, A wertilSi aRiZvreba da imave garemoSi gavrceldeba naxevarsferuli talRa, romlis radiusi AD iqneba BC -s toli. MN zedapiris sxva wertilebidan imave momentisaTvis gavrceldebian ufro

naklebi

radiusis

zedapirebis

mqone

momvlebi

naxevarsferuli

warmoadgens

talRebi.

areklili

naxevarsferuli

talRis

fronts.

mas

Seesabameba areklil sxivTa paraleluri kona, romelic zedapiris normalTan adgens

ADC  ABC

kuTxes.

mas

gverdebis

arekvlis tolobis

kuTxe gamo.

BAC  i1 , xolo DCA  i2 e.i. 125

ewodeba. maSasadame,

suraTidan

Cans,

BAC  DCA .

rom

magram

i1  i2

(44.1)

miRebuli toloba gamosaxavs arekvlis pirvel kanons, romlis Tanaxmadac, d a c e m is k u T x e are k v li s k u T x is t ol i a. garda amisa, agebidan gamomdinareobs, rom d ace m u l i s xiv i, ar e k v l il i s x iv i d a d ace m is w e r til S i a R m a rT u l i pe r pe n d ik u l ar i e r T s ir ty e Si m de b are ob s . es arekvlis meore kanonia.

§45. talRebis interferencia. mdgari talRebi. cdebi

gviCvenebs,

rom

sxvadasxva

wyaroebidan

gamosuli

taRlebi

vrceldebian erTmaneTisagan damoukideblad, e.i. TiToeuli talRa vrceldeba ise, rogorc gavrceldeboda, sxva talRebi rom ar yofiliyo. talRaTa gavrcelebis damoukideblobis princips yvelaze martivad davakvirdebiT wylis zedapirze tal



Rebis gavrcelebisas. Tu wyalSi CavagdebT or qvas da amiT SevqmniT or wriul talRas, SevamCnevT, rom TiToeuli talRa gaivlis

sur. 45.1

meoris

SigniT ise, TiTqos meore talRa saerTod ar arsebobda (sur. 45.1).

vinaidan talRebi vrceldebian erTmaneTisagan damoukideblad, anu isini ar urTierTqmedeben, amitom, Tu erT talRaSi garemos romelime wertilis  gadaxra aris S1 , es gadaxra am talRaSi ar Seicvleba, Tu amave garemoSi meore talRac gavrceldeba. aqedan gamodis, rom Tu garemoSi vrceldeba ori talRa, maSin maTi gadafarvis areSi SedegiTi gadaxra    S  S1  S 2

(45.1)

(45.1) toloba gamosaxavs talRaTa superpoziciis (anu zeddebis) princips: T u

ga re m oS i

v rce l de b a

o ri

( an

r am de n im e )

t al R a,

m a Si n

g a r e m os ne b is m ie r i naw il ak is Se de giT i g a da x ra t ol ia c a lk e u l i t a lR e b is S e s ab am is i g a d ax re b i s v e q t oru li j am is . Tu rxevebi erTi wrfis gaswvriv warmoebs, veqtoruli jami gadaiqceva algebrul jamad, e.i. moxdeba

amplitudebis

Sekreba

gamokleba,

Sesusteba. 126

anu

rxevebis

gaZliereba

t a lR e b is S e s u s te b is

ze dd e b is

m ov le n a s

d ro s

ta lR e b is

m aT i

u rT ie rT g aZ l ie r e b is a

i nte rf e re nc ia

e w o de b a.

gamoviyva-

noT talRebis gaZlierebisa da Sesustebis piroba, an, rogorc amboben xolme, interferenciuli suraTis maqsimumisa da minimumis piroba. vTqvaT, o1 da o2 toli

sixSiris

wertilebidan vrceldeba

erTmxriv

mimarTuli

(sur. 45.2), romelTa gantolebebia

S1  A1 cos t  kx1  ; S 2  A2 cos t  kx2  . ganvixiloT wertilSi.

39-e

Sesabamisad,

 A  A1  A2  ,

A 

maTi

paragrafSi

talRebi Tu

zeddeba

fazaTa

erTmaneTs sxvaoba

(45.2)

nebismier

miRebuli

talRebi

Sedegebis



sur.45.2

gaaZlierebs

 2  1  2n

erTmaneTs

Seasustebs

A1  A2  , Tu  2  1  2n  1 , sadac n  1, 2, 3, ....; n -s ewodeba interferenciis

rigi. magram (45.2)-dan 1  kx1 da  2  kx2 , amitom

 2  1  k x1  x2  

2



.

  x1  x2 sidides ewodeba talRebis s v l aT a s xv aob a . rogorc suraTidan Cans, i g i t oli a t a lR e b is m ie r S e xv e d ram de gav li li m an Z i le b is s xv aob is a. svlaTa sxvaobis SemoRebis Semdeg gaZlierebis anu maqsimumis piroba miiRebs saxes:   n  2n



2 xolo Sesustebis anu minimumis piroba iqneba:

  2n  1

(45.3)



. (45.4) 2 cxadia, talRebis gadafarvis areSi aRmoCndeba uamravi wertili, romelTaTvisac sruldeba rogorc (45.3), ise (45.4) piroba. amrigad, miRebuli Sedegi SeiZleba ase CamovayaliboT: e r T n a iri s i x Si ris t alR e b i ze d d e b is a s e rT m a ne T s aZ li e r e b e n a nu m iiR e b a in te r fe re n ciu l i s u ra T is m aq s im u m i s iv rc is i m w e r t ile b S i, r om e lT a T v is a c t al R e b s S or is s v l aT a s xv aob a t al 127

R is

s i grZ is

a s u s te b e n, rcis

n ax e v r is

an u

m i iR e b a

w e rt ile b S i,

luw i

ri c xv is

t ol ia,

int e r fe re n ciu li rom e lT a T v i s ac

x ol o

s u r aT is

t alR e b s

e r T m a ne T s

m in im u m i S or is

siv -

s v l aT a

s xv aob a t al R is s ig rZ is n ax e v r is k e n ti r ic xv i s to li a. svlaTa sxvaobis nebismieri sxva mniSvnelobebis dros SedegiTi rxevis amplitudas Sualeduri mniSvnelobebi eqneba, e.i. moTavsebuli iqneba

A1  A2 -

A1  A2 -s Soris.

sa da

mdgradi (dakvirvebadi) interferenciuli suraTis misaRebad saWiroa talRuri procesis SeTanxmebuli mimdinareoba: mocemul wertilSi talRebi erTmaneTs

mudam

unda aZlierebdnen an mudam

asustebdnen,

anu

mocemul

wertilSi amplituda drois mixedviT ar unda icvlebodes. amas ki, rogorc (39.3)

formula

fazaTa

sxvaoba

s i xS ir e e b i

gviCvenebs,

adgili

eqneba

drois

mixedviT

e rT n a ir ia ,

x olo

mxolod maSin, icvleba.

fa z aT a

roca

t al R e b s ,

s xv aob a

talRebis r om e l T a

m u dm iv i,

k oh e r e n -

t u l i * ta lR e b i e w ode b a, x ol o S e s ab am is w y ar oe b s _ k oh e re ntu l i w y ar oe b i .

maSasadame,

interferenciuli

suraTis

misaRebad

aucilebelia

koherentuli talRebis zeddeba. interferenciis erT-erTi mniSvnelovani magaliTia Sekreba ori brtyeli talRisa, romlebic urTierTSemxvedri mimarTulebiT vrceldeba (magaliTad, pirdapiri da raime dabrkolebidan areklili talRebi). am dros miiReba e.w. m d ga r i talRa, romelic k v a nZ e b iT a da b u r c ob e b iT xasiaTdeba. miviRoT mdgari talRis gantoleba. davuSvaT x RerZis pirdapiri da sawinaaRmdego mimarTulebiT vrceldeba ori erTnairi sixSirisa da amplitudis talRa. maTi gantolebebia

S1  A1 cos t  kx

da S 2  A1 cos t  kx .

SedegiTi talRis gantoleba miiReba S1 -isa da S 2 -is SekrebiT, romelsac cos is argumentis gaSlis Semdeg aqvs saxe:

S  S1  S 2  2 A1 cos kx  cos  t *

(45.5)

warmodgeba laTinuri sityvisagan da niSnavs kavSirSi myofs (erTmaneTTan dakavSirebuls).



Tu talRebi koherentuli ar aris, SedegiTi rxevis amplituda drois mixedviT icvleba. talRebis gadafarvis wertilebSi adgili aqvs gaZlierebisa da Sesustebis swraf monacvleobas da interferenciuli suraTis dakvirveba SeuZlebeli xdeba.

128

miRebuli

gantoleba

saSualebas

gvaZlevs

ganvsazRvroT

talRis

nebismieri x wertilis mdebareoba drois nebismier t momentSi. mas mdgari talRis gantoleba ewodeba. igi gviCvenebs, rom nawilakebis rxeva warmoebs imave  sixSiriT, xolo amplituda (45.6) A  2 A1 cos kx x koordinatis perioduli funqciaa. maSasadame, x wrfeze moiZebneba iseTi wertilebi,

romelTa

amplituda

nulis

tolia

iseTebic,

romlebic

irxevian udidesi amplitudiT. t a lR i s im w e rt il e b s , r om e lT a r xe v is am pl itu d a n u li s t o l i a ( A  0 ) , k v a n Z e b i e w od e b a . kvanZebis koordinatebs advilad miviRebT. marTlac, A  0 , roca cos kx  0 , anu kx  2n  1



xkv  2n  1



. magram k 

2



, amitom

(45.7) 4 m d ga ri ta R l is im w e rt ile b s , r om l e b i c ir xe v a u di de s i am p l i tu d iT

 A  2A1  ,

b u rco b e b i e w o de b a. (45.6)-is Tanaxmad, burcobebisa-

Tvis cos kx  1 , anu kx  n , amitom xb  n



 2n



(45.8) 2 4 (45.7) da (45.8) tolobebidan gamomdinareobs, rom manZili or mezobel kvanZs, an or mezobel burcobs Soris

 -is, xolo manZili kvanZsa da burcobs Soris 2

 -is tolia. 4 m an Z i ls or m e z ob e l k v a nZ s a n o r m e zob e l b u rc ob s S or is m d ga r i t alR is s i grZ e e w o de b a . amrigad, mdgari talRis sigrZe msrboli talRis sigrZis naxevris tolia. mdgari talRa miiReba, Tu zonris erT bolos kedelze uZravad davamagrebT, xolo meore bolos moviyvanT periodul rxevaSi. zonarze gavrceldeba pirdapiri (mTliani wiri) da damagrebuli

bolodan areklili (wyvetili wi-

ri) talRebi, romelTa SekrebiT (interferenciiT) miiReba mdgari talRa (sur.



gavixsenoT, rom harmoniuli rxevis amplituda ganisazRvreba cos  t -s win mdgomi droze damoukidebeli sididiT.

129

k1  .

45.1). arekvlis adgilas gveqneba pirveli kvanZi Soris, SuaSi, moTavsebuli iqneba mniSvnelovania

aRiniSnos,

rom

B1 , B2 , B3 ,

TiTqos

k1 , k 2 , k 3 , da a.S. kvanZebs

da a.S. burcobebi da piriqiT.

(45.5)

formulis

Tanaxmad,

talRis yvela wertili unda irxeodes erTnairi fazebiT ( cos  t

mdgari ar aris

damokidebuli x -ze). sinamdvileSi es ase ar aris: drois mocemul momentSi erTnairi faza aqvs or mezobel kvanZs Soris

moTavsebul

yvela

wertils

( cos  yvelasaTvis erTnairia), erTi da

imave kvanZis sxvadasxva mxares mdebare



wertilebi ki irxevian sawinaaRmdego

fazebSi (fazaTa sxvaoba  -s tolia),

radgan

cos kx

icvlis

niSans

nulze

gavlisas kvanZis wertilSi. mdgari talRa arsebiTad gansxvav-

sur. 45.1

deba msrboli talRisagan: 1. msrbol talRaSi yvela wertili erTnairi amplitudiT

irxeva. mdgar

talRaSi ki sxvadasxva wertili _ sxvadasxvanairi amplitudiT; 2. msrbol talRaSi sxvadasxva wertili sxvadasxva faziT irxeva, mdgar talRaSi ki erTi kvanZidan meoremde yvela wertili erTnair fazaSi irxeva; 3. msrbol talRas gavrcelebisas Tan gadaaqvs energia sivrceSi. mdgari talRis SemTxvevaSi ki energiis gadatanas ara aqvs adgili (amitom hqvia mas mdgari talRa), vinaidan or msrbol talRas (pirdapirs da areklils), romelTa

SekrebiTac

miiReba

mdgari

taRla,

gadaaqvT

toli

sididis

energia urTierTsawinaaRmdego mimarTulebiT. Cvens mier ganxilul magaliTSi arekvlis wertilSi miiReba kvanZi. es gasagebicaa, vinaidan kedeli ar aZlevs zonris damagrebul bolos gadaadgilebis saSualebas. sazogadod, sazRvarze, sadac xdeba talRis arekvla, SeiZleba warmoiSvas rogorc kvanZi, ise burcobi. Tu garemo, romelSic talRa vrceldeba,

metad

mkvrivia,

vidre

garemo,

romelic

talRas

areklavs

(magaliTad, myari sxeuli_haeri), maSin sazRvarze miiReba burcobi, winaaRmdeg SemTxvevaSi ki _ kvanZi.

130

VIII Tavi

s i T xi s a d a a i r i s me q an i k a $46. hidroaerostatikis ZiriTadi kanonebi meqanikis nawils, romelic Seiswavlis siTxeebsa da airebs, h i dr o ae r om e q anik a ewodeba. igi ori nawilisagan Sedgeba: h idro ae ros tatik a d a h idr o ae r odinam ik a.

pirveli Seiswavlis siTxis da airis wonasworobas,

xolo meore _ siTxis da airis moZraobas, agreTve maT urTierTqmedebas erTmaneTTan da myar sxeulebTan. hidroaeromeqanikis mTavari amocanaa Txevad da airad garemosa da masSi moZravi an uZravi sxeulebis urTierTqmedebis kanonzomierebis dadgena. siTxeebis da airebis damaxasiaTebeli Tviseba, rac ganpirobebulia, maTi nawilakebis moZraobis xasiaTiT ($92) aris d e nad ob a

e.i. Zvris deforma-

ciisadmi mcire winaaRmdegobis gaweva. siTxisa da airis fenebis erTmaneTis mimarT gadanacvleba ar iwvevs drekadi Zalebis warmoqmnas, ris gamoc siTxeebsa da airebs, myari sxeulisagan gansxvavebiT ar gaaCniaT formis drekadoba (ar gaaCniaT sakuTari forma) – isini advilad iReben im WurWlis formas, romelSic

arian

moTavsebuli.

rac

Seexeba

moculobiT

drekadobas,

igi

damaxasiaTebelia rogorc siTxis, aseve airisaTvis, Tumca siTxeSi igi bevrad metia _ siTxe rom mcired SeikumSos, masze sagrZnobi Zalis moqmedebaa saWiro. amis gamo iTvleba, rom siTxe praqtikulad ukumSvelia. siTxis (an airis) SekumSvisas warmoiSoba drekadi Zalebi, romlebic awonasworeben deformaciis gamomwvev Zalas. siTxisa da airis drekadi Tvisebebi mJRavndeba imiT, rom maTi calkeuli nawilebi moqmedeben erTmaneTze an (maTTan) SexebaSi myof sxeulebze ZaliT, romelic damokidebulia SekumSvis xarisxze. es moqmedeba xasiaTdeba fizikuri sididiT, romelsac wneva ewodeba.



miuxedavad didi kumSvadobisa cda da Teoria gviCvenebs, rom airic (siTxis msgavsad) SeiZleba CaiTvalos ukumSvelad, Tu misi moZraobis siCqare mcirea bgeris siCqaresTan SedarebiT (=333m/wm=1200k/sT!). amitom, Tu es piroba sruldeba, siTxis moZraobis ganxilvisas miRebuli Sedegebi marTebuli iqneba airisaTvisac.

131

ganvixiloT siTxe (airi) romelic imyofeba wonasworobaSi. es niSnavs, rom misi calkeuli nawilebi ar gadaadgildeba erTmaneTisa da mosazRvre sxeulebis mimarT.

Tu aseT

siTxeSi movaTavsebT Txel firfitas,

masze,

orientaciis miuxedavad, yoveli mxridan imoqmedebs zedapiris marTobulad mimarTuli Zalebi (sur. 46.1), vinaidan mxebi Zalebis moqmedeba gamoiwvevda misi nawilebis (nawilakebis) moZraobas. wneva ganimarteba rogorc zedapirze marTobulad moqmedi Zalis fardoba zedapiris farTobTan ($54). vnaxoT,

rogor

aris

ganawilebuli

wneva

wonasworobaSi myof (uZrav) siTxeSi. jer

ganvixiloT

s ak u T ar i lo b a Si rivad

w oniT ar

cilindri,

romlis

farTobi

(sur.

daSorebul

roca

s iT xis

w ne v a

m xe d v e -

F

gamovyoT siTxeSi azrob-

F

gam ow v e u li

m iiR e b a.

nebismierad

SemTxveva,

orientirebuli simaRlea 46.2).

wertilebSi

mcire

zomis

xolo

fuZis

vTqvaT, wneva

S

F

S

F

manZiliT

gansxvavdeba

p

sur.46.1

sididiT. maSin cilindris gaswvriv moqmedebs Zala

pS , rac gamoiwvevs siTxis moZraobas da wonasworoba dairRveva. maSasadame, siTxis wonasworobis dros, roca siTxis sakuTari woniT gamowveuli wneva ugulvebelyofilia, siTxis nebismier adgilas unda Sesruldes piroba saidanac

p  0, l

p  const. e.i. wonasworobis dros wneva siTxis mTel moculobaSi

mudmivia (erTnairia). metad mniSvnelovania sakiTxi imis Sesaxeb, Tu

p  p

rogor gadascems siTxe an airi maTze warmoebul wnevas. frangma mecnierma paskalma daadgina: s iT xe ze u c v le lad

airze

g adae ce m a

w arm oe b u li

y v e la

w ne v a

m i m ar T u le b iT

(siTxis an airis yovel wertils). paskalis kanoni, romlis Semowmeba SeiZleba

pS l

sur. 46.2 sur.



46.2 aerTianeben siTxesac siTxisa da airis bevri saerTo Tvisebis gamo, xSirad terminSi `siTxe~ da airsac. 132

uamravi (maT Soris Zalian martivi) cdiT, advilad aixsneba nawilakebis didi ZvradobiT:

dguSian

WurWelSi

moTavsebuli

airi

(an

siTxe)

imyofeba

garkveuli wnevis qveS. airis molekulebi Tanabradaa ganawilebuli WurWlis mTel moculobaSi (sur. 46.3 a). Tu dguSs swrafad davawvebiT, SeikumSeba uSualod mis qvemoT moTavsebuli airi da am adgilze molekulebi ufro mWidrod ganlagdeba (sur. 46.3 b). magram didi Zvradobis gamo, molekulebi yvela mimarTulebiT gadaadgildebian, ris gamoc isini kvlav Tanabrad, magram winandelze ufro mWidrod ganlagdebian (sur. 46.3 g), amitom airis wneva yvelgan (yvela wertilSi) erTnairad gaizrdeba. Tu magaliTad airis wneva uSualod dguSTan 1 paskaliT gaizarda, amdeniTve gaizrdeba wneva airis yvela

wertilSi

WurWlis kedlebzec. paskalis kanoni udevs safuZvlad

udevs

sxvada-

sxva hidravlikuri mowyobilobebis

(hidravlikuri

wnexi, hidravlikuri doma

b sur. 46.3

s im Z im is

Z alis

krati da sxv.) moqmedebis princips.

gaT v alis w ine b is

Se m T xve v aSi, cxadia, siTxis mTel

moculobaSi wnevis mudmivobis piroba darRveva, vinaidan siTxis zeda fenebi awveba qveda fenebs, rac warmoSobs damatebiT wnevas (gare Zalebis moqmedebis Sedegad siTxis SekumSviT gamowveuli wnevis garda). am SemTxvevaSi siTxis moculobaSi wnevis ganawilebis dasadgenad siTxeSi azrobrivad gamovyoT mcire S ganiveTiani h simaRlis cilindri, romelsac horizontaluri AB mdebareoba

ukavia

(sur. 46.4).

vinaidan

simZimis

Zala

vertikalurad

aris

mimarTuli, cilindris RerZis gaswvriv fuZeebze imoqmedebs mxolod ori Zala: FA da FB . siTxis wonasworobidan gamomdinareobs, rom FA = FB . magram

FB  PB S

da FA  PA S . amitom PA  PB . maSasadame, wonasworobis dros erT



cilindris gverdiT zedapirze moqmedi wnevis Zalebis gegmilebi RerZze (cilindris nebismieri orientaciis dros) yovelTvis nulis tolia.

133

doneze mdebare siTxis yvela wertilSi (anu horizontaluri sibrtyis yvela wertilSi) wneva erTnairia. gadaviyvanoT cilindri vertikalur CD mdgoma-

reobaSi.

Zalebis

garda

simZimis

h D FA

SemTxvevaSi

RerZis

imoqmedebs

`siTxis

F  mg  Vg  Shg ,

Zala

simkvrivea,

xolo

moculoba.

V  Sh

Zala

gaswvriv

sadac

siTxis

modebulia

wnevis

cilindris~



siTxis

cilindris

qveda

fuZeze da mimarTulia qveviT, amitom wonasworobis pirobas eqneba saxe:

FC  F  FD , an pC  S  ghS  p D S ,

sur. 46.4

saidanac p D  pC  gh

gh aris simZimis Zala siTxis h simaRlis svetisa, romlis ganikveTis farTobi erTis tolia. maSasadame, w ne v a s iT xis or s xv a das xv a done s Sor is g ans xv av de b a s i didi T , rom e lic am done e b s Soris m o T av s e b uli e r T e u lov ani gan ik ve T is s iT xis v e rtik alu ri s v e tis s im Z imis Z alis tol ia. Tu CD cilindrs vertikalurad avwevT ise, rom C aRmoCndes siTxis Tavisufal zedapiridan

zedapirze, h

siRrmeze

maSin

pC  0 

mdebare

nebismier

p D  gh . wertilSi

maSasadame, wneva

siTxis

gamoiTvleba

formuliT

p  gh

(46.1)

WurWlis forma da ganikveTi saerTod ar Sedis (46.1) formulaSi, rac niSnavs imas, rom simZimis ZaliT gamowveuli wnevis zom a siTxis nebismier wertilSi aris m xolo d v e rtik aluri m anZ ili ( h ) am wertilidan siTxis zedapiramde. raodenobrivad igi ganisazRvreba (46.1) formuliT. paskalis kanonis Tanaxmad, amave formuliT gamoiTvleba wneva imave doneze mdebare WurWlis kedlis wertilebSic. Tu cilindris zeda fuZe



atmosferul wneva mxedvelobaSi ar viRebT.

134

siTxis zedapirzea, xolo qveda fuZe WurWlis fskerze, maSin (46.1) formula gamosaxavs siTxis wnevas WurWlis fskerze. siTxis

simZimis

ZaliT

gamowveul

wnevas

h idros t atik u ri

w ne v a

ewodeba. simZimis velSi wnevis ganawilebis yvelaze cnobili Sedegia arqimedes am om dge b i

Z ala , romelic moqmedebs siTxeSi CaSvebul sxeulze. kanoni,

romelic gansazRvravs amomgdebi Zalis sidides, daadgina arqimedem da mis saxels atarebs. kanonis Camosayalibeblad simartivisaTvis davuSvaT, rom siTxeSi CaSvebul sxeulze aqvs marTkuTxa paralelepipedis forma. misi fuZis farTobi h1

aRvniSnoT

-iT, simaRle

-iT, xolo manZili

zeda fuZidan siTxis zedapiramde F1

-iT (sur.

46.5). paralelepipedze moqmedi wnevis Zalebis h

tolqmedi tolia mis gverdiT waxnagebze da fuZeebze

moqmedi

wnevis

Zalebis

jamisa.

gverdiT waxnagebze moqmedi Zalebi sididiT S

toli da sawinaaRmdegodaa mimarTuli, amitom F2

isini

0 wnevis

Zala

abaTilebs.

zeda

moqmedi wnevis Zala F1  p1 S  0 gh1 S ,

sur. 46.5

moqmedi

erTmaneTs

siTxis

F2  p2 S  0 g h1  hS .

simkvrivea, am

xolo

Zalebis

qveda

tolqmedi

fuZeze sadac fuZeze zeviTaa

mimarTuli da mas arqimedes amomgdebi Zala ewodeba. misi sidide F  F2  F1  0 gS h1  h - h1   0 gSh  0 gV

(46.2)

magram V  Sh - aris paralelepipedis moculoba, xolo  0 g - mis mier gamodevnili siTxis wona. amrigad, arqimedes kanoni Semdegnairad Camoyalibdeba: s iT xe S i (a n air Si) Ca Sv e b u l s xe u lze m oq m e deb s am om gde b i Z al a, r om e lic s xe u lis m ie r gam ode v nili s iT xis (an airis ) w onis toli a.

igi

m im arT u lia

v e rtik alu rad

g am ode v nili m ocu lob is s im Z im is ce ntrSi.

135

ze v iT

m ode b u lia

zogjer am amomgdeb (`amwev~) Zalas, siTxeSi sxeulis moZraobisas warmoqmnili amwevi Zalisagan gansxvavebiT h idros tatik u r (ae ros t atik ur ) am w e v Z alas uwodeben. Tu sxeuli siTxeSi nawilobrivaa CaZiruli, amomgdebi Zala CaZiruli nawilis mier gamodevnili siTxis wonis tolia. paskalis da arqimedes kanonebi hidroaeromeqanikis ZiriTadi kanonebia. amrigad, siTxeSi CaSvebul sxeulze moqmedebs ori Zala: romelic vertikalurad qveviTaa mimarTuli da

simZimis Zala,

amomgdebi Zala, romelic

vertikalurad zeviTaa mimarTuli. aq SeiZleba adgili qondes Semdeg SemTxvevebs: 1.

P  F - am dros sxeuli iZireba.

P  F - sxeuli amodis siTxis zedapirze, tivtivebs.

P  F - sxeuli `livlivebs~ anu SeuZlia wonasworobaSi iyos

siTxeSi nebismier siRrmeze. Tu sxeulis saSualo simkvrive naklebia siTxis simkvriveze, maSin sxeulis nawili amoweulia (amoSverilia) siTxis zedapiridan da miT ufro, rac metia gansxvaveba simkvriveebs Soris. am SemTxvevaSi amboben, rom sxeuli curavs.

§47. idealuri siTxe. siTxis stacionaruli dineba. uwyvetobis gantoleba hidrodinamika aris

hidroaeromeqanikis

nawili,

romelic Seiswavlis

siTxis moZraobas da zemoqmedebas mis mier garsSemodenil sxeulebze. siTxis moZraobas d in e b a ewodeba, xolo moZravi siTxis nawilakebis erTobliobas _ n ak a di (W av l i ). miuxedavad imisa, rom siTxisa da airis Tvisebebi arsebiTad gansxvavdeba erTmaneTisagan, rogorc wina paragrafSi iyo aRniSnuli, maT bevri saerTo Tvisebac aqvT, romelTagan mTavaria is, rom isini xasiaTdebian

sxeulis curva advili asaxsnelia: Tu sx.  siTx. , arqimedes Zalis moqmedebiT, romelic metia simZimis Zalaze, sxeuli iwyebs zeviT moZraobas. roca misi nawili amoiwevs siTxis zedapiridan, arqimedes Zala Semcirdeba, vinaidan Semcirdeba gamodevnili siTxis moculoba. sxeulis amoweva siTxidan maSin Sewydeba, roca sxeulis CaZiruli (siTxeSi darCenili) nawilis mier gamodevnili siTxis wona gautoldeba sxeulis simZimis Zalas. 

136

moculobis drekadobiT da ar gaaCniaT formis drekadoba: rogorc siTxis, ise airis SekumSvisas, maTSi aRiZvreba mniSvnelovani sididis drekadi Zalebi, maSin, rodesac fenebis urTierTgadanacvlebisas (`Zvris deformacia~) aseTi Zalebi ar aRiZvreba. amitomaa, rom siTxec da airic advilad iRebs im WurWlis

formas,

romelSic

arian

moTavsebulni.

marTalia,

siTxe

praqtikulad ukumSvelia, airi ki advilad ikumSeba, magram mtkicdeba, rom airis dinebisas misi kumSvadoba SeiZleba ugulebelvyoT, T u k i s i Cq ar e

m c ire a

b ge ris

s i Cq a re s T a n

Se d are b iT .

d in e b i s

saerTo Tvisebebis

arseboba ganapirobebs imas, rom Tu sruldeba zemoT moyvanili piroba, siTxis moZraobis ganxilvisas miRebuli Sedegebi marTebulia airisTvisac. siTxisa da airis fenebis urTierTgadaadgilebisas aRiZvreba Zalebi, romlebic fenebisadmi mxebadaa mimarTuli. magram es Zalebi, myari sxeulis Zvris

deformaciis

dros

aRZruli

mxebi

Zalebisagan

gansxvavebiT,

damokidebulia ara erTi fenis meoris mimarT wanacvlebis sidideze, aramed fenebis fardobiT siCqareze da xdeba nulis toli, rogorc ki es siCqare gautoldeba nuls. es Zalebi myari sxeulis moZraobisas warmoqmnili srialis xaxunis

Zalebis

analogiuria

maT

Sinagani

xaxunis

Zalebi

ewodeba.

Sinagani imitom, rom isini moqmedeben siTxis SigniT. siTxis moZraobis ganxilvisas, sakiTxis gamartivebis mizniT, SemoaqvT i de a l u r i s i T x is cneba. e s i s e T i s iT xe a, r om e l ic ar ik u m Se b a d a r om l is f e ne b s So ri s a c a dg il i a r a a q v s x ax u ns . mkacrad Tu vimsjelebT, idealuri siTxe bunebaSi ar arsebobs, magram aseTi modelis Semotana gamarTlebulia imiT, rom garkveuli SesworebiT aq miRebuli Sedegebi advilad vrceldeba realur siTxezec. garda amisa, nebismieri siTxe praqtikulad ukumSvelia. rac Seexeba Sinagani xaxunis Zalebs, isini yovelTvis arsebobs, magram Tu es Zalebi mcirea siTxeSi moqmed sxva ZalebTan (magaliTad, simZimis an wnevis ZalebTan) SedarebiT, SeiZleba isini ugulebelvyoT. siTxis moZraobis Teoriuli Seswavlis ori meTodi arsebobs _ lagranJis da eileris. pirveli meTodi gulisxmobs siTxis TiToeuli nawilakis moZraobis Seswavlas, anu misi siCqaris, aCqarebisa da traeqtoriis gansazRvras, rac did maTematikur siZneleebTanaa dakavSirebuli. meore meTodi gulisxmobs imis dadgenas, Tu ra xdeba s iv r ci s y ov e l w e rt il Si drois mixed137

viT, anu igi gulisxmobs sivrcis yovel wertilSi gamavali siTxis calkeuli (sxvadasxva) nawilakis siCqaris veqtoris gansazRvras drois yovel momentSi. sivrcis yvela wertilisaTvis gansazRvruli siCqaris veqtorTa erToblioba qmnis siCqareTa veqtorebis vels. am velis grafikuli gamosaxvisaTvis SemoaqvT de nis

w i re b is cneba. es iseTi wirebia, romelTa yovel wertilSi

gavlebul mxebs aqvs am wertilSi arsebuli siCqaris mimarTuleba (sur. 47.1). denis wirebs avleben ise, rom maTi saSualebiT gaigon siCqaris ara marto mimarTuleba,

aramed

sididec.

wirebisadmi

marTobulad

kerZod, dayenebul

farTis erTeulSi avleben imden wirs, risi tolic

aris

adgilas

siCqare.

maSasadame, iq, sadac siCqare didia, denis wirebi gaivleba ufro xSirad da piriqiT.

sur. 47.1

siTxis d am y a re b u l i a nu s t a c io n a r u l i dineba ewodeba iseT dinebas, romlis drosac mis yovel wertilSi nawilakTa

siCqare

icvleba

(e.i.

mocemul

wertilSi

gamaval

siTxis

sxvadasxva nawilaks erTnairi siCqare aqvs). stacionaruli dinebis dros denis wiris forma ucvlelia da igi Tanxvdeba nawilakis moZraobis traeqtorias. gamovyoT

siTxeSi

nebismieri

Sekruli

konturi

misi

TiToeuli

wertilidan gavavloT denis wirebi. am wirebis erToblioba SemosazRvravs siTxis nawils, romelsac de ni s m ili ewodeba (sur. 47.2).

ď ľ1

ď ľ2 sur. 47.3

sur. 47.2

vinaidan nawilakebis siCqare mxebia denis milis zedapirisa, stacionaruli dinebis dros siTxis verc erTi nawilaki ver gamova denis milidan da verc Seva masSi. e.i. siTxe ise moZraobs denis milSi, rogorc myarkedlebian milSi.

138

davuSvaT, mocemuli gvaqvs cvladi ganivkveTis denis mili (sur. 47.3). gamovyoT masSi ori kveTi, romelTa farTobebia S1 da S 2 xolo dinebis siCqareebi 1 da  2 (vinaidan siTxe idealuria, milis mocemul ganivkveTSi yvela nawilaks erTnairi siCqare aqvs). siTxis raodenoba (moculoba), romelic gaivlis (Seva) pirvel ganivkveTSi drois erTeulSi tolia

S11 -is, xolo

siTxis raodenoba, romelic gaivlis meore ganivkveTSi imave droSi (gamova am kveTidan) tolia S 22 -is. stacionaruli dinebis SemTxvevaSi

S11  S 2 2 .

(47.1)

winaaRmdeg SemTxvevaSi adgili eqneboda S1 da S 2 kveTebs Soris siTxis matebas an klebas* da stacionarulobis piroba dairRveoda. (47.1)-dan

1 S1  2 S2

(47.2)

e.i. dinebis siCqare ganivkveTis farTis ukuproporciulia: iq, sadac mili viwroa, dinebis siCqare aris didi da piriqiT. (47.1) gantolebas ewodeba siTxis Wavlis uwyvetobis gantoleba.

§48. bernulis gantoleba ganvixiloT idealuri siTxis stacionaruli dineba cvladi ganivkveTis denis daxril milSi (sur. 48.1). azrobrivad gavavloT masSi ori kveTi, romelTa farTobebia S1 da S 2 , xolo

1t

siCqareebi maTSi, Sesabamisad, 1 da  2 .

 2 t

S1 da S 2

kveTebs Soris moTavsebuli

siTxe ar aris izolirebuli: S1 kveTze

awarmoebs

S1 ' S2

S2 '

wnevas

mis

marcxniv

myofi

siTxe. Tu am wnevas aRvniSnavT

P1 -iT,

maSin kveTze moqmedi wnevis Zala iqneba

F1  P1S1 . aseve, S 2 kveTze awarmoebs P2

sur. 48.1 *

wnevas

mis

marjvniv

myofi

siTxe

siTxis raodenobis ganuwyveteli kleba aucileblad gamoiwvevs siTxis Wavlis gawyvetas.

139

masze moqmedebs wnevis Zala F2  P2 S 2 (romelic mimarTulia siTxis dinebis sawinaaRmdegod). vTqvaT, siTxis moculoba, romelic moTavsebulia S1 da S 2 kveTebs Soris gadaadgilda da mcire t droSi aRmoCnda S1 ' da S 2 ' kveTebs Soris. am gadaadgilebis dros misi energia Seicvleba da energiis mudmivobis kanonis Tanaxmad,

cvlileba

toli

iqneba

gareSe

Zalebis

mier

Sesrulebuli

muSaobisa: E  A .

(48.1)

gamovTvaloT jer energiis cvlileba E  t droSi S1 da S 2 kveTebSi gamavali siTxis moculobebia, Sesabamisad, S11t da S 2 2 t . siTxis Wavlis uwyvetobis pirobis Tanaxmad, es moculobebi tolia:

S11t  S 2 2 t  V . am tolobis  simkvriveze gamravleba mogvcems:

m1  m2  m . e.i. t droSi S1 kveTSi gasuli siTxis m1 masa udris imave droSi S 2 kveTidan gamosul siTxis m2 masas. vinaidan dineba stacionarulia, S1 ' ' da S 2 ' kveTebs Soris moTavsebuli siTxe araviTar cvlilebas ar ganicdis. ucvleli darCeba siTxis am nawilis energiac. amitom, SeiZleba davuSvaT, rom siTxis es nawili darCa Tavis adgilze, xolo m masis siTxe, romelic moTavsebuli iyo

S1 '

kveTebs

Soris,

aRmoCnda

gadatanili

daikava

axali

mdebareoba S 2 da S 2 ' kveTebs Soris. amrigad, E tolia m masis siTxis sruli

meqanikuri

energiis

cvlilebisa

pirveli

mdebareobidan

meoreSi

gadasvlisas:

m 22 m12  mgh2   mgh1 , (48.2) 2 2 aris simaRleebi, romlebzec mdebareoben S1 da S1 ' 'da S 2 da E 

sadac h1 da h2

S 2 ' kveTebs Soris moTavsebul siTxeTa simZimis centrebi. gamoviTvaloT A . S1 kveTze moqmedi F1  P1S1 wnevis Zalis moqmedebiT siTxe gadaadgilda 1  t manZilze. amitom, mis mier Sesrulebuli muSaoba

A1  P1 S11t  P1V . 140

kveTze

F2  P2 S 2

moqmedi

wnevis

Zalis

mier

Sesrulebuli

muSaoba

uaryofiTia, vinaidan igi mimarTulia dinebis (gadaadgilebis) sawinaaRmdegod:

A2   P1 S 2 2 t   P2 V . sruli muSaoba

A  A1  A2  P1  V  P2  V

(48.3)

Tu (48.1) tolobaSi SevitanT E -s da A -s mniSvnelobebs (48.2) da (48.3) formulebidan, martivi gardaqmnebis Semdeg miviRebT:

12 2

sadac  

 gh1  P1 

m siTxis simkvrivea. V

vinaidan gamosaxuleba

 2 2

kveTebi

22 2

 gh2  P2 ,

nebismierad

(48.4)

iyo

SerCeuli,

amitom

 gh  P mudmivi iqneba denis milis nebismieri kveTisaTvis.

e.i. SeiZleba davweroT:

 2

 gh  P  const . (48.5) 2 (48.5) gantolebas, an, rac igivea (48.4) gantolebas ewodeba b e r nu lis gantoleba. igi hidrodinamikis erT-erTi ZiriTadi gantolebaa da gamosaxavs energiis mudmivobis kanons moZravi siTxisaTvis. marTlac, ganvixiloT (48.5) gantolebis

TiToeuli

wevri.

 2

aris

siTxis

kinetikuri energia anu kuTri kinetikuri energia

erTeuli  jouli  ,   m3  

moculobis

gh

kuTri

potenciuri energiaa simZimis velSi, xolo P , romelsac igive ganzomileba

 niutoni jouli  aqvs    , SeiZleba CaiTvalos siTxeSi wnevis Zalebis kuTr m2 m3   energiad*.

maSasadame,

(48.5)-is

Tanaxmad,

siTxis

moculobis

erTeulis

moZraobisas, misi erTi saxis energia SeiZleba gaizardos, meore ki Semcirdes, magram ise, rom maTi jami darCes mudmivi.

igulisxmeba drekadad SekumSuli siTxis potenciuri energia. saqme isaa, rom wnevis Zalebis

moqmedebiT nebismieri siTxe umniSvnelod, magram mainc ikumSeba.

141

vinaidan (48.5) gantolebaSi yvela wevrs aqvs wnevis ganzomileba, igi SeiZleba Semdegnairadac CamovayaliboT: P -s ewodeba statikuri wneva, gh -s _ hidravlikuri wneva, xolo siTxeSi

sruli

wneva,

 2 2

-s _ dinamikuri wneva. (48.5)-is Tanaxmad, moZrav

romelic

Sedgeba

statikuri,

dinamikuri

hidravlikuri wnevebis jamisagan, mudmivi sididea. davuSvaT, mili horizontaluria, e.i. h1  h2 . maSin (48.5) gantolebaSi Sua wevri amovardeba da gveqneba:

 2

(48.6)  P  const . 2 miRebuli gantoleba gviCvenebs, rom siTxis wneva damokidebulia dinebis siCqareze. iq , s a d ac s i Cq a re m e t ia , w ne v a n ak le b i a d a pi riq iT . miRebuli

Sedegi

martivad

aixsneba

uwyvetobis

(47.1)

gantolebis

safuZvelze. marTlac, am gantolebis Tanaxmad, milis ganieri nawilidan viwroSi gadasvlisas siTxis dinebis siCqare izrdeba, e.i. s iT xe m oZ r a ob s a C q a r e b u l ad. es ki, niutonis meore kanonis Tanaxmad niSnavs, rom siTxeze moqmedebs

Zala,

romelic

mimarTulia

didi

kveTidan

mcire

kveTisaken.

maSasadame, milis ganier nawilSi wnevis Zala (da wnevac) meti unda iyos, vidre viwro nawilSi (milis viwro nawilSi wneva SeiZleba imdenad Semcirdes, rom atmosferul wnevaze naklebic ki gaxdes. es faqti safuZvlad udevs s iT xi s W av l is S e m w ov m oq m e de b as ). igive gamodis energiis mudmivobis kanonidanac: vinaidan milis viwro nawilSi gadasvlisas izrdeba siTxis siCqare da, maSasadame, kuTri kinetikuri energia,

unda

Semcirdes

wnevis

Zalebis

kuTri

potenciuri

energia

da,

maSasadame, P wnevac. miuxedavad siTxisaTvis

imisa,

(kerZod,

rom

bernulis

mxedvelobaSi

gantoleba iyo

gamoyvanilia

miRebuli

idealuri

Sinagani

xaxunis

daZlevaze Sesrulebuli muSaoba), igi warmatebiT gamoiyeneba mcire siblantis mqone realuri siTxeebisTvisac (wyali, spirti, benzoli da sxv.). ufro metic, bernulis gantoleba gamoiyeneba airebisTvisac, Tu maTi dinebis siCqare bgeris siCqareze naklebia. airis SemTxvevaSi bernulis gantolebis mTavar Sedegs _ wnevis Semcirebas siCqaris gazrdiT _ advilad SevamowmebT 142

umartivesi cdiT: Tu qaRaldis furcels xelSi daviWerT da zedapiris gaswvriv SevuberavT, furclis Tavisufali bolo zemoT aiwevs. es gamowveulia qaRaldis zemoT haeris nakadSi wnevis SemcirebiT (atmosferul wnevasTan SedarebiT, romelic qaRaldis qvemoTaa). haeris nakadSi wnevis Semcireba misi siCqaris gazrdisas, safuZvlad udevs amwevi Zalis warmoqmnas: TviTmfrinavis frTis forma da dayenebis kuTxe (`ieriSis kuTxe~) isea SerCeuli, rom moZraobisas Semxvedri nakadis siCqare frTis zemoT ufro metia, vidre qvemoT. maSasadame, frTis zemoT wneva ufro naklebi iqneba, vidre qvemoT, rac warmoSobs amwev Zalas.

ยง49. realuri siTxis dineba. siblante realuri siTxe ewodeba siTxes, romelic ikumSeba da romlis fenebs Sorisac adgili aqvs xaxuns. magram aRmoCnda, rom wnevis sakmarisad did farglebSi realuri siTxis kumSvadoba metismetad mcirea*. amitom Tvlian, rom realuri siTxe praqtikulad ukumSvadia da misi realuroba ganpirobebulia mxolod fenebs Soris xaxuniT. fenebs Soris xaxuni ganapirobebs imas, rom siTxis sxvadasxva fena moZraobs

sxvadasxva

siCqariT.

ki,

Tavis

mxriv,

siTxeSi

warmoSobs

S i n ag a ni xa xu nis Zalebs. am Zalebis moqmedeba vlindeba imiT, rom nela moZravi fenebi amuxruWeben maT mezobel ufro swrafad moZrav fenebs da piriqiT. Sinagani xaxunis Zalebi fenebisadmi mxebadaa mimarTuli. Sinagan xaxuns xSirad s ib l a nte s a c uwodeben, xolo realur siTxes _ b l an t siTxes. siblante, kerZod, vlindeba imiT, rom siTxis (airis) moZraoba, am moZraobis gamomwvevi mizezis Sewyvetis Semdeg, TandaTan mcirdeba da bolos wydeba. amis magaliTia WiqaSi siTxis moZraoba kovziT morevis Sewyvetis Semdeg. Sinagani xaxunis Zalebis warmoqmna siTxeSi, ZiriTadad, ganpirobebulia mezobeli fenebis molekulebs Soris mizidviT. airSi es mizidva mcirea, molekulebis Zvradoba ki _ didi. amitom aq Sinagani xaxunis Zalebi, ZiriTadad, warmoiqmneba mezobel fenebs Soris molekulebis urTierTgacvliT (ยง93). *

magaliTad, wnevis 1 atmosferodan 100 atmosferomde gazrdiT, wylis simkvrive mxolod

0,5%-iT izrdeba.

143

cdebiT

dadgenilia,

rom

miuxedavad

gansxvavebuli

mizezebisa,

Sinagani

xaxunis Zala siTxeSi da airSi erTi da imave kanoniT gamoisaxeba. siTxis

(airis)

siblante

xasiaTdeba

siblantis koeficientiT ( ). ra aris igi?

2

vTqvaT, siTxis dineba warmoebs dz

1

gaswvriv,

xolo fenis

siCqare

RerZis

icvleba

RerZis gaswvriv. gamovyoT siTxeSi ori fena, romelTa Soris manZilia dz . fenebis siCqare-

ebi aRvniSnoT 1 -iT da  2 -iT (sur. 49.1). pirveli fenidan meoreze gadasvlisas anu

sigrZeze siCqare icvleba  2  1  d

sur. 49.1

sidi-

diT. siCqaris cvlileba sigrZis erTeulze toli iqneba

d -is. mas siCqaris gr ad ie n ti ewodeba. igi axasiaTebs siCqaris dz

cvlilebis siswrafes fenidan fenaze gadasvlisas. n i u to nm a daadgina, rom Sinagani xaxunis Zala proporciulia siCqaris gradientisa da fenebis Sexebis S farTobisa:

d (49.1) S dz proporciulobis  koeficients ewodeba siblantis koeficienti* (sibFx  

lante). misi mniSvneloba damokidebulia siTxis gvarobaze da mdgomareobaze (pirvel rigSi _ temperaturaze). (49.1) formulidan Cans,  ricxobrivad im Zalis tolia, romelic moqmedebs moZravi siTxis erTeulovan farTze, roca siCqaris gradienti erTis tolia. SI

sistemaSi

siblantis

erTeulia

p a s k al i. w m .

iseTi

siTxis

siblantea, romlis fenis 1 m2 farTobze moqmedebs 1 niutoni Zala, roca fenis siCqare 1 m sigrZeze 1 m/wm-iT icvleba:



sazogadod,

simkvrivesTan

F  dz n m n  2  2  wm  pa  wm S  d m  m m wm

 -s uwodeben dinamikuri siblantis koeficients,      _ kinematikuri siblantis koeficients.   144

xolo

mis

fardobas

siTxis siblante bevrad aRemateba airisas. magaliTad, oTaxis temperaturaze wylis siblante η 1,0 103 pa.wm , xolo haeris (normalur pirobebSi) _ 1,8.10-5 pa.wm-s. siTxis dineba SeiZleba iyos l am in ar u li da t u rb u le ntu ri . laminaruli anu fenobrivi dineba ewodeba iseT dinebas, rodesac siTxis fenebi erTimeoris

mimarT

misrialeben.

dros

xdeba

siTxis

nawilakebis

gadasvla erTi fenidan meoreSi da, maSasadame, fenebis urTierTSereva. stacionaruli dineba SeiZleba mxolod laminaruli

iyos. fenebis Seurevobas

SeiZleba davakvirdeT, Tu ramdenime santimetri diametris mqone minis milSi SevqmniT wylis sust nakads da misi RerZis gaswvriv viwro milis saSualebiT SeviyvanT Seferadebul siTxes: milis mTel sigrZeze warmoiqmneba wvrili Seferadebuli nakadi, romelsac aqvs mkafio sazRvari wyalTan. laminaruli dineba damaxasiaTebelia mcire siCqariT moZravi siTxisaTvis. Tu gavzrdiT dinebis siCqares, siCqaris garkveuli mniSvnelobidan dawyebuli (romelsac kritikuli siCqare ewodeba), dinebis xasiaTi mkveTrad icvleba: nawilakebis traeqtoriebi ukve aRaraa milis RerZis paraleluri. dineba xdeba mouwesrigebeli.

warmoiqmneba

nakadis

marTobulad

mimarTuli

siCqaris

mdgenelebi (amboben, rom warmoiqmneba grigalebi) da iwyeba fenebis erTmaneTSi Sereva. aseT dinebas turbulenturi dineba ewodeba. turbulenturi dinebis dros nawilakis siCqare sivrcis yovel wertilSi uwesrigod icvleba, e.i. dineba stacionaruli ar aris. turbulenturi dinebis dros fenebis urTierTSerevas advilad davakvirdebiT zemoT aRweril cdaSi: Tu milSi SevqmniT turbulentur nakadsa

cdas

gavimeorebT,

Seferadebuli

siTxe,

Seyvanis

adgilidan

mcireodeni moSorebiT, Tanabrad ganawildeba milis mTel ganivkveTSi. ingliselma

mecnierma

re in ol ds m a

daadgina,

rom

dinebis

xasiaTs

gansazRvravs ganyenebuli sidide (reinoldsis ricxvi):

Re 

 l 

(49.2)

sadac  siTxis (airis) simkvrivea,  _ milis ganivkveTSi nakadis saSualo siCqare,  _ siblantis koeficienti, xolo l _ milis ganivkveTis damaxasiaTebeli wiriTi zoma (wriuli milis SemTxvevaSi _ diametri). dineba laminarulia, roca Re < 2300 da turbulenturi _ roca Re > 2300. 145

§50. sxeulebis moZraoba realur siTxeSi. stoqsisa da niutonis formulebi realur siTxeSi (airSi) moZrav sxeuls ekvris siTxis (airis) Txeli fena,

romelic moZraobs

masTan

erTad,

rogorc

erTi

mTliani.

Sinagani

xaxunis gamo es Txeli fena moZraobaSi iTrevs (waritacebs) mis mimdebare fenebs. sxeulis zedapiridan daSorebasTan erTad watacebuli fenebis siCqare TandaTan mcirdeba da bolos, zedapiridan raRac manZilze, xdeba nulis toli. am manZilidan dawyebuli, sxeulis moZraoba araviTar gavlenas aRar axdens siTxeze. amrigad, moZravi sxeuli garSemortymulia siTxis (airis) feniT,

romelSic siCqare

swrafad

icvleba.

fenas

s as a zR v r o

fe n a

ewodeba*. sasazRvro fenaSi moqmedi Sinagani xaxunis Zalebi saboloo jamSi modebulia sxeulze da warmoqmnis e.w. S u b l a winaaRmdegobas**. Subla winaaRmdegobis Zalis sidide mcire zomis sferuli formis sxeulisaTvis daadgina ingliselma fizikosma stoqsma: (50.1) F  6r , sadac  siblantis koeficientia, r _ birTvis radiusi,  _ misi moZraobis siCqare. stoqsis formula saSualebas gvaZlevs, gamoviTvaloT siTxeSi Tanabrad vardnili birTvis siCqare. marTlac, am dros birTvze moqmedebs sami Zala: vertikalurad qveviT mimarTuli simZimis Zala mimarTuli arqimedes amomgdebi

Fa 

P 

da vertikalurad zeviT

da stoqsis winaaRmdegobis

F 

Zalebi.

Tavdapirvelad, birTvis siCqare mcirea, ris gamoc mcirea stoqsis winaaRmde-

Teoriulad dadgenilia, rom sasazRvro fenis sisqe

formuliT



l , sadac Re

sxeulis

 

SeiZleba miaxloebiT gamoiTvalos

damaxasiaTebeli

wiriTi

zomaa,

xolo

reinoldsis ricxvi. ** galileis fardobiTobis principis Tanaxmad, sxeulze moqmedi Zala ar Seicvleba, Tu CavTvliT, rom igi uZravia, xolo siTxe (airi) garsedineba mas toli da sawinaaRmdegod mimarTuli siCqariT, oRond sxeuli did manZilze unda imyofebodes WurWlis kedlebidan. sakiTxis aseTi Sebrunebis SesaZleblobiT sargebloben, kerZod, TviTmfrinavis an gemis modelis Semowmebis dros. moZrav modelze moqmedi Zalebis gazomva Znelia. amitom, models uZravad aTavseben aerodinamikur milSi, airi an siTxe ki garsedineba mas.

146

gobis Zalac. amitom P  Fa  F

da birTvi moZraobs qveviT aCqarebulad. es

iwvevs

Sesabamisad,

birTvis

siCqarisa

da,

stoqsis

winaaRmdegobis

Zalis

zrdas. bolos dadgeba momenti, roca qveviT da zeviT mimarTuli Zalebi gaawonasworeben erTmaneTs, anu Sesruldeba piroba P  Fa  F

(50.2)

4 da birTvi daiwyebs vardnas Tanabari siCqariT. magram P  mg  Vg   r 3 g , 3 4 Fa   0Vg   r 3  0 g , sadac r birTvis radiusia, xolo  da  0 _ Sesabamisad 3

birTvis masalisa da siTxis simkvrive. P –s,

Fa -s da

F –is mniSvnelobebi

SevitanoT (50.2)-Si: 4 3 4  r g   r 3  0 g  6r 3 3

saidanac

 cdis

saSualebiT

birTvis

2 g    0  2 r 9 Tanabari

(50.3) vardnis

siCqaris

gansazRvra

siZneles ar warmoadgens. amitom (50.3) formulas xSirad iyeneben siTxis siblantis gasazomad. garda amisa, Tu garemos siblante cnobilia, igive formuliT SeiZleba ganvsazRvroT patara sferos (magaliTad, wveTis) radiusi. magram aRmoCnda, rom garemos siblantis gavlena xaxunis winaaRmdegobis (stoqsis Zalis) warmoqmniT ar amoiwureba. sasazRvro fenis arsebobis gamo mkveTrad icvleba siTxis mier sxeulebis garSemovla. kerZod, sruli garSemovla xdeba SeuZlebeli*. saC A

fenaSi

xaxunis

moqmedeba

iwvevs

imas,

rom

Zalebis nakadi

scildeba sxeulis zedapirs, ris ga-

D D

moc sxeulis ukan warmoiqmneba grigalebi (sur. 50.1). es grigalebi mih-

sur. 50.1

sazRvro

yvebian nakads da TandaTan miilevi-

ufro zustad, aseT garSemovlas adgili aqvs metismetad mcire siCqareebis dros.

147

an xaxunis gamo. maTi energia ixarjeba siTxis gaTbobaze. sxeulis ukan warmoqmnil grigalur ubanSi wneva ufro mcirea (grigaliseburad moZravi nakadis didi siCqaris gamo), vidre win. amitom sxeulze imoqmedebs nakadis gaswvriv mimarTuli Zala. davuSvaT axla, rom siTxe uZravia, xolo sxeuli moZraobs masSi imave siCqariT. maSin sxeulis ukan warmoqmnili grigalebi, ganapirobeben

wnevis

Semcirebas,

gamoiwveven

sxeulze

moZraobis

sawinaamRdegod mimarTuli Zalis warmoqmnas. es Zalac Subla winaaRmdegobis Zalaa,

magram

vinaidan

igi

gamowveulia

wnevaTa

sxvaobiT,

w nev is

w ina aR m de gob a e w odeb a . avxsnaT Tvisobrivad sxeulis ukan grigalebis warmoqmna misi blanti siTxis garsdenisas (sur. 50.1).

wertilTan siTxis nakadi orad iyofa:

nawili sxeuls uvlis zevidan, meore nawili ki _ qvevidan. A -dan C -sken da A -dan D -sken moZravi sasazRvro fenis nawilakebis siCqare TandaTan izrdeba,

radgan aq wneva mcirdeba (denis xazebis SemWidroebis gamo). magram xaxunis gamo,

wertilebSi siTxis nawilakebs iseTi siCqare ar eqnebaT,

rogoric iqneboda idealuri siTxis SemTxvevaSi. C -dan B -sken moZraobisas sasazRvro fenis nawilakebis siCqare TandaTan mcirdeba, rogorc wnevis gazrdis (denis xazebis SemWidroeba mcirdeba), ise xaxunis Sedegad. amitom nawilakebi ver aRweven B wertilamde. isini jer Cerdebian, Semdeg brundebian ukan. e.i. sxeulis ukan, C da B wertilebs Soris sasazRvro fenaSi, warmoiSoba nakadis sawinaaRmdegod mimarTuli dineba, romelic ZiriTad nakadTan Sexvedris

Sedegad

dagrigaldeba

saaTis

isris

moZraobis

mimarTulebiT.

analogiur movlenas eqneba adgili sxeulis qveviT, D da B wertilebs Soris,

oRond

grigals

eqneba

saaTis

isris

moZraobis

sawinaaRmdego

mimarTuleba. warmoqmnil grigalebs TandaTan emateba axladmosuli siTxis nawilakebi.

amitom

grigalebi

izrdeba,

wydebian

sxeulis

zedapirs

mihyvebian nakads. maT adgilas warmoiSoba axali grigalebi da a.S. niutonma daadgina, rom wnevis winaaRmdegobis Zala gamoiTvleba formuliT:

Fwn  C

sadac 

 2

S 2 siTxis simkvrivea,  _ sxeulis moZraobis siCqare, S

ganivkveTis

proeqcia

moZraobisadmi marTobul 148

sibrtyeze,

(50.4) _ sxeulis

xolo

proporciulobis koeficienti

uganzomilebo

ewodeba.

misi

koeficienti.

mniSvneloba

mas

Subla

damokidebulia

winaaRmdegobis

sxeulis

formaze,

zedapiris mdgomareobaze da garemos siblanteze. 50.2 suraTze mocemulia C -s mniSvnelobebi sxvadasxva formis sxeulebisaTvis. sxeulebs erTnairi ganivkveTi aqvT. suraTidan Cans, rom yvelaze naklebi C (da maSasadame Fwn ) aqvs kargi garsSemodenili formis (wveTis formis) sxeuls. naxevarsferoebis magaliTi gviCvenebs, rom ZiriTad gavlenas Subla winaaRmdegobis Zalaze axdens sxeulis ara wina, aramed ukana nawilis forma, romlis iqiTac Cndeba grigali. amrigad, C 1,32

Subla 1,12

disko

winaRoba 0,34

ikribeba 0,24

naxevarsferoebi

xaxunis

winaRobisa

wnevis

Zalas

Soris

0,04

sfero

kargi garsSemodenili formis sxeuli

sur. 50.2 winaRobis

Zalebisagan.

Tanafardoba

winaaRmdegobis

ganisazRvreba reinoldsis ricxviT (49.2), oRond, am SemTxvevaSi, l _ sxeulis damaxasiaTebeli zomaa, xolo  _ sxeulis siCqare siTxis (airis) mimarT (an sxeulze mimwydomi nakadis siCqare).

Re –is mcire mniSvnelobis dros (e.i.

mcire  -sa da l -is dros) ZiriTad rols asrulebs xaxunis winaaRmdegoba. amasTan, siblantis zrda Fx -is fardobiT rols zrdis. reinoldsis ricxvis zrda

iwvevs

wnevis

winaaRmdegobis

zrdas

Re -is

sakmaod

didi

mniSvnelobebisas wnevis winaaRmdegoba sWarbobs xaxunis winaaRmdegobas. Cven ganvixileT simetriuli formis sxeulis (sferos) moZraoba. Tu sxeuli asimetriuli formisaa (magaliTad, TviTmfrinavis frTa), masze, garda Subla

winaaRmdegobisa,

moqmedebs

vertikalurad

romelsac amwevis Zala ewodeba.

149

zeviT

mimarTuli

Zala,

§51. realuri siTxis dineba wriuli ganivkveTis mqone milSi. puazeilis formula viciT (§49), rom Sinagani xaxunis gamo realuri siTxis sxvadasxva fena moZraobs sxvadasxva siCqariT. vTqvaT, siTxe miedineba wriuli ganivkveTis mqone milSi. im fenis siCqare, romelic uSualod ekvris milis kedels, nulis tolia. momdevno fenebis siCqare TandaTan izrdeba da centrSi xdeba maqsimaluri. centraluri fenis siCqare maqsimaluria imitom, rom igi yvelaze metadaa daSorebuli milis kedlidan da, maSasadame, yvelaze naklebad ganicdis mis moqmedebas. siCqareTa ganawileba milis ganivkveTSi mocemulia 51.1 suraTze. gamoviyvanoT siCqareTa ganawilebis kanoni. davuSvaT, horizontalur wriul milSi,

romlis

radiusia

miedineba

siTxe marcxnidan marjvniv. dineba stacionarulia da, maSasadame, laminaruli. mi-

sur. 51.1

lis SigniT azrobrivad gamovyoT siTxis elementi

r -radiusiani da

l -sigrZis cilindris saxiT. wnevebi cilindris

marcxena da marjvena fuZeebze aRvniSnoT Sesabamisad P1 -iT da P2 -iT (sur. 51.2). maSin fuZeebze moqmedi wnevis Zalebi iqneba cilindris

fuZis

farTobia.

pirobis

Tanaxmad,

P1 r 2 da P2 r 2 , sadac  r 2

P1  P2 .

amitom

elementze

moqmedi jamuri wnevis Zala (igi mimarTulia dinebis gaswvriv) iqneba:

Fwn  P1  P2  r 2 .

(51.1)

garda amisa, am cilindris gverdiT zedapirze

mis

gareT

moqmedebs

Sinagani

myofi

xaxunis

siTxe ZaliT,

romelic 49.1 formulis Tanaxmad, tolia (siCqare icvleba milis radiusis gaswvriv. d d amitom Secvlilia -iT): dz dr Fx  

sadac 2 rl

d  2 rl , dr

Fx P

r 

Fx l

sur. 51.2 (51.2)

gamoyofili cilindris gverdiTi zedapiris farTia. 150

stacionaruli

dinebis

SemTxvevaSi,

elementi

moZraobs

aCqarebis

gareSe (vinaidan milis yvela kveTSi siCqare erTnairia). maSasadame, masze modebuli Zalebis jami nulis tolia, e.i.

 saidanac

d  2 rl  P1  P2  r 2  0 , dr

P  P2 d  1 r . dr 2l

anu

d  

P1  P2  rdr . 2l

am gantolebis integrebis Sedegad miviRebT: P P    1 2 r 2  c . 4l integrebis

(51.3)

mudmivas davadgenT cdiseuli faqtis gamoyenebiT, rom

kedelze `mikruli~ siTxis fena uZravia, e.i. roca r  R , fenis siCqare   0 . (51.3)-dan:

0

P1  P2 2 R  c , 4l

c

saidanac

P1  P2 2 R . 4l

Tu c -s miRebul mniSvnelobas SevitanT (51.3)-Si, sabolood gveqneba:







P1  P2  R2  r 2 . 4l

(51.4)

miRebuli formula gviCvenebs, rom milis raR

diusis

gaswvriv

siCqare

paraboluri

kanoniT

icvleba, rac asaxulia (51.1) suraTze. (51.4) formulis saSualebiT SeiZleba gamoviTvaloT siTxis nakadi Q , anu siTxis is raodenoba (moculoba), romelic gaivlis milis ganivkveTSi sur. 51.3 Ti dr

drois erTeulSi. amisaTvis davyoT milis ganivkve-

sisqis koncentrul wriul rgolebad (sur. 51.3). siTxis raodenoba,

romlic gaivlis r -radiusian aseT wriul rgolSi drois erTeulSi tolia

dQ    2 r dr ,

(51.5)

sadac dS  2 r dr wriuli rgolis farTia, xolo  _ dinebis siCqare milis RerZidan r manZilze. SevitanoT  -s mniSvneloba (51.4)-dan: 151

dQ 

 P1  P2  2  R  r 2 rdr . 2l

(51.6)

milis mTel ganivkveTSi drois erTeulSi gasuli siTxis raodenoba miiReba (51.6) gamosaxulebis integrebiT 0-dan R

Q   dQ  0

 R



magram

 r 2 rdr 





R4 , 4

Q

 P1  P2  4 R . 8l

(51.7) formulas ewodeba pu a ze i lis metismetad

R -mde:

 P1  P2  R 2 2  R  r rdr . 2l 0

amitom, sabolood

nakadi

Zlier

aris

(51.7)

f o rm u l a . igi gviCvenebs, rom

damokidebuli

milis

radiusze.

kerZod,

radiusis 2-jer Semcireba iwvevs Q -s 16-jer Semcirebas. bunebrivia, rom Q

 P  P2  proporciulia wnevis gradientis  1  da ukuproporciulia siblantis.  l  Tu miRebul ASedegs gamoviyenebT sisxlZarRvebSi sisxlis dinebisaTvis, * miviRebT, rom sisxlZarRvis radiusis orjer Semcireba gamoiwvevs gasuli sisxlis raodenobis 16-jer Semcirebas. e.i. guls mouwevs 16-jer meti muSaobis Sesruleba, raTa uzrunvelyos saWiro sisxlis

raodenobis

mimoqceva

organizmSi. sisxlZarRvis

Seviwroebas

iwvevs

sxvadasxva

spazmebi da rac gardauvalia, sisxlZarRvis kedelze qolesterinis** daleqva, romelic asakTan erTad matulobs. amitomaa, rom xandazmul adamians uWirs mZime fizikuri samuSaos Sesruleba, vTqvaT, kibeebze asvla an sirbili – mas ewyeba qoSini.

puazeilis gamodenili

formulas

siTxis

iyeneben

raodenobis

cilindruli

mixedviT

siTxis

milidan

siblantis

erT

wamSi

koeficientis

gansasazRvravad (ostvaldis viskozimetri). puazeilis formula samarTliania mxolod laminaruli dinebisaTvis.

aseTi gamoyeneba SeiZleba mxolod miaxloebiT, vinaidan sisxlZarRvi ar warmoadgens

myarkedlebian mils. **

qolesterini cximovani nivTierebaa, romelsac Seicavs sisxli.

152

m e o r e

n a w i l i

molekuluri fizikisa da Termodinamikis safuZvlebi airis Tvisebebi ยง52. Sesavali. molekuluri fizikisa da Termodinamikis sagani da Seswavlis meTodebi kursis

mocemul

Termodinamikis

nawilSi

safuZvlebs.

SeviswavliT

molekuluri

fizikisa

Termodinamikis

Seswavlis sagani erTi da igivea, saxeldobr, erTica da meorec swavlobs makroskopuli sxeulis (e.i. iseTi sxeulis, romelic Sedgeba Zalian didi raodenobis atomebisa da molekulebisagan) Tvisebebs airad, Txevadsa da myar mdgomareobaSi. magram gansxvavdebian isini Sesaswavli sakiTxebisadmi sxvadasxva midgomiT _ Seswavlis meTodebiT. m ole k u lu r i

fi zik a

(romelsac

xSirad

nivTierebis

aRnagobis

molekulur-kinetikur Teoriasac uwodeben) gamodis nivTierebis atomurmolekuluri aRnagobidan da siTbos ganixilavs rogorc atomebisa da molekulebis

uwesrigo,

makroskopuli

qaosur

Tvisebebis

moZraobas.

(temperatura,

igi Seiswavlis wneva,

nivTierebis

drekadoba,

siblante,

Tbogamtaroba da a.S.) damokidebulebas mis Sinagan aRnagobaze, amasTan am Tvisebebs ganixilavs rogorc molekulebis jamuri moqmedebis (moZraobisa da urTierTqmedebis) gamovlenis Sedegs. TeoriaSi

gamoiyeneba

statistikuri meTodi,

romelsac

safuZvlad

udevs Semdegi mtkicebani: 1. didi raodenobis molekulebis erTobliobas (makrosxeuls) iseTi Tvisebebi aqvs, rogoric ar gaaCnia calke aRebul TiToeul molekulas. magaliTad,

molekulebis

erTobliobis

Tvisebebia

temperatura,

wneva,

Tbogamtaroba, siblante da sxv. cxadia, ar SeiZleba vilaparakoT erTi molekulis temperaturaze, wnevasa an siblanteze. 2. arsebobs garkveuli raodenobrivi kavSiri makrosxeulis Tvisebebsa da

calkeuli

sidideebis Soris.

molekulebis

(siCqare,

magaliTad,

impulsi, sxeulis

moZraobis energia

temperatura 153

damaxasiaTebeli sxv.)

saSualo

proporciulia

fizikuri

mniSvnelobebs molekulebis

qaosuri moZraobis saSualo kinetikuri energiisa, airis wneva WurWlis kedlebze ganisazRvreba molekulebis saSualo impulsiT da a.S. swored

makroskopuli

sxeulis

Tvisebebis

aseTi

aRwera

misi

Semadgeneli nawilakebis moZraobis damaxasiaTebeli sidideebis saSualo mniSvnelobebis

mixedviT,

aris

mniSvnelobebis

gansazRvra

statistikuri xdeba

meTodis

statistikuri

arsi.

saSualo

meTodis

ZiriTadi

kanonebis _ nawilakebis ganawilebis (siCqareebis, impulsebis, energiebis mixedviT) kanonebis meSveobiT. statistikur meTodze gadasvlis aucileblobaSi advilad davrwmundebiT Semdegi magaliTiT.

davuSvaT,

gvaqvs

didi

raodenobis

nawilakebisagan

Semdgari

sistema

(koleqtivi). rogor SeviswavloT es sistema, rogor ganvsazRvroT misi mdgomareoba drois yovel momentSi? erTi SexedviT SeiZleba mogveCvenos, rom amisaTvis saWiroa SeviswavloT sistemis

yoveli

koordinatebisa

nawilakis

moZraoba,

ganvsazRvroT

impulsis

mdgenelebis

mixedviT.

nawilakis

magram,

jer

moZraoba erTi,

maTi

nawilakebis

kolosaluri raodenobis gamo es praqtikulad SeuZlebelia, vinaidan amisaTvis saWiro iqneba normalur pirobebSi 1 sm3 airisaTvis (sawyisi pirobebis gaTvaliswinebiT) ~1019 gantolebis amoxsna. meorec, aseTi samuSaos Catarebas (SesaZlebelic rom iyos) araviTari azri ara aqvs, radgan wonasworobaSi myofi aseTi sistemis Tvisebebi ara Tu ar aris damokidebuli

molekulebis

koordinatTa

impulsebis

mdgenelebis

sawyis

mniSvnelobebze, aramed arc ki icvleba (miuxedavad imisa, rom calkeuli molekulebis koordinatebi da impulsi ganuwyvetliv icvleba). aqedan gamomdinareobs, rom sistema, rogorc mTliani, Tvisobrivad gansxvavdeba calkeuli nawilakebisagan da igi sul sxva kanonzomierebebs emorCileba. swored es kanonzomierebania statistikuri kanonzomierebani.

T e r m o d in am ik a, piriqiT, ar akeTebs araviTar daSvebas nivTierebis aRnagobisa da siTbos bunebis Sesaxeb. igi siTbos ganixilavs rogorc raRac Sinagani moZraobis saxes, magram ar akonkretebs Tu ra moZraobaa es. makrosistemis

Seswavlis

fenomenologiur,

anu

Termodinamikuli aRweriT

meTodsac

meTodi,

romelsac

uwodeben,

xSirad

makrosistemas

ganixilavs rogorc mTlians, uwyvets, da aRwers mis Tvisebebs ara misi mikrostruqturisa

masSi

mimdinare

procesebis

safuZvelze,

aramed

eqsperimentze dayrdnobiT. rogor xdeba am meTodiT sxeulis Tvisebebis Seswavla? pirvel yovlisa, ganisazRvreba am Tvisebebis damaxasiaTebeli sidideebi: temperatura, wneva da sxv. magaliTad, temperatura ganisazRvreba rogorc

sxeulis

mTbarobis

damaxasiaTebeli

sidide

izomeba

TermometriT, pirobiT SerCeuli skalis meSveobiT. wneva ganisazRvreba rogorc zedapiris farTobis erTeulze moqmedi wnevis Zala da a.S. Semdeg 154

uSualod cdiT myardeba kavSiri am sididebs Soris, da miiReba saTanado kanonebi. Termodinamika emyareba or ZiriTad kanons (princips), romlebic miRebulia kacobriobis mravalsaukunovani praqtikuli gamocdilebisa da dakvirvebebis safuZvelze. amgvarad,

makrosxeulis

Tvisebebis

Seswavla

SesaZlebelia

ori

meTodiT _ fenomenologiuri da statistikuri meTodebiT. fenomenologiuri, anu Termodinamikuli meTodis upiratesobaa misi zogadoba. igi ar aris damokidebuli makrosxeulis Sinagani agebulebis ama Tu im modelze da amitom am meTodiT dadgenili kanonzomierebani marTebuli iqneba miuxedavad imisa, Tu sxeulis Sinagani aRnagobis Sesaxeb romeli Tvalsazrisi aRmoCndeba swori. garda amisa, es meTodi gamoirCeva simartiviT, rac saSualebas gvaZlevs nivTierebis atomebis an molekulebis Sesaxeb raime informaciis gareSe garkveuli maTematikuri operaciebis SesrulebiT

amovxsnaT

dinamikuli

meTodis

amocanebis

amoxsnisas

mTeli

rigi

upiratesobaa, (teqnikuri

konkretuli

amocanebi.

gansakuTrebiT

Termodinamika,

teqnikuri

Tboteqnika).

TermoxasiaTis

magram

meTods aqvs naklic, rac imiT gamoixateba, rom movlenaTa Termodinamikuli ganxilvisas ar xdeba misi Sinagani meqanizmis gaxsna. amitom TermodinamikaSi azri ara aqvs kiTxvas `ratom?~ Tu, magaliTad, Termodinamikuli meTodiT dadgenilia faqti, rom swrafad gaWimvisas spilenZis mavTuli civdeba, rezinis zonari ki Tbeba, Cven unda davkmayofildeT am SedegiT, xolo fizikuri meqanizmi, romelic am Sedegs iwvevs, rCeba gauxsneli. statistikuri, anu molekulur-kinetikuri meTodi ufro Rrmaa, vidre fenomenologiuri, vinaidan igi movlenas ganixilavs misi Sinagani meqanizmis gaTvaliswinebiT da asabuTebs yvela Termodinamikul kanonzomierebas maTi dayvaniT atomur da molekulur urTierTqmedebaTa kanonebze. amitom aq kiTxva `ratom?~ azrs moklebuli aRar aris. magaliTad, fenomenologiuri

meTodi

eqsperimentulad

amyarebs

kavSirs

idealuri

airis

wnevas,

temperaturasa da moculobas Soris:

pV ď&#x20AC;˝ RT , statistikur meTods ki Teoriulad gamohyavs es kavSiri, vinaidan rogorc wnevas, ise temperaturas igi ganixilavs rogorc molekulis siCqaresTan an kinetikur energiasTan dakavSirebul sidideebs. statistikuri meTodi saSualebas iZleva principulad amoixsnas mTeli rigi amocanebi, romlebsac ver xsnis Termodinamikuli meTodi (mag.: 155

siTbotevadoba, gamosxivebis Teoriis zogierTi sakiTxi da sxv.). sabolood, statistikuri meTodi ara marto asabuTebs Termodinamikul kanonzomierebebs, aramed adgens maTi gamoyenebis sazRvrebsac, ganixilavs ra Termodinamikuli wonasworobis mdgomareobis TviTnebur darRvevebs, anu fluqtuaciebs, romlebic miT ufro cxadad vlindeba, rac ufro mcirea sistemis zomebi. statistikuri meTodis naklia misi hipoteturi xasiaTi: igi eyrdnoba nivTierebis Sinagani aRnagobis garkveul models da amitom mis mier dadgenili kanonzomierebis konkretuli saxe mxolod imdenad iqneba marTebuli, ramdenadac marTebulia es modeli. marTalia, amJamad nivTierebis atomurmolekuluri aRnagoba araviTar eWvs aRar iwvevs, magram meTods mainc SemorCa hipoteturi elementi, vinaidan misi gamoyeneba moiTxovs garkveul daSvebebs molekulebis moZraobisa da urTierTqmedebis Sesaxeb, gamartivebuli, gaidealebuli molekuluri modelebiT sargeblobas (mag. molekulebs ganixilavs rogorc drekad birTvebs da sxv.), rac mxolod nawilobriv asaxavs realuri sxeulis Tvisebebs. saerTod, mkacri zRvari Seswavlis statistikur da Termodinamikul meTodebs Soris ar arsebobs. es meTodebi Seiswavlis wonasworobaSi myof makroskopuli sxeulis Tvisebebs* da avsebs erTmaneTs. amasTan xSirad laparakoben erTian statistikuri Termodinamikis meTodze.

IX Tavi

Â§53. molekulur-kinetikuri Teoriis ZiriTadi debulebani nivTierebis molekulur-kinetikur Teorias safuZvlad udevs Semdegi sami debuleba: 1. y v e l a s xe u l i Se dge b a Z a li a n d i di r ao de nob is u m cir e s i n aw il ak e b is a ga n (atomebis, molekulebis, an ionebisagan). warmodgena nivTierebis wyvetili, diskretuli agebulebis Sesaxeb axali ar aris, mas Rrma istoriuli warsuli aqvs. jer kidev V saukuneSi Cvens welTaRricxvamde, gamoCenili berZeni moazrovne demokritosi (460-370 ww. Cv. á&#x192;Ź. aR.-mde) Tvlida, rom yvela nivTiereba Sedgeba umciresi nawilakebisagan _ atomebisagan, romelTa Semdgomi gayofa aRar SeiZleba. amasTan, sityva atoms, romelic berZnulad niSnavs `ganuyofels~, eniWeboda *

arawonasworul sistemaSi mimdinare procesebs Seiswavlis fizikuri kinetika (XIII Tavi).

156

is mniSvneloba, rac dRes _ sityvas `molekula~*. magram Zveli saberZneTis gamoCenili moazrovnisa da filosofosis aristoteles (380-322 ww. Cv w. aR.-mde) zegavleniT, romelic ar iziarebda atomistur Sexedulebebs, igi didi xniT iqna miviwyebuli da aRorZinda mxolod me-17 saukuneSi, ukve rogorc ara gamoCenili moazrovnis genialuri mixvedra, aramed rogorc mecnieruli hipoTeza. kerZod, frangma mecnierma gasendim (1592-1655) Tavis naSromSi ganaviTara Sexeduleba nivTierebis atomur-molekuluri aRnagobis Sesaxeb. manve SemoiRo

ufro

(warmodgeba

Tanamedrove

laTinuri

terminologia,

sityvisagan

`moles~

pirvelad _

masa).

Semoitana

`molekula

sityva Sedgeba

`molekula~ atomebisagan,

rogorc sityva asoebisagan~, _ werda is Tavis naSromSi. atomisturma Teoriam Semdgomi ganviTareba hpova XVIII-XIX saukuneSi lomonosovis, daltonis, krenigis, klauziusis, bolcmanis, maqsvelis da sxvaTa SromebSi.

marTalia, sxeulis Semadgenel nawilakebs maTi zomis simciris gamo ver vxedavT da amitom nebismieri sxeuli gveCveneba mTliani, uwyveti, magram

maTi arseboba dasturdeba mravali cdiseuli faqtiT, rogorc

pirdapiri, ise arapirdapiri gziTac. pirdapirs uSualo

damzera

mikroskopiT.

Tanamedrove

miekuTvneba molekulis

eleqtronuli

mikroskopis

gamadidebloba iseTia, rom igi saSualebas gvaZlevs davinaxoT didi zomis molekula. arapirdapirs miekuTveneba, magaliTad, myari sxeulisa da siTxis xsnadoba, airis (iseve rogorc myari sxeulisa da siTxis) kumSvadoba da sxva.

kerZod,

kumSvadoba

miuTiTebs

faqts,

rom

sxeuli

Sedgeba

calkeuli nawilakebisagan, romelTa Soris aris Tavisufali adgilebi _ SoriseTebi da kumSva xdeba am SoriseTebis Semcirebis xarjze. 2. s xe u li s

Se m a dg e ne li

n aw ile k e b i

g an u w y v e t la d

u w e s r i g od (q aos u r ad ) * * m oZ r aob e n. am moZraobaze dakvirveba SeiZleba nebismier dros, misi mospoba ki araviTari saSualebiT ar SeiZleba, amitom moZraobas ganixilaven, rogorc materiis pirvelad Tvisebas. qaosuri moZraobis intensivoba damokidebulia mxolod sxeulis mTbarobis xarisxze. sxeulis

temperaturis

gazrda

iwvevs

nawilakebis

moZraobis

siCqaris

gazrdas, amitom qaosur moZraobas xSirad siTbur moZraobasac uwodeben. molekulebis moZraobiT aixsneba myari sxeulisa da siTxis aorTqleba, airis gafarToeba da sxv. dedamiwis atmosferos arsebobac haeris molekulebis moZraobis Sedegia. *

sityva atomma dakarga Tavisi mniSvneloba am sityvis Tavdapirveli gagebiT, vinaidan amJamad cnobilia, rom atomi rTuli aRnagobisaa, magram molekuluri fizika Semoifargleba im sakiTxebis ganxilviT, sadac atomis struqturuli elementebi ar vlindeba. ** qaosi berZnuli sityvaa da niSnavs srul uwesrigobas.

157

molekulebis moZraobis yvelaze naTeli dadasturebaa brounis moZraoba da difuzia. Tu mikroskopiT davakvirdebiT emulsias, e.i. siTxeSi Setivtivebuli nivTierebis

mcire

zomis

( d  0,5  1 mkm)

nawilakebs,

romlebic masSi

ixsneba, SevamCnevT, rom es nawilakebi ganuwyvetliv da qaosurad moZraoben. am moZraobas brounis moZraoba ewodeba, vinaidan igi pirvelad 1827 wels aRmoaCina ingliselma botanikosma brounma mcenaris preparatebze* mikroskopiT dakvirvebis dros. rac ufro naklebia siTxis siblante, miT ufro swrafia brounis moZraoba masSi. brounis moZraoba gacilebiT ufro intensiuria airSi. temperaturis gazrdiT brounis moZraobis intensiuroba matulobs. brounis moZraobis intensiuroba damokdebulia agreTve brounis nawilakis zomaze. zomis gazrdiT intensiuroba mkveTrad mcirdeba da ukve d  5 mkm zomis nawilakebisaTvis praqtikulad nulis toli xdeba.

ra aris brounis mozraobis mizezi? brounis moZraobis asaxsnelad aucilebelia davuSvaT, rom TviT siTxe (an airi), romelSic Setivtivebulia brounis nawilakebi, Sedgeba ganuwyvetlad moZravi molekulebisagan. es molekulebi ejaxebian brounis nawilakebs, aamoZraveben maT da amiT arapirdapiri gziT amJRavneben Tavis arsebobas. molekulebis zoma Zalian mcirea, Cveulebrivi (optikuri) mikroskopiT maT ver vxedavT, amitom siTxe gveCveneba mTliani, uwyveti. samagierod mikroskopiT vxedavT brounis nawilakebs, romelTa zomebi aTasjer da ufro metjer metia, vidre molekulisa. siTxis molekulebi brounis nawilaks ejaxeba yoveli mxridan, magram vinaidan siTxis molekulebis moZraoba qaosuria, amitom dajaxebaTa ricxvi xan erTi mxridanaa meti, xan meore mxridan, ris gamoc misi moZraobis mimarTuleba uwesrigod icvleba (sur. 53.1). roca siTxeSi (an airSi) Setivtivebuli

nawilakis

zoma

didia,

brounis moZraobas ar asrulebs, vinaidan am SemTxvevaSi biZgebis (dajaxe-

brouni akvirdeboda wyalSi Setivtivebul yvavilis mtvers.

158

sur. 53.1

bebis) ricxvi imdenad didia, rom maTi umniSvnelo gansxvaveba, romelic SemTxveviT SeiZleba aRmoCndes erT mxares, ver aamoZravebs sxeuls*. difuzia

ewodeba

erTmaneTTan

Semxebi

sxvadasxva

nivTierebis

urTierTSerevis process**. CavataroT cda. minis cilindrul WurWelSi Suamde CavasxaT Sabiamnis xsnari da frTxilad davasxaT wyali, ise rom maT Soris mkveTri sazRvari iyos, ramdenime xnis Semdeg SevamCnevT, rom wyali TandaTan galurjdeba da gamyofi sazRvris simkveTre Semcirdeba, bolos ki sazRvari sruliad gaqreba _ miviRebT erTgvarovan lurj narevs. miuxedavad imisa, rom Sabiamnis xsnaris simkvrive metia wylisaze, molekulebis moZraobis gamo moxda maTi sruli urTierTSereva. difuziis procesi mimdinareobs myar sxeulebSic, Tumca aq igi ufro nelia. sainteresoa oTaxis temperaturaze Catarebuli Semdegi cda: tyviisa da oqros gluvzedapiriani firfitebi erTmaneTze moaTavses da amyofes didi dawolis qveS. 5 wlis Semdeg es firfitebi Seezarda erTmaneTs, yofili gamyofi sazRvris maxloblad miRebul iqna 1 mm sisqis oqrosa da tyviis Sereuli fena. airSi difuzia gacilebiT swrafad mimdinareobs, vidre myar sxeulsa da siTxeSi. cnobilia, Tu ra swrafad vrceldeba daRvrili sunamos suni, romelic gadaaqvT sunamos aorTqlebul nawilakebs. temperaturis

gazrdiT

difuziis

procesi

Cqardeba,

rac

aixsneba

nawilakebis siTburi moZraobis siCqaris zrdiT. difuziis movlena miuTiTebs, rom yvela nivTiereba Sedgeba ganuwyvetlad

moZravi

nawilakebisagan,

romelTa

Soris

aris

Tavisufali

adgilebi _ SoriseTebi. sxva nivTierebis nawilakebis SeWra xdeba swored am SoriseTebSi. amiT aixsneba, kerZod, spirtisa da wylis narevis saerTo moculobis Semcireba: Tu minis viwro grZel milSi CavasxamT wyals da zemodan frTxilad davasxamT spirts, davinaxavT, rom ramdenime xnis Semdeg siTxis done milSi daiwevs.

siTxis molekulebis moZraoba TavisTavad ver gamoiwvevda brounis moZraobas, adgili rom ar hqondes siTxis simkvrivis fluqtuaciebs (saSualo mniSvnelobidan SemTxveviT gadaxrebs), rac iwvevs brounis nawilakze sxvadasxva mxridan moqmedi wnevis fluqtuaciebs. gamoTvlebi gviCvenebs, rom nawilakze moqmedi wnevis saSualo mniSvnelobidan garkveuli sididiT gadaxris albaToba mkveTradaa damokidebuli nawialkis zomaze. Tu, magaliTad, 0,5 mkm zomis nawialakisaTvis wnevis saSualo mniSvnelobidan 10%-iT gadaxris albaToba aris daaxloebiT 0,2, 5 mkm zomis nawilakisaTvis imave sididis fluqtuaciis albaToba Seadgens 2.10-8-s. ** difuzia berZnuli sityvaa da niSnavs gavrcelebas, SeRwevas.

159

cxadia, rom difuziis movlenas adgili eqneba ara marto sxvadasxva nivTierebis Sexebisas, aramed erTsa da imave nivTierebaSic. amis Sedegia, magaliTad simkvrivis gaTanabreba airis mTel moculobaSi. 3 . s x e u lis S e m a d ge ne l n aw i lak e b s S or is m oq m e de b s m iz i d v is a da g an zi dv is Z a le b i . es Zalebi ZiriTadad eleqtruli bunebisaa (gamowveulia nawilakebSi Semavali eleqtruli muxtebis urTierTqmedebiT) da moqmedebs metad mcire _ TviT nawilakis zomis rigis manZilze. nawilakebs Soris urTierTqmedebis Zalis sidide ganapirobebs nivTierebis ama Tu im agregatul mdgomareobas. m y ar

m d gom ar e ob a Si

nawilakebs Soris manZili Zalian mcirea,

urTierTqmedebis Zalebi ki _ Zalian didi. es Zalebi saSualebas ar aZlevs myari sxeulis nawilakebs Seasrulon gadataniTi moZraoba. maT garkveuli mdebareoba ukaviaT sivrceSi da asruleben mxolod qaosur rxevebs am (wonasworuli) mdebareobis mimarT. amiT aixsneba, rom myar sxeuls aqvs sakuTari formaca da moculobac. T xe v a d m d gom are ob a Si c nawilakebs Soris, maTi siaxlovis gamo, urTierTqmedebis Zalebi didia, magram es Zalebi mainc ar aris sakmarisi imisaTvis, rom siTxis nawilakebs garkveuli mdebareoba ekavoT sivrceSi. siTxis nawilakebi ufro Tavisufalia, vidre myari sxeulisa. isini, myari sxeulis nawilakebis msgavsad, asruleben rxeviTs moZraobas garkveuli wonasworobis mdebareobis mimarT, magram es wonasworobis mdebareobebi ucvleli (uZravi) ki ar aris, aramed ganuwyvetliv da swrafad icvleba. siTxis molekulebs Soris mcire manZili, e.i. maTi mWidro ganlageba ganapirobebs siTxis ukumSvelobas (ufro zustad, metad mcire kumSvadobas), radgan siTxis moculobis sul mciredi Semcirebac ki dakavSirebulia TviT molekulebis deformaciasTan, risTvisac Zalian didi Zalebia saWiro. a i r ad m d gom a re ob a S i manZili nawilakebs Soris gacilebiT metia, vidre myarsa da Txevad mdgomareobaSi. * amitom urTierTqmedebis Zalebi Zalian mcirea. es Zalebi imdenad sustia, rom isini faqtiurad sruliad ar zRudavs nawilakebis moZraobas. airis nawilakebi wrfivad moZraoben manam, sanam isini ar daejaxebian erTmaneTs an WurWlis kedlebs. susti urTierTmizidvis Zalebs ar SeuZlia Seakavos airis molekulebi erTmaneTis axlos, amitom

airs

usazRvrod

gafarToebis

unari

aqvs.

amiT

aixsneba

airis

airSi, myari sxeulisa da siTxisagan gansxvavebiT, manZili nawilakebs Soris bevrad aRemateba TviT nawilakebis zomebs. amiT aixsneba, kerZod, airis didi kumSvadoba.

160

Tviseba _ daikavos WurWlis mTeli moculoba. airi ver inarCunebs verc formas, da verc moculobas. naTqvamidan Cans, rom nawilakebis urTierTqmedeba da moZraoba yvelaze

martivia

airSi.

amiT

aixsneba,

rom

molekulur_kinetikuri

Teoria

sakmaod dasrulebuli formiT Camoyalibebulia mxolod airisaTvis. sxeulis, kerZod airis, mocemuli masis mdgomareoba ZiriTadad xasiaTdeba sami sididiT, anu parametriT: moculobiT, wneviTa da temperaturiT. ganvixiloT es sidideebi da maTi sazomi erTeulebi.

Â§54. moculoba da wneva. erTeulebi 1 . m o cu l ob a . rogorc aRvniSneT, airis moculoba ganisazRvreba im WurWlis moculobiT, romelSic igia moTavsebuli. SI sistemaSi moculobis erTeulad miRebulia k u b u r i m e t ri (1 m3), anu iseTi kubis moculoba, romlis wibos sigrZea 1 metri. CGS sistemaSi moculobis erTeulad miRebulia k u b u r i s a nt im e tr i (sm3). es iseTi kubis moculobaa, romlis wibos sigrZea 1 santimetri. 2. w n e v a . yoveli airi axdens wnevas im WurWlis kedlebze, romelSic igia moTavsebuli. airis wneva WurWlis kedlebze aixsneba molekulebis siTburi moZraobiT. siTburi moZraobisas molekulebi ejaxebian WurWlis kedlebs da gadascemen garkveul impulss. sxeulis (kedlis) impulsis cvlileba ki (erTeulovan droSi) gansazRvravs masze moqmed Zalas. cxadia, rom Zala, romelic moqmedebs kedelze, damokidebulia kedlis farTobze _ rac metia farTobi, miT metia Zala. amitom miRebulia, rom airis moqmedeba WurWlis kedelze daxasiaTdes ara ZaliT (romlis sidide damokidebulia iseT SemTxveviT faqtorze, rogorc kedlis farTobia), aramed wneviT, romelic warmoadgens Zalis marTobul erTeul farTobze moqmed saSualo Zalas. Tu saSualo Zalis sidides aRvniSnavT F -iT, Zalisadmi marTobuli zedapiris farTobs S -iT, maSin wneva pď&#x20AC;˝

F S

(54.1)

(54.1) formulidan Cans, rom wnevis erTeulad miRebulia erTeulovani Zalis wneva erTeulovan farTobze.

161

SI sistemaSi wnevis erTeulia 1 n/m2, anu niutoni Zalis wneva 1 m2 farTobze. am erTeuls ewodeba pa s k al i (pa). CGS sistemaSi Zalis erTeuliა dini, farTobis erTeuli _ sm2, amitom wnevis erTeuli iqneba dn/sm2. es aris dn Zalis wneva 1 sm2 farTobze. davamyaroT damokidebuleba paskalsa da dn/sm 2-s Soris: 1 pa=1 n/m2=105 dn/104 sm2=10 dn/sm2. xSirad ixmareba sistemgareSe erTeulebi _ fizikuri atmosfero (atm), teqnikuri atmosfero (at) da vercxliswylis svetis milimetri (tori). f i zik u ri , (saSualo)

anu

wneva zRvis

n orm al u ri doneze.

at m os fe ro

rogorc

toriCelim

aris

atmosferos

aCvena,

igi tolia

h  76 sm simaRlis vercxliswylis svetis wnevisa. gamovTvaloT am wnevis

sidide. vercxliswylis svetis ganikveTis farTobi aRvniSnoT S -iT, xolo simkvrive  -Ti, maSin svetis wona P  gSh , xolo wneva p

F P gSh    gh . S S S

(54.2)

verxcliswylis simkvrive   13595 kg/m3=13,595 g/sm3; g  9,80665 m/wm2=980,665 sm/wm2.

am mniSvnelobebis CasmiT (54.2) formulaSi miviRebT: 1 atm=1,013·106 dn/sm2=1,013·105 n/m2=1,013·105 pa=1013 hpa (heqtopaskali), t e q n ik u r i

atm os fe ro

aris 1 kilogramZalis (kgZ) wneva 1 sm 2

farTobze: 1 at=1 kgZ/sm2; davamyaroT damokidebuleba teqnikur atmosferosa da paskals Soris. amisaTvis gavixsenoT, rom 1 kgZ=9,80665 n (kgZ aris Zala, romelic 1 kg masis sxeuls 9,80665 m/wm2 aCqarebas aniWebs): 1 at= 1kgZ/sm2=9,80665 n/10-4 m2=0,980665·105 pa. rogorc vxedavT, teqnikuri atmosfero umniSvnelod (2%-iT) gansxvavdeba fizikuri atmosferogan. mcire wnevis gasazomad miRebulia vercxliswylis svetis milimetri 1 mm simaRlis vercxliswylis svetis wneva. am erTeuls tori ewodeba: 1 tori=1/760 atm=133,322 pa=1333,22 dn/sm2.

162

ยง55. temperatura da misi gazomva temperatura aris erT-erTi ZiriTadi sidide molekulur fizikasa da TermodinamikaSi. temperaturis cneba SemoRebulia sxeulis mTbarobis xarisxis dasaxasiaTeblad. Tavdapirvelad es cneba mecnierebaSi Semovida adamainis mier `civisa~ da `cxelis~ SegrZnebis safuZvelze; magram cxadia, rom Cveni SegrZnebis safuZvelze ar SeiZleba dadgindes sxeulis mTbarobis xarisxis raodenobrivi zoma, vinaidan es SegrZneba uaRresad subieqturia da damokidebulia ara marto sxeulis mTbarobis xarisxze, aramed Cveni organizmis mdgomareobazec. magaliTad, Tu marjvena xels CavyofT cxel wyalSi, marcxenas _ civSi, garkveul dros gavaCerebT da Semdge orive xels CavyobT oTaxis

temperaturis

wyalSi,

marjvena

xelze

SevigrZnobT

sicives,

marcxenaze ki _ siTbos. temperaturis SegrZnebiTi Sefaseba Zlier aris damokidebuli sxeulis Tbogamtarobazec. oTaxis temperaturaze liTonis sxeuli ufro civi gveCveneba, vidre xisa, xolo Tu am sxeulebs gavaxurebT, SegrZneba sapirispiroTi Seicvleba. amrigad, sxeulis mTbarobis xarisxis raodenobrivi zomis dadgenas e.i. temperaturis raodenobriv gansazRvras, agreTve zusti temperaturuli skalis agebas, safuZvlad unda daedos Cveni subieqturi SegrZnebebisagan damoukidebuli obieqturi fizikuri movlenebi da faqtebi. temperaturis cneba SeiZleba ganvixiloT sxvadasxva TvalsazrisiT. Termodinamikis TvalsazrisiT temperatura aris

sidide,

romelic

axasiaTebs Tbogadacemis (siTbos gacvlis) mimarTulebas. Tu ori sxeulis Sexebisas energia siTbos saxiT gadadis erTidan meoreSi, maSin amboben, rom

pirveli

sxeulis

temperatura

ufro

maRalia,

vidre

meoresi.

procesis Sedegad garkveuli drois Semdeg sxeulebs Soris myardeba s iT b u r i w o na s w o rob a, ro c a e ne r gi is W arb i g a da s v l a e rT i s xe u l id a n m e o re Si ar xd e b a . aseTi sxeulebis Sesaxeb amboben, rom m a T i t e m p e r atu re b i e rT n a iri a . TermometriT temperaturis gazomvas safuZvlad udevs swored es faqti _ urTierTSemxebi sxeulebis temperaturebis toloba siTburi wonasworobis damyarebis Semdeg. siTburi wonasworobis damyarebamde am sxeulebis temperaturaTa sxvaoba miT metia, rac metia, gadaxra siTburi wonasworobis mdgomareobidan, e.i. temperaturaTa sxvaoba

163

SeiZleba ganvsazRvroT rogorc siTburi wonasworobis mdgomareobidan gadaxris zoma. c de b iT d a dg e n il ia , r om T u ori s xe u li s iT b u r w o na s w o r o b a Si a m e s am e s T a n, m a S in is i ni s iT b u r w o n as w o rob a S i iq ne b i an e r T m ane T T a n ac . es mniSvnelovani daskvna aris bunebis erT-erTi ZiriTadi kanoni. molekulur-kinetikuri Teoriis TvalsazrisiT te m pe r a tu r a s i di de ,

rom e li c

ax a s i aT e b s

s xe u li s

(i de al u ri

ar i s

a ir is )

n aw il ak e b is g a d at a niT i m oZ ra ob i s s a Su a lo k i ne t ik u r e ne r gi a s . temperaturis am ganmartebas safuZvlad udevs is faqti, rom erTnairi temperaturis mqone ori sxvadasxva airis nawilakebis saSualo kinetikuri energiebi erTnairia. amrigad, molekulur-kinetikuri Teoria xsnis temperaturis fizikur arss, ukavSirebs ra mas sxeulSi mimdinare fizikur process _ nawilakebis siTbur moZraobas. temperaturis gasazomad ZvelaTaganve sargeblobdnen imiT, rom sxeulis temperaturis cvlilebisas icvleba misi Tvisebebi da maSasadame, am Tvisebebis damaxasiaTebeli fizikuri sidideebic (wneva, moculoba, winaRoba da sxva). amitom, temperaturis gasazom xelsawyos _ Termometrs _ Semdegnairad amzadeben: irCeven raime nivTierebas (Termometruli nivTiereba an Termometruli sxeuli) da am nivTierebis Tvisebebis damaxasiaTebel raime fizikur sidides (Termometruli sidide). rogorc erTis, ise meoris arCeva savsebiT nebismieria. magaliTad, sayofacxovrebo Termometrul nivTierebad aRebulia vercxliswyali (ufro iSviaTad _ spirti), xolo Termometrul sidided _ vercxliswylis svetis simaRle. airian TermometrSi Termometruli sxeulia airi, xolo Termometruli sidide _ airis wneva an moculoba da a.S. imisaTvis, rom garkveul temperaturas eTanadebodes garkveuli ricxviTi

mniSvneloba,

unda

gamoviyenoT

Termometrul

sididesa

temperaturas Soris arsebuli raime damokidebuleba. magaliTad, siTxian TermometrSi

aseT

damokidebulebad

irCeven

temperaturasa

svetis

simaRles (siTxis moculobas) Soris arsebul wrfiv damokidebulebas. amis Semdeg saWiroa dadgindes temperaturis erTeuli _ gradusi. misi sididis dadgenac

nebismierad

xdeba:

irCeven

temperaturas

(maT

reperuli

wertilebi ewodeba), Cveulebriv esaa yinulis dnobisa da wylis duRilis 164

temperaturebi intervals

normaluri

yofen

temperaturidan

tol

wnevis

dros

nawilebad

miaweren

garkveul

miRebul

gradusebad. ricxviT

temperaturul

erT-erTs

mniSvnelobas,

ori riTac

ganisazRvreba meore temperaturisa da yvela Sualeduri temperaturis ricxviTi

mniSvnelobebi.

miiReba

garkveuli

te m p e r at u ru li

skala.

cxadia, rom aRwerili meTodiT SeiZleba uamravi temperaturuli skalis miReba. aRmoCnda, rom sxvadasxva Termometris Cvenebebi, romlebSic Termometrul sxeulad gamoyenebulia sxvadasxva nivTiereba, graduirebis miRebuli wesis gamo reperul wertilebSi Tanxvdeba erTmaneTs, magram sxva temperaturaze ara. es gansakuTrebiT exeba siTxian Termometrs. maSasadame, temperaturis calsaxad gansazRvrisaTvis ar aris sakmarisi graduirebis wesis dadgena _ unda SevTanxmdeT Termometruli sxeulis arCevazec. Tanamedrove Termometria dafuZnebulia i d e a lu r i a ir is s k al a z e c , romlis dadgenac xdeba a i ri a ni T e rm om e tr is saSualebiT. airiani Termometri aris idealuri airiT avsebuli daxSuli WurWeli, romelsac aqvs manometri. Tu aRvniSnavT yinulis dnobis temperaturas T0 –iT, wylis duRilis temperaturas Tk –Ti, xolo Sesabamis wnevebs p 0 –iTa da p k –Ti da davuSvebT, rom wnevasa da temperaturas Soris wrfivi damokidebulebaa, SeiZleba davweroT:

p k Tk  p0 T0 fardoba

(55.1)

pk T martivad ganisazRvreba cdiT. maSasadame cnobili iqneba k -is p0 T0

mniSvnelobac. mravalricxovani gazomvebiT dadgenilia, rom

p k Tk   1,3661. p0 T0

gradusis zoma, cxadia, damokidebuli iqneba imaze, Tu ramden nawilad davyofT temperaturul intervals T0 -dan Tk -mde. miRebulia am intervalis dayofa 100 tol nawilad, e.i. Tk  T0  100 ori ukanaskneli tolobidan miiReba, rom idealuri airis skaliT yinulis

dnobis temperetura T0  273,15 graduss, xolo wylis duRilis temperatura 373,15 graduss. airiani TermometriT romelime sxeulis temperaturis gasazomad igi unda SevaxoT am sxeuls da siTburi wonasworobis damyarebis Semdeg 165

aviTvaloT airis wneva TermometriT. sxeulis temperaturas gamoviTvliT formuliT

273,15 p p0 moTavsebuli

T sadac

mdnobare

yinulSi

(55.2) Termometris

airis

wnevaa.

rogorc (55.2) formula gviCvenebs, Cvens skalaze nulovani temperatura es is temperaturaa, romlis drosac idealuri airis wneva iqneba nulis toli*. temperaturul skalas, romlis nulze Termometruli sidide xdeba nulis toli, ewodeba ab s o lu t u ri s k a l a , xolo am skalaze aTvlil temperaturas ewodeba a b s o lu tu ri

te m pe ra tu r a . amrigad Cven mier

aRwerili airiani Termometris skala aris absoluturi skala. mas xSirad kelvinis skalas uwodeben ingliseli mecnieris kelvinis pativsacemad **. absoluturi temperaturis erTeuls ewodeba kelvini, aRiniSneba

K -Ti.

(adre am erTeuls ewodeboda kelvinis gradusi, aRiniSneboda ° K -Ti). teqnikasa da yofa-cxovrebaSi ufro xSirad gamoiyeneba celsiusis skala, romelic absoluturi skalisagan imiT gansxvavdeba, rom yinulis dnobis

temperaturad

(gradusis

imave

sididis

dros)

miCneulia

nuli.

celsiusis skalze aTvlili temperatura aRiniSneba t  C -iT. cxadia, rom celsiusis skalaze aTvlil temperaturasa ( t ) da absolutur skalaze aTvlil temperaturas ( T ) Soris Semdegi damokidebuleba arsebobs:

t  T  273,15; T  t  273,15. realur pirobebSi airian TermometrSi Termometrul sxeulad aRebulia ara idealuri airi (romelic bunebaSi ar arsebobs), aramed realuri. (55.1) toloba zustad sruldeba mxolod idealuri airisaTvis, magram Zlier gaiSviaTebuli realuri airic sakmaod didi sizustiT akmayofilebs am tolobas, amitom airiani Termometris sizuste Zalian maRalia. Cveulebriv, dabali temperaturis gasazomad Termometrul sxeulad iyeneben heli*

es ar niSnavs, rom idealuri airi marTlac SeiZleba ise gavacivoT, rom misi wneva nulis toli gaxdes. ** airiani Termometris skala unaklo ar aris, vinaidan misi Cveneba, marTalia sustad, magram mainc damokidebulia imaze, Tu romeli airia aRebuli Termometrul sxeulad. 1848 wels kelvinma SemoiRo e.w. T erm odin amik uri skala, romelsac ara aqvs zemoaRniSnuli nakli, vinaidan mas safuZvlad udevs karnos Seqcevadi cikli, romlis mqk damokidebulia mxolod saxureblisa da macivris temperaturaze da ar aris damokidebuli muSa sxeulis gvarobaze. Termodinamikuli skaliT temperaturis gansazRvrisaTvis saWiro iqneboda karnos Seqcevadi ciklis Catareba da saxureblisgan miRebuli da macivrisaTvis gadacemuli siTbos raodenobis gazomva, rac praqtikulad ganuxorcilebelia da aucilebelic ar aris, vinaidan Termodinamikuli skala umniSvnelod gansxvavdeba airiani Termometris skalisagan. SeiZleba imis damtkiceba, rom es ori skala zustad Tanxvdeba erTmaneTs, roca Termometrul sxeulad aRebulia idealuri airi.

166

ums, maRali temperaturisaTvis ki _ azots (vinaidan maRal temperaturaze xdeba heliumis gaJonva WurWlis kedlebSi). miuxedavad didi sizustisa, praqtikaSi airiani TermometriT temperaturas TiTqmis ar zomaven, vinaidan igi mouxerxebelia. mas akisria ufro mniSvnelovani

roli

igi

aris

temperaturis

gasazomi

p i rv e la d i

instrumenti, romlis saSualebiTac xdeba meoradi Termometrebis (siTxiani, winaRobis da sxv.) dagraduireba (romlebic naklebad zustia, magram praqtikulad mosaxerxebeli). meoradi Termometrebis dagraduireba xdeba an uSualod airiani Termometris meSveobiT, an, ufro xSirad, e.w. s ae rT aS o r is o p r aq tik u li te m pe r at u r aT a s k al is gamoyenebiT. es skala SemoRebulia 1968 wels zoma-wonis XIII saerTaSoriso konferenciis mier da dafuZnebulia kargad ganmeorebad 11 reperul wertilze, romelTa mniSvnelobebi dadgenilia airiani Termometris meSveobiT (wyalbadis sammagi wertili, wyalbadis duRilis wertili, neonis duRilis wertili, Jangbadis duRilis wertili, wylis sammagi wertili, wylis duRilis wertili da sxv.). SemdegSi (§59) vaCvenebT, rom absoluturi temperatura proporciulia idealuri airis nawilakebis gadataniTi moZraobis saSualo kinetikuri energiisa*: W 

sadac W 

m 2 2

3 kT , 2

T

2 W, 3k

(55.3)

aris nawilakebis gadataniTi moZraobis saSualo kinetiku-

ri energia, xolo

2 3k

_ gadamyvani koeficienti energiis erTeulebidan

(ergi, jouli) temperaturis ereulebSi (gradusi an, rac igivea, kelvini, K ). k -s bolcmanis mudmiva ewodeba: k  1,38  10 16 ergi/KK 1,38  10 23jouli/K .

(55.3) formulis Sesabamisad klasikuri fizika absolutur nuls gansazRvravda rogorc temperaturas, romlis drosac sxeulSi wydeba yovelgvari atomur-molekuluri moZraoba. sinamdvileSi es ase ar aris. am dros wydeba nawilakebis ara yovelgvari, aramed m x ol od s iT b u r i (q a os u r i )

es Sedegi, Tu temperatura Zalian dabali ar aris, marTebulia agreTve realuri airis, siTxisa da myari sxeulisaTvisac.  atomebisa da molekulebis sruli uZraobis daSveba ewinaaRmdegeba haizerbergis ganuzRvrelobis princips.

167

moZraoba. absolutur nulzec sxeulis Semadgeneli nawilakebi sakmaod intensiurad moZraoben _ asruleben e.w. nulovan rxevebs. am moZraobas Seesabameba

minimaluri

energia,

romelic

ukve

aRar

SeiZleba

sxeuls

waerTvas. am energias nulovani energia ewodeba. amrigad, a b s o lu tu ri

nu li

Se iZ le b a

g an im ar tos

r o gor c

i s e T i te m pe r at u r a, rom li s dro s ac s x e u l S i w y de b a Se m a d g e ne l i naw il ak e b is ( at om e b is , io ne b is , m ole k u le b is ) s iT b u r i m oZ r ao b a

rCe b a

m x ol o d

nu l ov an

e ne r gi as T a n

d ak av Si re b u l i

m oZ r a ob a . absoluturi nuli aris yvela SesaZlo temperaturaze dabali da `civi~. dadgenilia, rom absoluturi nuli principulad miuRwevelia, Tumca SesaZlebelia misgan nebismierad mcired gansxvavebuli temperaturis miReba. amJamad miRweuli umciresi temperatura 10 ď&#x20AC;6 K -is rigisaa. (55.3)

formulidan

gamomdinareobs,

rom

absoluturi

temperatura

SeiZleba iyos mxolod dadebiTi, vinaidan kinetikuri energia arsebiTad dadebiTi sididea. marTalia, zogjer, arawonasworuli kvanturi sistemebis aRwerisas formalurad SemoaqvT uaryofiTi temperaturis cneba, magram igi ar

aris

dakavSirebuli

sistemis

mTbarobis

xarisxTan.

ufro

metic,

uaryofiTi absoluturi temperaturis energetikuli are, rogorc es kursis meore tomSi iqneba naCvenebi, mdebareobs ara absoluturi nulis qvemoT, aramed usasrulod maRali temperaturis zemoT. vinaidan temperatura ganisazRvreba nawilakebis moZraobis saSualo energiiT, igi, iseve rogorc wneva, statistikuri sididea _ ar SeiZleba vilaparakoT erTi an ramdenime molekulis temperaturaze. imis gamo, rom temperatura metad mniSvnelovani sididea, misi erTeuli, kelvini, iseve rogorc masis, sigrZisa da drois erTeulebi SI sistemis ZiriTadi erTeulia (temperaturis ganzomileba aRiniSneba ď &#x2018; asoTi). magram saerTaSoriso SeTanxmebiT SI sistemis erTeulis _ kelvinis dadgena xdeba ara

temperaturuli

intervalis,

`mdnobare

yinuli

mduRare

wyali~,

safuZvelze, aramed `absoluturi nuli _ wylis sammagi wertili~ intervalis

safuZvelze*.

gamowveulia imiT,

rom

wylis

sammagi

wertilis

gansazRvra SesaZlebelia gacilebiT ufro meti sizustiT, vidre yinulis dnobis an wylis duRilis wertilebisa. wylis sammag wertils miewereba

wylis sammagi wertili ewodeba temperaturas, romlis drosac wyali, wylis orTqli da yinuli wonasworobaSia.

168

temperatura 273,16K (zustad). amrigad, kelvini ganimarteba, rogorc wylis sammagi wertilis Sesabamisi temperaturis 1/273,16 nawili. vinaidan wylis sammagi wertili tolia 0,01  C -is, gradusis zoma celsiusisa

kelvinis

temperatura

SeiZleba

skalebiT

gamoisaxos

erTnairia.

celsiusis

amitom

nebismieri ( C )

saxis

gradusebiT

kelvinebiT ( K ).

§56. atomisa da molekulis masa da zoma bunebaSi

arsebuli

yvela

sxeuli

Sedgeba

sxvadasxva

atomisagan*. nivTierebas, romelic Sedgeba erTi saxis atomebisagan qimiuri elementi

ewodeba.

qimiuri

elementebia:

wyalbadi,

heliumi,

naxSirbadi,

TuTia, rkina, oqro da sxv. cxadia, maTi raodenobac 89-is tolia. atomebi, erTmaneTTan

daaxloebisas,

mizidvis

Zalis

moqmedebiT

qmnian

mdgrad

sistemas, romelsac molekula ewodeba**. nivTierebas, romelic Sedgeba erTnairi molekulebisagan, romlebic Tavis mxriv Sedgebian sxvadasxva atomebisagan, qimiuri naerTi ewodeba. naerTebia: wyali H2O, sufris marili NaCl, meTani CH4 da mravali sxva***. molekula aris nivTierebis umciresi nawilaki, romelic inarCunebs am

nivTierebis

damaxasiaTebel

yvela

qimiur

Tvisebas.

azri

ara

aqvs

romelime nivTierebis, magaliTad wylis, molekulis naxevarze laparaks da ara imitom, rom misi gayofa ar SeiZleba, aramed imitom, rom gayofis Sedegad miRebuli nawilebi wylis nawilakebs aRar warmoadgenen. imis oratomian,

mixedviT,

samatomian

ramdeni atomi Sedis da

a.S.

molekulas.

molekulaSi, magaliTad:

H2,

vRebulobT O2,

NaCl

oratomiani molekulebia, H2O _ samatomiania, xolo CH4 _ xuTatomiani. xSirad (gansakuTrebiT organul nivTierebebSi) gvxvdeba iseTi molekulebi, romlebic Sedgebian atomebis metad didi ricxvisagan. *

amJamad cnobilia sxvadasxva saxis 111 atomi, magram bunebaSi gvxvdeba mxolod 89, danarCeni ki miRebulia xelovnurad, amaCqareblebSi da atombirTvul reaqtorebSi. ** molekulis Semadgenel atomebs Soris urTierTqmedebis Zala eleqtruli bunebisaa, magram molekulis warmoqmna miekuTvneba kvanturmeqanikur movlenebs da klasikur fizikaSi ar SeiZleba ganxilul iqnes. *** zog SemTxvevaSi molekula SeiZleba Sedgebodes erTnairi atomebisagan. aseTia magaliTad zogierTi airis _ wyalbadis (H2), Jangbadis (O2), azotis (N2), molekulebi; qimiuri elementi _ wyalbadi aris wyalbadis molekulebis erToblioba, Jangbadi _ Jangbadis molekulebisa da a.S.

169

rogorc aRvniSneT, bunebaSi atomTa raodenoba 89-s ar aRemateba, molekulebis raodenoba ki milionobiTaa. amrigad, Cvens irgvliv arsebuli nivTierebebis udidesi mravalsaxeoba iqmneba bunebaSi arsebuli SedarebiT mcirericxovani sxvadasxva saxis atomebis kombinaciebis Sedegad. Tanamedrove

eqsperimentuli

meTodebiT

didi

sizustiT

SeiZleba

molekulisa da masSi Semavali atomebis masisa da zomis gansazRvra. magram atomisa da molekulis masa Zalian mcirea, amitom miRebulia maTi gazomva fardobiTi erTeulebiT. kerZod, nebismieri atomis an molekulis masas adareben naxSirbadis (naxSirbadis C12 izotopis) atomis masis 1/12 nawils. naxSirbadis atomis masis 1/12 nawils ewodeba m a s i s at om u r i e r T e u l i (mae). qimiuri

elementis

fardobiTi

atomuri

masa

(A)

(SemoklebiT

atomuri masa) an nivTierebis fardobiTi molekuluri masa (M) (SemoklebiT _ molekuluri masa)* ganisazRvreba, rogorc am elementis atomis masis an nivTierebis  m A  mae 

molekulis

da M 

masis

fardoba

masis

atomur

erTeulTan

m  . am erTeulebiT wyalbadis atomuri masa AH  1,00797 mae 

(rac imas niSnavs, rom wyalbadis atomis masa 1,00797-jer metia naxSirbadis C12

izotopis atomis masis 1/12-ze), Jangbadis atomuri masa

AO  15,9994,

wyalbadis molekuluri masa M H 2  2  1,00797  2 , Jangbadis molekuluri masa M O2  2  15,9994  32, wylis molekuluri masa 18 da a.S. cdebiT dadgenilia,

rom

1 mae  1,66  10 24 g  1,66  10 27 kg. amitom kg-ebiT gamosaxuli nebismieri atomis (molekulis) masa m  A  1,66  10 24 g  A  1,66  10 27 kg.

nivTierebis masa, romelic Seicavs n molekulas, tolia m  n  M  1,66  10 24 g  n  M  1,66  10 27 kg.

Tu aviRebT nebismier or nivTierebas iseTi raodenobiT, rom maTi m1 da m2 masebi proporciuli iyos maTi molekuluri M 1 da M 2 masebisa e.i.

m1 M 1  , m2 M 2 *

adre am sidideebs ewodeboda atomuri da molekuluri wonebi

170

(56.1)

maSin am nivTierebebSi toli raodenobis molekulebi iqneba. marTlac

n1  M 1  1,66  10 24 g n1  M 1 m1   m2 n2  M 2  1,66  10 24 g n2  M 2 da (56.1) tolobis Sesasruleblad saWiroa, rom n1  n2 . mocemuli nivTierebis iseT raodenobas, romlis masa, gamosaxuli gramebiT, am nivTierebis molekuluri masis tolia, grammolekula, anu moli ewodeba, aRinisneba  -Ti. amrigad, nivTierebis molis  masa ricxobrivad misi molekuluri M masis tolia, Tumca maT Soris arsebobs gansxvavebac:

 -s aqvs ganzomileba kg/moli, xolo M uganzomilebo sididea. radgan sxvadasxva nivTierebis molis masebis fardoba maTi molekuluri masebis fardobis tolia, amitom, zemoTqmulis Tanaxmad, nebismieri nivTierebis erTi moli Seicavs molekulaTa erTsa da imave ricxvs. es ricxvi universaluri mudmivaa da mas avogadros ricxvi ( N A ) ewodeba. cdebiT dadgenilia, rom avogadros ricxvi N A  6,023  10 23moli1 .

amrigad, moli SeiZleba ganvsazRvroT rogorc gansakuTrebuli sididis _ nivTierebis raodenobis _ erTeuli: 1 m o li niv T ie r e b is is r a ode n o b a a, r o m e l ic Se i c av s av o ga d ros r ic xv is t ol i r ao de n ob i s n a w il a k e b i s (m ole k u le b is , a tom e b i s , io ne b is ) ri cxv s . * zoma-wonis XIV generalurma konferenciam (1971 w.) nivTierebis raodenobis erTeuli _ moli _ miiRo SI sistemis meSvide ZiriTad erTeulad (sigrZis, masis, drois, temperaturis, denis Zalia da sinaTlis Zalis erTeulebis Semdeg). avogadros ricxvis saSualebiT SeiZleba ganisazRvros nebismieri molekulis m absoluturi masa (masa kg-obiT). amisaTvis 1 molis masa  unda gavyoT N A -ze:

m wyalbadisaTvis μ ≃ 2  10 3

 NA

kg.

kg , amitom wyalbadis molekulis masa moli

nivTierebis raodenobis am erTeulis moZvelebuli saxelwodebebi: gramatomi, grammolekula, gramioni, gramekvivalenti Secvlilia universaluri saxelwodebiT moli.

171

m

2  10 3 ≃ 0,38 1026 kg . 6,023  10 23

JangbadisaTvis μ ≃ 32 103

kg moli

32 ≃ 5,3 1026 kg. 6,023  10 23 SevafasoT molekulis zoma. Cven aRvniSneT, rom siTxeSi molekulebs m

Soris manZili mcirea _ siTxis molekulebi mWidrodaa erTmaneTTan ganlagebuli. amitom erTi molekulis moculoba miaxloebiT SeiZleba gavigoT, Tu romelime siTxis, magaliTad, wylis 1 molis moculobas, gavyobT molekulebis raodenobaze molSi, e.i. avogadris ricxvze. wylis simkvrivea 10 3

kg , amitom 1 moli (e.i. 0,018 kg) wyali daikavebs 0,018·10-3 m3 moculobas m3

da, maSasadame, erTi molekulis moculoba

0,018  10 3 ≃ 30 1030 m3 . 23 6,023  10 wylis molekulis wiriTi zoma daaxloebiT iqneba V

30  10 30 m 3 ≃ 3 1010 m.

§57. idealuri airi. idealuri airis cdiseuli kanonebi. nivTierebis Semadgeneli nawilakebi (atomebi, molekulebi, ionebi) rTuli eleqtruli sistemebia, romelTa Soris moqmedebs urTierTmizidvisa da ganzidvis Zalebi. am Zalebis sidide damokidebulia manZilze da manZilis gazrdiT swrafad utoldeba nuls. magram, rogorc aRvniSneT, airSi manZili molekulebs Soris Zalian didia _ bevrad aRemateba TviT molekulebis zomebs. amitom SeiZleba pirvel miaxloebaSi ugulebelvyoT rogorc molekulebis urTierTqmedeba (miviCnioT, rom urTierTqmedebas, drekadi birTvebis msgavsad, adgili aqvs mxolod drekadi urTierTdajaxebis dros), aseve molekulebis sakuTari moculoba (airis saerTo moculobasTan

SedarebiT,

romelic

ganisazRvreba

WurWlis

moculobiT),

e.i.

ganvixiloT isini rogorc nivTieri wertilebi. airs,

r om lis

m ole k u l aT a s ak u T ar i m ocu l ob a d a u r T i -

e r T m i zi dv is a d a g an zi dv i s Z ale b i u gu le b e ly o f il i a, i de a l u r i a ir i e w od e b a . 172

cxadia,

rom

idealuri airi bunebaSi ar

arsebobs,

magram

aseTi

modelis SemoReba mizanSewonilia, vinaidan igi airis garkveul kanonzomierebaTa dadgenis saSualebas iZleva. airis gaiSviaTebasTan erTad manZili molekulebs Soris kidev ufro izrdeba da Cveni

daSvebebi

sul

ufro marTebuli

xdeba.

amitom,

yoveli realuri

airi

sakmao

gaiSviaTebisas Tavisi TvisebebiT uaxlovdeba idealur airs. zogierTi airi, mag. Jangbadi, azoti, haeri, Cveulebriv pirobebSic ki (oTaxis temperaturisa da atmosferuli wnevis dros),

mcired gansxvavdeba

idealuri

airisagan.

Tavisi TvisebebiT

idealur

airTan

gansakuTrebiT axlosaa wyalbadi da heliumi.

fizikur

sidides,

romelic

axasiaTebs

sistemis

mdgomareobas,

p a r am e tr i ewodeba. miRebulia, rom mocemuli masis airis mdgomareoba daxasiaTdes misi wneviT, moculobiTa da temperaturiT. airis mdgomareobis Secvlas,

romelic

Tan

sdevs

parametrebis

cvlilebas,

ai ru li

p r oc e s i ewodeba, process, romelic mimdinareobs mudmiv temperaturaze (am dros icvleba airis moculoba da wneva) izoTermuli procesi ewodeba. temperaturis cvlilebiT airis moculobis cvlilebis process mudmivi wnevis

pirobebSi

iz ob aru li

procesi

ewodeba.

xolo

temperaturis

cvlilebiT airis wnevis cvlilebis process mudmivi moculobis pirobebSi i zoq or u li procesi ewodeba. molekulur-kinetikuri Teoriis safuZvelze airis Tvisebebis Seswavlamde, mokled ganvixiloT e.w. airis cdiseuli kanonebi (boil-mariotis, geilusakis, Sarlis, daltonisa da avogadrosi), romlebic dadgenil iqna empiriulad molekulur-kinetikuri Teoriis Camoyalibebamde gacilebiT adre. aRsaniSnavia, rom es kanonebi airis Tvisebebs zustad asaxavs mxolod iseT pirobebSi, romlebic Zalian ar gansxvavdeba normalurisagan. metad dabali temperaturisa da maRali wnevis pirobebSi adgili aqvs sagrZnob gadaxrebs. 1 . b o i l -m ar io ti s

k an on i .

1662 w. ingliselma mecnierma boilma

(1627-1691) da 1667 w. frangma mecnierma mariotma (1620-1684) erTmaneTisagan damoukideblad daadgines mudmivi temperaturis dros airis wnevasa da moculobas Soris damokidebulebis kanoni: a i r i s m oce m u li m as is w ne v a m u d m iv i te m pe r a tu r is

d r os

i c v l e b a m o cu l ob i s u k u p ro po rci u l a d: p1 V2 ď&#x20AC;˝ , p 2 V1

anu 173

(5 7 . 1 )

p1V1  p2V2 ,

pV  const.

(57.2)

boil-mariotis kanonis grafiki ( pV ) koordinatebiT aris hiperbola. mas izoTerma ewodeba. 57.1 suraTze mocemulia sxvadasxva temperaturaze agebuli izoTermebis ojaxi. 2. g e i - lu s ak i s k an on i. 1802 wels frangma mecnierma gei-lusakma (1778-1850) daadgina kanoni, romlis Tanaxmadac a i r is m o ce m u li m a s is m o c u l ob a m u dm iv i w ne v is dr os te m p e -

sur. 57.1

r a tu r is m i xe dv iT w r f iv a d ic v le b a V  V0 1  α t ,

(57.3)

sadac V0 aris airis moculoba 0 C -ze, xolo V _ moculoba t  C -ze.  -s 

ewodeba moculobiTi gafarToebis koeficienti. (57.3)-dan



V  V0 , V0 t

e.i.  ricxobrivad tolia 0  C temperaturis

airis

erTeuli

moculobis

namatisa

1 C -iT gaTbobis dros. gei-lusakma daadgina, rom m o cu l ob iT i g a f a r T oe b is

k oe f ici e n ti m u dm iv i w ne -

sur. 57.2

v is dr os y v e l a a ir is aT v is e rT i d a i gi v e a d a r ic xob riv a d

1 grad1. 273,15 57.2. suraTze gamosaxulia gei-lusakis kanonis grafiki, romelsac izobara



ewodeba. izobara temperaturaTa RerZs kveTs t  273,15 C wertilSi. am temperaturas absoluturi nuli vuwodeT. suraTidan Cans, rom absolutur nulze Vt  0, e.i. nivTiereba qreba. es aSkarad araswori Sedegi kvlav adasturebs zemoT naTqvams imis Sesaxeb, rom airis cdiseuli kanonebis gamoyeneba Zalian dabali temperaturis dros ar SeiZleba (dabali temperaturis areSi grafiki amitomaa wyvetili), nivTiereba am dros airad mdgomareobaSi aRar aris. (57.3)

formula

ufro

gamartivdeba,

celsiusis

temperaturas

SevcvliT absoluturi temperaturiT

1 273,15  t T   V  V0 1  t   V0 1  t   V0  V0T  V0 , 273,15 T0  273,15  174

(57.4)

sadac T0 



 273,15K Seesabameba 0  C -s celsiusis skalaze. (57.4)-dan

V T  , anu, sazogadod, V0 T0 V1 T1  V2 T2 da

V  const , T e . i. i zob ar u li pr oce s is dr os ai ri s m oc e m u l i m a s is m o c u l ob a i c v l e b a m is i ab s o lu t u r i t e m p e r a tu r is p ro po r c iu l a d. 3 . S ar li s k a no ni. 1800 wels frangma mecnierma Sarlma (1746-1823) daadgina kanoni, romlis Tanaxmadac m u dm iv i

m o cu l ob i s

dr os

a i r is

m oce m u l i

te m pe r a tu r is

m as is

m i xe dv i T

w ne v a

w r f iv ad

icvleba:

p  p0 (1  t ), sadac

(57.5)

p 0 -aris airis wneva 0  C -ze, xolo

p _ wneva t  C -ze.

 -s ewodeba

wnevis Termuli koeficienti. (57.5)-dan

 e.i. 

p  p0 , p0 t

ricxobrivad tolia 0  C -ze wnevis

TiToeuli erTeulis namatisa 1 -iT gaTbobis

dros.

aRmoCnda,

p r o ce s is d ros sur. 57.3



rom

i z oq oru li

y v e l a ai ris aT v is

e rT i d a i giv e a da r i c xob r iv a d

1 grad1 . 273,15 aqac, Tu celsiusis temperaturas SevcvliT absoluturi temperaturiT,



miviRebT:

p1 T1  p 2 T2 anu

e . i.

p  const , T izoq oru li p roc e s i s

d r os

a ir i s m oce m u l i m as is w ne v a i c sur. 57.4

v l e b a m is i a b s o lu t u ri t e m p e r at u ri s pr o po rci u l a d. 175

57.3

suraTze

mocemulia

Sarlis

kanonis

grafiki,

mas

izoqora

ewodeba. izobaruli da izoqoruli procesebi grafikulad SeiZleba gamoisaxos pV koordinatebiTac*.

cxadia, am sistemaSi izobara moculobaTa

RerZis paraleluri wrfea (sur. 57.4 a), xolo izoqora _ wnevaTa RerZis paraleluri wrfe (sur. 57.4 b). 4. d a l to nis k an on i. davuSvaT, garkveul WurWelSi moTavsebulia airTa meqanikuri narevi (romlebic erTmaneTTan qimiurad ar moqmedeben), vTqvaT, haeri. WurWlidan gamovdevnoT yvela airi erTis garda. davuSvaT, es airia Jangbadi. maSin igi daikavebs WurWlis mTel moculobas. am airis wnevas WurWlis kedlebze pa rc i alu r i (kerZo) wneva ewodeba. e.i. parcialuri wneva ewodeba im wnevas, romelsac awarmoebda narevis TiToeuli mdgeneli WurWelSi sxva airi rom ar yofiliyo. 1801 wels ingliselma mecnierma dalto nm a (1766-1844) daadgina kanoni, romlis Tanaxmadac: a ir T a n a r e v i s s ru l i w ne v a to li a m as S i S e m av a li c alk e u li a ire b is p a r c i alu ri w ne v aT a j am is a p   pi ,

sadac p narevis sruli wnevaa, xolo p i _ misi i -uri komponentis parcialuri wneva. 5.

a v o g a dro s

k a no ni.

1811

wels

italielma

mecnierma

a v o g a dr om (1776-1856) daadgina kanoni, romlis Tanaxmadac: e r T n a iri w ne v i s a d a te m pe r atu r is

pi rob e b S i ne b is m ie r i

a i r i s 1 m o li ik av e b s e rT s a d a im av e m o cu l ob a s . kerZod, normalur pirobebSi t  0  C, p  1atm  es moculoba

V 0

l m3 3  22,42  22,42  10 . moli moli

avogadros kanoni saSualebas gvaZlevs gamovTvaloT normalur pirobebSi molekulebis n0 ricxvi nebismieri airis moculobis erTeulSi n0 

6,023  10 23 moli1 NA   2,7  10 25 m 3  2,7  1019 sm 3 3  3 V μ0 22,42  10 m moli

am ricxvs lo Sm it is ricxvi ewodeba.

nebismieri porocesis gamosaxva ( p, V ) koordinatebiT gansakuTrebiT mosaxerxebelia procesis dros Sesrulebuli muSaobis ganxilvisas.

176

§58. idealuri airis mdgomareobis gantoleba. airis universaluri mudmiva boil-mariotis, gei-lusakisa da Sarlis kanonebi asaxavs airuli procesis kerZo SemTxvevebs (e . w . i zo p ro ce s e b s ), roca airis mdgomareobis damaxasiaTebeli sami sidididan erTi mudmivia, danarCeni ori ki icvleba. axla ganvixiloT airuli procesis zogadi SemTxveva, roca mdgomareobis damaxasiaTebeli sami sidididan samive icvleba da davamyaroT maT Soris kavSiri. gantolebas, romelic amyarebs damokidebulebas wnevas, moculobasa da temperaturas Soris Soris

m d gom a re ob is

g an to le b a

e w ode b a. aseTi gantoleba idealuri airisaTvis miiRo 1834 wels frangma mecnierma k la pe ir onm a (1799-1864) boil-mariotisa da gei-lusakis kanonebis gaerTianebiT. miviRoT es gatoleba. vTqvaT, mocemuli gvaqvs airis ori mdgomareoba _ sawyisi (I), romlis parametrebia (I)

p1 ,V1 , T1 , da saboloo (II), romlis parametrebia

(II)

p2 ,V2 , T2 . gadaviyvanoT

airi

izobarulad

mdgomareobidan

Sualedur

(I’)

mdgomareobaSi, romlis parametrebia

p1 ,V ' , T2

(I’)

radgan es gadasvla izobarulia, amitom marTebulia gei-lusakis kanoni, V1 T1  , V ' T2

saidanac

T2 (58.1) . T1 vinaidan Sualedur (I’) da saboloo (II) mdgomareobaTa temperaturebi V '  V1

erTnairia, SeiZleba gamoviyenoT boil-mariotis kanoni: p1  V '  p2  V2

SevitanoT V ' -is mniSvneloba (58.1) gantolebidan:

p1  V1

T2  p 2V2 , T1

anu

p1V1 p 2V2  . T1 T2 177

(58.2)

gantoleba,

romelic

warmoadgens

boil-mariot-gei-lusakis

gaerTianebul kanons, zogadad ase Caiwereba: pV  const. T Tu am mudmivas mocemuli masisaTvis aRvniSnavT B –Ti, miviRebT: pV (58.3)  B. T (58.3) gantoleba gamosaxavs idealuri airis mdgomareobas. mas klapeironis

gantoleba ewodeba. klapeironis gantolebas aqvs nakli, rac imiT gamoixateba, rom

mudmivas mniSvneloba sxvadasxva airisaTvis sxvadasxvaa*. am naklis gamosasworeblad 1875 wels rusma mecnierma mendeleevma (1834-1907) ramdenadme saxe ucvala klapeironis gantolebas, gaaerTiana ra igi avogadros kanonTan. davuSvaT, mocemuli gvaqvs 1 moli airi. misi moculoba aRvniSnoT

V  –Ti. avogadros kanonis Tanaxmad erTnairi wnevisa da temperaturis pirobebSi V  yvela airisaTvis erTnairia. maSasadame, Tu (58.3) gantolebas davwerT airis ara nebismieri masisaTvis, aramed erTi molisaTvis, maSin B mudmivas

mniSvneloba

yvela

airisaTvis

erTnairi

iqneba.

mudmiva

aRiniSneba R -iT da mas airis universaluri mudmiva ewodeba. amrigad, (58.3) gantoleba, 1 moli airisaTvis miiRebs saxes:

pV

 R,

(58.4)

pV  RT

(58.5)

T anu (58.5)

gantolebas

ewodeba

k la pe ir on -m e nd e le e v is

ga nt ole b a.

ganvazogadoT es gantoleba airis nebismieri masisaTvis: Tu airis masa aris

m , xolo 1 molis masa _  , maSin fardoba



iqneba molebis ricxvi. (58.5)

tolobis orive mxare gadavamravloT am ricxvze:

pV V 







RT .

 V aris m masis moculoba, amitom

pV 



RT .

(58.6)

vinaidan, erTnairi masis sxvadasxva airi, erTnairi wnevisa da temperaturis pirobebSi sxvadasxva moculobas ikavebs.

178

(58.6)

gantoleba

warmoadgens

klapeiron-mendeleevis

gantolebas

airis

nebismieri masisaTvis. (58.6) gantolebidan SeiZleba miviRoT airis simkvrivis gamosaxuleba: m p  , V RT e.i. airis simkvrive pirdapirproporciulia wnevisa da ukuproporciulia



absoluturi temperaturisa. gamovTvaloT R mudmivas ricxviTi mniSvneloba: avogadros kanonis Tanaxmad, nebismieri airis 1 moli normalur pirobebSi ikavebs 22,42

l moli

moculoba, e.i. Tu p  p0  1 atm da T  T0  273,15K , maSin

l moli SevitanoT es mniSvnelobebi (58.4) tolobaSi: Vμ0  22,42

R

p0V0 T0



1atm  22,42

moli l  atm  0,082 273,15K K  moli

SI sistemaSi: 1atm  1,013  10 n/m , 5

V 0

m3  22,42  10 , moli 3

amitom

1,013  10 5 n/m 2  22,42m 3 /moli j  8,31 . 273,15K K  moli airis universaluri mudmiva SeiZleba gamovsaxoT siTburi erTeuleR

biT. cnobilia (§72), rom

1kal  4,1868j.

e.i. 1j 

1 kal, 4,1868

amitom

kal . K  moli vinaidan klapeiron-mendeleevis gantoleba aris izoprocesebis Sesa-

R ≃2

bamisi gantolebebis gaerTianeba, amitom, bunebrivia, rom aqedan, rogorc kerZo SemTxveva, gamodis idealuri airis izoprocesebis samive kanoni. marTlac, Tu T  const , maSin (58.6)-dan

pV  const , miviReT boil-mariotis kanoni. Tu p  const , maSin 179

V mR   const , T p rac gei-lusakis kanons gamosaxavs. Tuki V  const , maSin

p mR   const _ Sarlis kanoni. T V

§59. airis kinetikuri Teoriis ZiriTadi gantoleba yoveli airi awarmoebs wnevas im WurWlis kedlebze, romelSic is imyofeba. airis wneva WurWlis kedlebze aixsneba molekulebis siTburi moZraobiT. moZraobisas molekulebi ejaxebian WurWlis kedels da gadascemen mas moZraobis raodenobas (impulss), ris gamoc kedelze moqmedebs Zala. calkeuli molekulis mier gadacemuli impulsi, misi masis simciris gamo, cxadia, mcirea. magram vinaidan airi Seicavs molekulebis kolosalur raodenobas, xolo dajaxebebi Zalian xSiria, kedelze sakmaod didi Zala moqmedebs. gamovTvaloT airis wneva WurWlis kedlebze molekulur-kinetikuri Teoriis safuZvelze. winaswar SevniSnoT, rom molekulebis dajaxebas WurWlis kedlebze aqvs drekadi xasiaTi, e.i. am dros icvleba molekulis siCqaris mxolod mimarTuleba, sidide ki ara. marTlac, molekulis dajaxeba rom aradrekadi iyos, igi, TiToeuli dajaxebisas WurWlis kedels gadascemda energiis nawils, ris Sedegadac molekulis saSualo kinetikuri energia Semcirdeboda da airi gacivdeboda, rac sinamdvileSi ar xdeba. vTqvaT, airi moTavsebulia V moculobis kuburi formis WurWelSi (sur. 59.1). Tu kubis wibos aRvniSnavT l -iT, maSin TiToeuli waxnagis farTobi

S  l 2 , xolo WurWlis mocu-

loba V  l 3 . koordinatTa RerZebi mivmarToT kubis

wiboebis

gaswvriv.

kubSi

molekulebi

moZraoben sruliad qaosurad*, moZraobis arc sur. 59.1

erT mimarTulebas ara aqvs upiratesoba, e.i.

saSualod, yoveli mimarTulebiT moZravi molekulebis raodenoba erTna*

molekulaTa moZraobis upiratesi mimarTuleba rom arsebobdes, maSin am mimarTulebis marTobul kedelze airis wneva meti iqneboda, vidre sxva kedlebze. rac sinamdvileSi ar xdeba, cdebiT dadgenilia, rom airis wneva sxvadasxvanairad orientirebul zedapirze erTnairia.

180

iria. amitom SeiZleba davuSvaT, rom molekulebis saerTo raodenobis 1/3 moZraobs

RerZis

gaswvriv

(marcxnidan

marjvniv

marjvnidan

marcxniv), 1/3 moZraobs Oy RerZis gaswvriv, xolo 1/3 Oz RerZis gaswvriv. Tu molekulebis saerTo raodenobas aRvniSnavT

n -iT, maSin erT-erTi

RerZis gaswvriv moZravi molekulebis raodenoba 1 n'  n . 3

sazogadod,

molekulebis

siCqare

sxvadasxvaa.

aRvniSnoT

isini

1 , 2     n -iT. avirCioT erT-erTi molekula, romelic moZraobs marjvena '

kedlisken 1 siCqariT. kedelTan dajaxebis Semdeg (vinaidan es dajaxeba

1

drekadia) molekula ukuiqceva imave sididis

siCqariT*. molekulis

impulsis cvlileba iqneba:

 m1  m1  2m1 . niutonis

III kanonis Tanaxmad kedlis

impulsi

Seicvleba

 2m1

sididiT. amrigad, 1 molekulis erTxel dajaxebis Sedegad kedlis impulsi icvleba 2m1 sididiT. Tu dajaxebaTa ricxvs wamSi aRvniSnavT Z -iT, maSin erTi molekulis dajaxebis Sedegad kedlis impulsis cvlileba wamSi iqneba 2m1 Z . impulsis cvlilebis kanonis Tanaxmad

m  Ft  ,

sxeulis impul-

sis cvlileba 1 wm-Si am sxeulze moqmedi Zalis tolia. maSasadame, erTi molekulis 1 wm-Si dajaxebaTa Sedegad dartymis saSualo Zala kedelze

F1  2mυ1  Z . gamovTvaloT

(59.1)

Z . molekula daejaxeba ra marjvena kedels, misgan

ukuiqceva, daejaxeba mopirdapire kedels, misgan ukuiqceva da mxolod amis Semdeg (e.i. 2l

manZilis gavlis Semdeg) kvlav ejaxeba marjvena kedels.

maSasadame, dajaxebaTa ricxvi rom miviRoT, erT wamSi gavlili manZili (e.i. molekulis siCqare 1 ) unda gavyoT kedelze or momdevno dajaxebebs Soris gavlil manZilze (e.i. 2l -ze): Z

1 2l

SevitanoT (59.1)-Si:

aq vixilavT SemTxvevas, roca molekula kedels ejaxeba marTobulad. suraTi ar icvleba molekulis iribi dakaxebisas.

181

2m12 m12 F1   . 2l l F1 aris saSualo Zala, romelic moqmedebs kedelze erTi molekulis dajaxebis Sedegad. magram marjvena kedels ejaxeba n ' molekula, romelTa  siCqareebia 1 , 2     n' , amitom saSualo jamuri F Zala, romelic moqmedebs kedelze yvela n ' molekulis dajaxebis Sedegad, toli iqneba:

m n' m 2 m12 m 22 F      1   22       n2' l l l l 2





tolobis marjvena mxare gavamravloT da gavyoT n ' –ze: 2 2 2 n ' m 1   2       n' F  . l n'

(59.2)

sidide

12   22       n2

2

aris molekulebis siCqareTa kvadratebis saSualo mniSvneloba. kvadratul fesvs am gamosaxulebidan, e.i.

  2

12   22       n2

s a Su al o k v a dr a tu l i s i Cq are ewodeba*, aRiniSneba igi υkv -iT. 1 SevitanoT (59.2)-Si  2 da gaviTvaliswinoT, rom n '  n , miviRebT: 3 F

1 nm 2  . 3 l

tolobis orive mxare gavyoT kedlis farTobze l 2 –ze:

F 1 nm 2   . l2 3 l3

saSualo kvadratuli siCqare

 kv

gansxvavdeba saSualo ariTmetikuli



 siCqarisagan,

romelic miiReba yvela molekulis siCqareTa jamis gayofiT maT ricxvze. Tu molekulebis moZraoba sruliad qaosuria,

saSualo kvadratuli siCqare

daaxleobiT 9%-iT metia

saSualo ariTmetikulze. saerTod, saSualo kvadratuli siCqaris Semotana imiT aris gamowveuli, rom romelime RerZis paralelurad moZravi molekulebis siCqareTa jami SeiZleba

nulis

toli

aRmoCndes,

vinaidan

qaosurobis

gamo

maTi

moZraoba

urTierTsapirispiro mimarTulebiT erTnairad albaTuria, siCqareTa kvadratebis jami ki nulis toli ar SeiZleba iyos, vinaidan siCqaris kvadrati dadebiTi sididea.

182

F p l2

aris wneva, xolo

n  n0 l3

_ molekulebis raodenoba moculobis

erTeulSi, amitom

1 p  n0 m 2 . 3

(59.3)

amrigad, airis wneva WurWlis kedelze ganisazRvreba molekulis masiT, molekulebis kvadratis

raodenobiT

saSualo

erTeul

mniSvnelobiT.

moculobaSi (59.3)

formulas

maTi mivceT

siCqareebis sxva

saxe.

amisaTvis misi marjvena nawili gavamravloT da gavyoT 2-ze:

p magram

2 m 2 ;  n0  3 2

m 2  W aris molekulis gadataniTi moZraobis saSualo kinetikuri 2

energia, amitom

p

2 n0W , 3

(59.4)

e.i. airis wneva proporciulia erTeul moculobaSi molekulaTa raodenobisa da molekulis saSualo kinetikuri energiisa. (59.3) gantolebas an mis ekvivalentur (59.4) gantolebas ewodeba a ir is k in e ti k u r i T e or i is Z ir iT ad i g an to le b a . * SevniSnoT, rom es gantoleba Cven gamoviyvaneT statistikuri meTodis gamoyenebiT: airi ganvixileT rogorc qaosurad moZravi Zalian didi raodenobis molekulebisagan Sedgenili makroskopuli sistema

misi

mikroprocesebis

makroskopuli

maxasiaTebeli

damaxasiaTebeli sididis

wneva

gamovsaxeT

(mikrosididis) _ molekulis

saSualo kinetikuri energiis meSveobiT. (59.4) gantolebis orive mxare gavamravloT 1 molis V  moculobaze: 2 n0W V μ . 3 aris molekulebis raodenoba 1 molSi, anu avogadros pV μ 

vinaidan n0V  N A ricxvi, amitom

2 N AW . 3 klapeiron-mendeleevis gantolebis Tanaxmad pV  RT da miviRebT: pV μ 

ZiriTadi mas ewodeba imitom, rom igi erTgvari xidia mikroskopul da makroskopul sidideebs Soris – calkeuli nawilakebis damaxasiaTebel mikroskopul sidideebs akavSirebs maroskopul sididesTan, wnevasTan.

183

2 (59.5) N AW  RT . 3 (59.5) gantoleba akavSirebs molekulis gadataniTi moZraobis saSualo pV μ 

kinetikur energias airis makroskopul parametrebTan _ wnevasTan, moculobasTan da absolutur temperaturasTan. (59.5)-dan

3 R (59.6) T. 2 NA mudmivi sidideebia, maTi fardobac mudmivi iqneba. W 

vinaidan R da N A

igi aRiniSneba k -Ti da ewodeba bolcmanis universaluri mudmiva 8,32

j K  moli

j ergi R .   1,38  10 23  1,38  10 16 23 1 N A 6,023  10 moli K K SevitanoT (59.6) formulaSi bolcmanis mudmiva k

3 kT . 2 miviReT mniSvnelovani Sedegi: m o le k u le b i s W 

(59.7) g ad a ta ni T i

m oZ r aob is

s aS u a l o k i ne tik u ri e ne rg i a d am ok i de b u l i a m xo lo d t e m p e r a t u r az e , s a xe l d ob r pi r da p ir pr opo rc iu l i a ab s o lu t u ri te m pe r a t u r i s a . amitomaa, rom absoluturi temperaturis zomad miRebulia molekulebis gadataniTi moZraobis saSualo kinetikuri energia. Tu (59.7) formuliT gamosaxul W -s SevitanT (59.4) formulaSi, airis kinetikuri Teoriis ZiriTadi gantoleba miiRebs saxes: e.i.

airis

wneva

p  n0 kT . proporciulia molekulaTa

raodenobisa

(59.8) erTeul

moculobaSi da absoluturi temperaturisa. (59.4)

formuliT

gamosaxuli

airis

kinetikuri

Teoriis

ZiriTadi

gantoleba Cven miviReT Teoriulad. Teoriuli daskvnebis siswore ki yovelTvis mowmdeba eqsperimentiT. Cven gavecaniT airis cdiseul kanonebs. Tu airis kinetikuri Teoriis ZiriTadi gantoleba sworad asaxavs sinamdviles, misgan unda gamomdinareobdes zemoxsenebuli cdiseuli kanonebi. magaliTad, (59.4) formulidan miviRoT boil-mariotis kanoni. amisaTvis masSi SevitanoT mniSvneloba n0  p

n , maSin V

2n W, 3V

saidanac pV 

2 nW . 3

184

(59.9)

airis mocemuli masisaTvis n  const , mudmiv temperaturaze molekulis saSualo kinetikuri energiac W  const , e.i. (59.9) tolobis marjvena mxare mudmivia da, maSasadame, miviReT,

rom

airis

mocemuli

pV  const . masisaTvis

mudmivi

temperaturis

dros

moculobisa da wnevis namravli mudmivi sididea _ boil-mariotis kanoni. (59.4) formulidan aseve martivad miiReba daltonis kanonic: davuSvaT, gvaqvs airTa narevi. erTeul moculobaSi pirveli gvaris airis molekulaTa raodenoba aRvniSnoT n01 -iT, meore gvarisa _ n02 -iT, mesamesi _ n03 -iT da a.S. maSin,

cxadia,

rom

molekulaTa

saerTo

raodenoba

moculobis

erTeulSi toli iqneba n0  n01  n02  n03    

airTa narevis wnevisaTvis miviRebT: p

2 2 2 n01W  n02W  n03W     3 3 3

(59.10)

magram

2 n01W  p1 3 aris pirveli gvaris airis parcialuri wneva. analogiurad

2 n02W  p 2 , 3

2 n03W  p3 3

da a.S. aris meore, mesame gvaris da a.S. airis parcialuri wnevebi. amitom (59.10)-dan viRebT: p  p1  p2  p3    

e.i. airTa narevis sruli wneva masSi Semavali calkeuli airis parcialur wnevaTa jamis tolia _ daltonis kanoni. sabolood, gamovTavloT saSualo kvadratuli siCqaris mniSvneloba. ganmartebis Tanaxmad

kv   2 , magram

3 2  kT 2W 3kT 2   2  , m m m amitom

185

(59.11)

3kT . m

υkv  gamosaTvlelad SevcvaloT k 

ufro

mosaxerxebeli

(59.12) formulis

misaRebad

(59.12)-Si

R  da m  , miviRebT: NA NA 3RT . μ gviCvenebs, υkv 

miRebuli

formulebi

rom

molekulaTa

saSualo

kvadratuli siCqaris sidide miT metia, rac metia airis temperatura da rac ufro msubuqia igi. aqedan cxadia, rom yvelaze meti siCqare mocemul temperaturaze aqvs wyalbadis molekulas. magaliTad, saSualo

kvadratuli

siCqare

toli

iqneba

T  300K -ze misi

(   2  10 3

kg/moli,

R  8,32 j/ K mol.) υkv ≃ 2103 m/wm.

Jangbadis moluri masa (   32  10 3 kg/moli) 16-jer metia wyalbadisaze, amitom Jangbadis molekulis saSualo kvadratuli siCqare 4-jer naklebi iqneba.

§60. molekulebis ganawileba siCqareTa mixedviT. maqsvelis kanoni airis kinetikuri Teoriis ZiriTadi gantolebis gamoyvanisas visargebleT

molekulebis

saSualo

kvadratuli

siCqaris

cnebiT.

saSualo

kvadratuli siCqare aris molekulaTa moZraobis s t at is t ik u ri maxasiaTebeli (e.i. miRebulia didi raodenobis sxvadasxva sididis siCqareebis gasaSualoebiT), romlis Semotanas safuZvlad udevs daSveba,rom mocemul temperaturaze airis sxvadasxva molekulis siCqare gansxvavdeba erTmaneTisagan. molekulaTa siCqareebis sxvadasxvaoba (romelic dasturdeba mravali cdiseuli faqtiT) ganpirobebulia sistemaSi (airSi) mimdinare mikroprocesebiT, kerZod, qaosurad moZravi molekulebis urTierTdajaxebebiT, romlis drosac xdeba energiebis gadanawileba da icvleba TiToeuli molekulis siCqaris rogorc sidide, ise mimarTulebac. muxedavad calkeuli molekulebis siCqareebis ganuwyveteli cvlilebisa, rodesac airi im-

186

yofeba wonasworul mdgomareobaSi*, masSi yovelTvis arsebobs molekulaTa garkveuli stabiluri ganawileba siCqareTa mixedviT. rogoria es ganawileba? gansakuTrebuli yuradReba unda mivaqcioT ganawilebis Sesaxeb amocanis arss. molekulebis siCqareTa mixedviT ganawilebis kanonis dadgena ar niSnavs SegveZlos zustad (maTematikurad zustad)

mocemuli siCqariT

imas,

rom

moZravi molekulebis

raodenobis gageba. sakiTxis ase dasmas azri ara aqvs, vinaidan yovel mocemul momentSi aseTi molekulebis raodenoba SeiZleba nulis toli iyos (vinaidan molekulaTa raodenoba sasruloa, sxvadasxva SesaZlo siCqareTa mniSvnelobebi ki usasruloa). molekulebis siCqareTa mixedviT ganawilebis kanonis dadgena niSnavs molekulaTa im raodenobis

 -dan

gansazRvras,

romelTa siCqareebi moTavsebulia siCqareTa garkveul SualedSi

  d  -mde.

magaliTad, qveynis mosaxleobis aRwerisas interesdebian mcxovrebTa im

ricxviT, romelTa asakia 0-dan 10 wlamde. 10-dan 20 wlamde da a.S. da ara zustad mocemuli asakis (amdeni wlis, Tvis, dRis da a.S.) mcxovrebTa raodenobiT, vinaidan aseTi asakis moqalaqe SeiZleba saerTod ar aRmoCndes qveyanaSi.

siCqareTa

mixedviT

molekulebis

ganawilebis

sakiTxi

1860

wels

Teoriulad Seiswavla ingliselma mecnierma maqsvelma (1831-1879). sakiTxis dasma Semdegnairad xdeba. davuSvaT, idealuri airis molekulaTa raodenoba aris

n . ganvsa-

zRvroT im molekulaTa raodenoba, romelTa siCqareebi moTavsebulia  -sa da   d  -s Soris. cxadia, aseTi molekulebis dn raodenoba miT meti iqneba, rac metia molekulaTa saerTo n raodenoba da siCqaris d intervali e.i. dn ~ ndυ .

garda amisa, cxadia, dn damokidebuli iqneba im  siCqareze, romlis maxlobladac

aris

arebuli

d

Sualedi (magaliTad,

molekulaTa

raodenoba, romelTa siCqareebi moTavsebulia 95 m/wm÷100 m/wm SualedSi, gansxvavdeba im molekulebis raodenobisagan, romelTa siCqareebi moTavsebulia, vTqvaT, 495 m/wm÷500 m/wm SualedSi), amitom marjvena mxare unda Seicavdes  -ze damokidebul garkveul f   funqcias:

dn  nf υdυ .

(60.1)

fizikuri sistemis Termodinamikuri wonasworoba ewodeba mis iseT mdgomareobas, rodesac sistemis maxasiaTebeli parametrebi inarCunebs ucvlel mniSvnelobas, Tu masze gare zemoqmedeba ar xdeba (§71).

187

f   -s g a n aw ile b is fu nq c i a ewodeba. Tu d  1 , maSin f    gviCvenebs,

molekulaTa

saerTo

raodenobis

dn , e.i. f   n

nawili

moZraobs

siCqaris erTeulovan intervalSi  siCqaris maxloblad (  ,  1 SualedSi). maqsvelma albaTobaTa Teoriis safuZvelze ganawilebis funqciisaTvis miiRo Semdegi gamosaxuleba: 3

sadac

temperatura. Tu raodenobis

  d 

m 2

4  m  2 2  2 kT f    ,    e   2kT  molekulis masaa, k _ bolcmanis mudmiva, T

(60.2) _ absoluturi

f   -s mniSvnelobas SevitanT (60.1) formulaSi, maSin dn

molekulebisaTvis,

romelTa

siCqareebi

moTavsebulia

,

intervalSi, miviRebT: 3

m 2

4  m  2 2  2 kT dn  d .   n e   2kT 

(60.3)

(60.2) formuliT mocemuli funqciis grafiki (ganawilebis mrudi) gamosaxulia 60.1 suraTze. rogorc

vxedavT,

grafiki

asimetriulia,

gadis koordinatTa saTaveze, aqvs maqsimumis wertili

asimptoturad

uaxlovdeba

abscisaTa

RerZs. daStrixuli svetis farTobi tolia fuZisa da simaRlis namravlisa

 f  d  ,

da (60.1) tolobis

Tanaxmad gamosaxavs molekulaTa saerTo raode-

 dn  nobis im nawils   , romelTa siCqareebic moTav n 

sur. 60.1

sebulia siCqareTa mocemul d intervalSi. cxadia, mocemuli grafikiTa da abscisaTa RerZiT SemosazRvruli farTobi ricxobrivad toli iqneba molekulaTa sruli n raodenobisa. ganawilebis funqciis grafikis meSveobiT SeiZleba Semdegi daskvnebis gamotana: 1. idealuri airis molekulebi moZraoben sxvadasxva siCqariT. 2. uZravi molekulebi airSi ar aris (grafiki gadis koordinatTa saTaveze). 3. rogorc metad mcire, aseve metad didi siCqariT moZravi molekulebis raodenoba airSi metad mcirea.

188

4. yovel mocemul temperaturaze arsebobs garkveuli  m siCqare, romliTac moZraobs airis molekulebis umetesoba. mas u alb aT es i s iCq are ewodeba. 5. ualbaTesi siCqaris maxlobeli siCqariT moZraobs airis molekulaTa didi umravlesoba. 6. mocemuli siCqariT moZravi molekulebis raodenoba miT metia, rac ufro axlosaa igi ualbaTes siCqaresTan. 7. molekulaTa raodenoba, romelTa siCqareebi  m -ze metia, sWarbobs molekulaTa im raodenobas, romelTa siCqareebi  m -ze naklebia (grafikis marjvena nawili abscisaTa RerZTan

adgens ufro met farTobs, vidre

marcxena). ualbaTesi siCqaris mniSvnelobas, romelsac Seesabameba ganawilebis funqciis maqsimumi, miviRebT pirobidan: df  0, dυ

anu

d  2  2 kT υ e dυ   mυ 2

   0.  

aqedan m 2 2 kT

m 2

2m  2 kT 2 e  e  0, 2kT m 2   m 2    0 . 2e 2 kT 1  2 kT   

(60.4) toloba sruldeba roca

υ  0,

υ   , an

(60.4)

 mυ 2  1    2kT   0 . cxadia, rom  

pirveli ori piroba ar Seesabameba ganawilebis mrudis maqsimums, amitom ualbaTesi siCqare υm unda ganisazRvros mesame pirobidan

1

mυm2  0, 2kT

saidanac

υm  (60.5)

formula

gviCvenebs,

2kT  m rom

2 RT . μ temperaturis

(60.5) gazrdiT

ualbaTesi

siCqare izrdeba, e.i. ganawilebis mrudis maqsimumis wertili gadainacvlebs marjvniv, didi siCqareebisaken. magram, vinaidan mrudiT SemosazRvruli farTobi gamosaxavs molekulaTa saerTo raodenobas da, maSasadame, mudmivia, amitom mrudis maqsimumis wertili Tan dadabldeba (sur. 60.2). e.i. tem189

peraturis gazrdiT ualbaTesi siCqare izrdeba, magram am siCqariT moZravi molekulebis raodenoba mcirdeba (ganawileba ufro Tanabari xdeba). zogierTi

amocanis

mosaxerxebelia, kanonSi

maqsvelis

siCqareebs

erTeulebiT,

amosaxsnelad siCqareTa

gamovsaxavT

aramed

ara

fardobiTi

ufro

ganawilebis Cveulebrivi

erTeulebiT,

sadac

siCqaris erTeulad miRebuli iqneba ualbaTesi siCqare

m .

maSasadame, fardobiTi siCqare

u

sur. 60.2



sadac siCqare mocemuli



siCqare ualbaTes

m

 m

(60.6)

- molekulis mocemuli siCqarea. e.i. fardobiTi gviCvenebs,

ramdenjer

metia

naklebia

siCqareze.

Tu (60.3) tolobaSi gaviTvaliswinebT, rom

m 1 2 d   m du da  2  u 2 m ,  2, 2kT  m miviRebT im molekulaTa dn raodenobas, romelTa fardobiTi siCqareebi moTavsebulia u sa da

u  du  -s

Soris: u 2

m 2  4  1  4 2 m 2 2 2 u 2 dn  nu  e  du  nu e du . m m 2    m  

(60.7)

aseTi CaweriT ganawilebis kanoni ar aris damokidebuli arc airis gvarobaze da arc temperaturaze.

siCqareTa

ganawilebis

kanonidan

SeiZleba

miviRoT

molekulebis

kinetikur energiaTa mixedviT ganawilebis kanoni. aRvniSnoT molekulis kinetikuri energia W -Ti. maSin molekulaTa dn raodenoba, romelTa energiebi moTavsebulia W -sa da W  dW  -s Soris miiReba, Tu (60.3) formulaSi gaviTvaliswinebT, rom

m 2 W , 2

anu

2 

2W ; m

2W m



2kT 

3 2

dn  (60.8)

gantoleba

warmoadgens



 nW e 12

idealuri



W kT

1 2 1 2 W dW : 2 m

dW .

airis

energiaTa mixedviT ganawilebis maqsvelis kanons.

190

d 

(60.8) molekulebis

kinetikur

§61. molekulaTa saSualo ariTmetikuli siCqaris gamoTvla maqsvelis

ganawilebis

funqciis

meSveobiT

SesaZlebelia

saSualo

sidideebis gamoTvla. magaliTad, gamovTvaloT molekulaTa saSualo ariTmetikuli siCqare. igi

miiReba

yvela

molekulis

siCqareTa

jamis

gayofiT

molekulaTa

raodenobaze. molekulaTa

dn raodenoba, romelTa siCqareebi moTavsebulia  ,   d 

_ SualedSi, (60.1) formulis Tanaxmad tolia

dn  nf  d . aseTi molekulebis siCqareTa jami iqneba nf  d . imisaTvis, rom miviRoT nebismieri siCqaris molekulebis siCqareTa jami, unda movaxdinoT miRebuli gamosaxulebis integrireba yvela SesaZlo siCqareebis mixedviT, e.i. 0-dan

 -mde. amrigad, yvela molekulis siCqareTa jami tolia 

 nf  d , 0

xolo saSualo ariTmetikuli siCqare

 Tu

gamosaxulebaSi



1 nf  d   f  d . n 0 0

SevitanT

f   -s

mniSvnelobas

(60.2)

formulidan,

saTanado gamoTvlebis Sedegad miviRebT:

8kT . m miviReT, rom saSualo ariTmetikuli siCqare metia molekulebis υm ualbaTes

 

siCqareze. es asec unda yofiliyo, vinaidan, rogorc aRvniSneT, ganawilebis mrudis asimetriulobis gamo molekulaTa is raodenoba, romelTa siCqareebi υm -ze metia, sWarbobs molekulaTa im raodenobas, romelTa siCqareebi υm -ze

naklebia. saerTod, Tu erTmaneTs SevadarebT Cvens mier ganxilul siCqareebs: 1. saSualo kvadratuls

υkv 

3kT kT RT ,  1,73  1,73 m m μ 191

2. ualbaTess

υm 

2kT kT RT ,  1,41  1,41 m m μ

3. saSualo aritmetikuls

8kT kT RT ,  1,60  1,60 πm m μ vnaxavT, rom maT Soris umciresia ualbaTesi, xolo udidesi _ saSualo υ

kvadratuli siCqare υm  υ  υkv .

azotis molekulis siCqareebisaTvis oTaxis temperaturaze gveqneba  j kg   23  T  300 K , R  8,32  . ,   28  10  K  moli moli   υkv 517 m/wm; υm 421 m/wm; υ 478 m/wm.

§62. maqsvelis siCqareTa ganawilebis kanonis eqsperimentuli Semowmeba. Sternis cda siCqareTa ganawilebis kanoni maqsvelma gamoiyvana Teoriulad. 1920 wels germanelma fizikosma Sternma (1888-1969) Caatara eqsperimenti molekulaTa siCqaris

gasazomad,

romlis

Sedegebma

daadastura

maqsvelis

siCqareTa

ganawilebis kanonis siswore. ganvixiloT Sternis cda (sur. 62.1). aRebulia erTmaneTTan xistad dakavSirebuli ori

r - da R - radiusiani TanaRerZuli cilindri, romelTa RerZis gaswvriv gaWimulia movercxlili platinis 00’ mavTuli. Siga cilindrs aqvs msaxvelis paraleluri viwro

Wrili.

platinis sur. 62.1

mavTulSi eleqtruli denis

gatarebisas igi xurdeba, misi zedapiridan vercxli orTqldeba da Siga cilindrSi miiReba vercxlis erTatomiani airi (vercxlis molekula, iseve 192

rogorc sxva liTonisa, Sedgeba erTi atomisagan). vercxlis atomebi moZraoben

yvela

mimarTulebiT,

cilindris

Siga

zedapirs

maTi

nawili

vercxlis

gava

viwro,

WrilSi

Tanabari

daefineba

sisqis

gare

zolad,

romelsac Wrilis forma aqvs _ warmoadgens Wrilis gegmils (62.1 b. suraTze gamosaxulia xelsawyos ganikveTi). mTel sistemaSi Seqmnilia maRali vakuumi, raTa haeris molekulebma xeli ar SeuSalos vercxlis atomebis moZraobas. Tu

amis

Semdeg

orive

cilindrs



vabrunebT

kuTxuri

siCqariT,

danaleqi gare cilindris Siga zedapirze gadainacvlebs, Tanac igi ukve iqneba ara Tanabari sisqis viwro zoli, aramed sagrZnoblad gaganierebuli da cvladi sisqisa (sur. 62.1 g). zolis gadanacvleba gamowveulia imiT, rom vidre vercxlis atomebi gaivlian

manZils, gare cilindri garkveuli

kuTxiT Semobrundeba. zolis gafarToeba miuTiTebs, rom vercxlis sxvadasxva atomi sxvadasxva siCqariT moZraobs da, maSasadame,

manZilis gavlas

sxvadasxva dros andomebs (cxadia, rom yvelaze swrafi atomebi moxvdebian b sTan axlos, yvelaze neli ki misgan Sors, d wertilSi). zolis sxvadasxva sisqe miuTiTebs imas, rom sxvadasxva siCqariT moZravi atomebis raodenoba sxvadasxvaa.

cxadia,

rom

atomebis

yvelaze

didi

raodenoba

moZraobs

wertilis Sesabamisi siCqariT. es  m ualbaTesi siCqarea. cb -s Soris xvdeba is atomebi, romelTa siCqare ualbaTes siCqareze metia, cd -s Soris ki isini, romelTa siCqare ualbaTes siCqareze naklebia. amrigad,

Sternis

cdis

Sedegad

miRebuli

vercxlis

fenis

SemomsazRvreli mrudis forma TiTqos imeorebs ganawilebis mrudis formas, rac miuTiTebs (Tvisobrivad mainc) ganawilebis kanonis siswores. SemdgomSi, molekulaTa siCqaris gazomvis ufro zusti meTodebis gamoyenebiT dadgenil iqna, rom es kanoni raodenobrivadac kargad Tanxvdeba eqsperiments. υm

ualbaTesi

siCqare

gaizomeba

Semdegnairad:

dro,

romlis

 ac   im drois tolia, ganmavlobaSic υm siCqaris atomi gadis ac manZils  t  υm  

romlis ganmavlobaSic υ wiriTi siCqariT moZravi b wertili gadis S  bc rkals bc / υ , e.i.

ac bc  , υm υ 193

magram υ  ωR ,

amitom

ac bc ,  υm ωR saidanac

ac  ωR . bc gare cilindris radiusi R da ac  R  r winaswar gaizomeba,  υm 

cdis

Semdeg gavzomavT

wanacvlebas

da (62.1) formuliT

(62.1) cnobilia, gamovTvliT

ualbaTes υm siCqares. Sternma miiRo, rom cdis cdomilebis gaTvaliswinebiT υm

unda imyofe-

bodes 635 m/wm da 670 m/wm Soris. meore mxriv, Tu mxedvelobaSi miviRebT Zafis temperaturasa da vercxlis atomur masas, (60.5) formulis Tanaxmad ualbaTesi siCqaris mniSvneloba iqneba 660 m/wm. rogorc vxedavT, Tanxvedra Teoriul da cdiseul Sedegebs Soris sakmaod kargia _ Teoriulad miRebuli υm moTavsebulia cdiT miRebul sazRvrebs Soris.

$63. idealuri airis Sinagani energia Tavisuflebis xarisxTa ricxvi sxeulis Sinagani energia ewodeba misi Semadgeneli nawilakebis qaosuri moZarobis kinetikuri da urTierTqmedebis potenciuri energiebis jams. * idealuri airis molekulebi, ganmartebis Tanaxmad, ar urTierTqmedeben. amitom idealuri airis Sinagani energia tolia misi molekulebis kinetikuri energiebis jamisa.

ufro zustad, sxeulis Sinagani energia ewodeba misi Semadgeneli nawilakebis y v e l a

s a x i s e n e r g i a T a j a m s . masSi, garda zemoT dasaxelebulisa Sedis atomTa eleqtronebis kinetikuri da potenciuri energia da SidaatombirTvuli energia. magram am ukanasknelTa cvlilebis

pirobebi

molekulur

Cveulebriv,

igulisxmeba,

rom

fizikasa

TermodinamikaSi

ganixileba.

amitom,

sxeulis Sinagani energia aris nawilakTa kinetikuri da

potenciuri energiebis jami.

194

risi tolia idealuri airis molekulis saSualo kinetikuri energia? viciT

($53),

kinetikuri

rom

molekulis

energia

W 

3 kT . 2

gadataniTi magram,

moZraobis

vinaidan

Sesabamisi

molekula

saSualo

atomebisagan

Sedgenili rTuli sistemaa, amitom mas gadataniTi moZraobis garda SeuZlia Seasrulos moZraobac.

brunviTi

moZraobac,

TiToeuli am

saxis

xolo

atomebma

moZraobas

mis

SigniT

rxeviTi

garkveuli energia Seesabameba,

amitom, sazogadod, molekulis sruli kinetikuri energia Sedgeba rogorc gadataniTi, aseve brunviTi da rxeviTi moZraobebis energiebisagan. molekulis

sruli

kinetikuri

energiis

gamosaTvlelad

SemoviRoT

T av i s u fle b is x a ris xT a r ic xv i s cneba. sxeulis Tavisuflebis xarisxTa

ricxvi ewodeba im damoukidebel

koordinatTa ricxvs, romlebic gansazRvraven sxeulis mdebareobas sivrceSi. an, sxvanairad, Tavisuflebis xarisxTa ricxvi ewodeba im damoukidebel moZraobaTa ricxvs, romelTa Sesrulebac sxeuls SeuZlia. magaliTad, nivTieri wertilisaTvis

i  3,

vinaidan

nivTieri

wertilis

mdebareoba

sivrceSi

savsebiT ganisazRvreba sami ( x, y, z ) koordinatiT (an, rac igivea nivTieri wertilis nebismieri gadadgileba sivrceSi SeiZleba daiSalos sam damoukidebel moZraobad Ox , Oy , Oz RerZebis gaswvriv). Tu am wertils SeuZlia mxolod sibrtyeze moZraoba, Tavisuflebis xarisxi mcirdeba oramde, xolo mxolod wrfeze moZraobis SemTxvevaSi _ erTamde. absoluturad myari sxeulisaTvis i  6 , vinaidan sivrceSi myari sxeulis mdebareobis gansasazRvravad saWiroa 6

koordinatis codna, romelTagan sami gansazRvravs sxeulis masaTa c centris mdebareobas, ori _ sxeulSi gavlebuli nebismieri RerZis orientacias ( ,  kuTxeebi), xolo erTi _ sxeulis mobrunebis 

kuTxes am RerZis mimarT

(marTlac, myar sxeuls gadataniTi moZraobis garda, romelic xasiaTdeba sami Tavisuflebis

xarisxiT,

SeuSlia

brunva

sami

urTierTmarTobi

RerZis

garSemo, rac ganapirobebs kidev sam `brunviTs~ Tavisuflebis xarisxs). erTatomiani airis (He, Ar da sxv.) molekulis Tavisuflebis xarisxi i  3 , vinaidan atoms, nivTieri wertilis msgavsad, mxolod gadataniTi moZraobis Sesruleba SeuZlia, romelic 3 koordinatiT xasiaTdeba. rac Seexeba brunvas, cdebi da Teoria gviCvenebs, rom atomi centze gamavali RerZis garSemo 195

brunavs metad maRal temperaturaze, Cveulebriv temperaturaze atomi ar brunavs.* oratomiani airis xist molekulebs (e.i. iseT molekulebs, romlis atomebs Soris manZili ar icvleba) aqvs 5 Tavisuflebis xarisxi, romelTagan sami

gadataniTia,

ori

brunviTi.

marTlac

aseTi

sistemis

mdebareobis

gansasazRvravad sivrceSi saWiroa vicodeT misi simZimis centris mdebareoba ( xc , yc , z c _ koordinatebi) da atomebis SemaerTebeli wrfis orientacia ( ,  , kuTxeebi). meeqvse koordinati _ am wrfis irgvliv molekulis Semobrunebis  kuTxe

aRar

aris

saWiro,

vinaidan,

rogorc

aRvniSneT,

atomebis

brunva

centrze gamavali RerZis garSemo ar xdeba (aseTi Semobruneba kidevac rom moxdes,

simetriulobis

gamo

gamoiwvevs

sistemis

mdebareobis

konfiguraciis Secvlas sivrceSi.) Y

↺

↺ 0○

○

○ X

X H○

○H

Z a

b suraTi 63.1

63.1 a suraTze sqematuradaa gamosaxuli oratomiani molekula, romelsac aqvs 5 Tavisuflebis xarisxi: mas SeuZlia moZraoba urTierTmarTobi RerZis gaswvriv da brunva y da z RerZis garSemo. cxadia, rom 5 Tavisuflebis xarisxi eqneba iseT mravalatomian molekulasac, romlis atomebi erT

wrfe-

zea ganlagebuli. Tu mravalatomiani molekulebis atomebi erT wrfeze araa ganlagebuli, maSin mas, myari sxeulis msgavsad, eqvsi Tavisuflebis xarisxi eqneba. 63.1 b suraTze sqematuradaa gamosaxuli samatomiani airis (wylis

aseT brunvas kidevac rom hqondes adgili, misi Sesabamisi kinetikuri energia

I 2 iqneboda 2

Zalian mcire, vinaidan atomis zomis simciris gamo misi inerciis momenti Zalian mcirea.

196

orTqlis)

molekula,

romelsac

aqvs

Tavisuflebis

xarisxi

sami

gadataniTi da sami brunviTi. energiis ra raodenoba modis TiToeul Tavisuflebis xarisxze? amaze pasuxs gvaZlevs b o lcm a ni s T e ore m a , romelic mtkicdeba molekulebis moZraobis srul qaosurobaze dayrdnobiT da ase gamoiTqmis: T u m ole k u l aT a s is t e m a s i T b u r w o na s w o rob a Si a T t e m pe r a tu r a ze , m ole k u li s T i T o e u l T av is u f le b is x ar is xze m os u l i k ine tik u ri e ne r gi is r a o d e n ob a e rT i d a i giv e a da tol ia Tavisuflebis

xarisxTa

ricxvis

1 kT – s i . (am daskvnas xSirad uwodeben 2

mixedviT

energiis

Tanabrad

ganawilebis

kanonsac)*. marTlac, viciT, rom gadataniTi moZraobis Sesabamisi saSualo kinetikuri energia aris

3 kT . magram gadataniTi moZraoba xasiaTdeba sami Tavisuf2

lebis xarisxiT, amitom energia, romelic modis erT Tavisuflebis xarisxze, iqneba samjer naklebi, e.i. amrigad, energiis

ukve

1 kT . 2

SegviZlia

gamomsaxveli

davweroT

formula.

molekulis

sruli

kinetikuri

Tavisuflebis

xarisxTa

ricxvia i , maSin i kT . 2 airis Sinagani W 

erTi

moli

idealuri

(63.1) energiis

misaRebad

(63.1)

gamosaxuleba unda gadavamravloT molekulebis ricxvze erT molSi e.i. avogadros N A ricxvze: i i (63.2) kN AT  RT , 2 2 airis universaluri mudmivaa. Tu airi Seicavs n mols, maSin U μ  N AW 

sadac R  kN A

misi Sinagani energia

unda aRiniSnos, rom energiis Tanabrad ganawilebis kanoni ar aris universaluri kanoni. igi

irRveva,

magaliTad,

metad

dabal

temperaturaze,

roca

xdeba

brunviTi

Tavisuflebis

xarisxTa `gayinva~ da molekula mxolod gadataniTad moZraobs. am movlenas axsna, romelic SesaZlebelia mxolod kvanturi Teoriis safuZvelze, scileba Cveni kursis farglebs.

197

(63.3)

formula

gamosaxavs

i mi U  n RT  RT . 2 2 nebismieri m masis

(63.3) idealuri

airis

Sinagan

energias. Cven aqamde ganvixilavdiT xist molekulebs. SedarebiT dabal temperaturaze marTlac SeiZleba CaiTvalos, rom molekulaSi atomebi ar irxevia da maT Soris manZili ar icvleba. maRal temperaturaze molekulebis `sixiste~ irRveva, misi Semadgeneli atomebi iwyeben rxevas da Tavisuflebis xarisxTa ricxvi izrdeba _ gadataniTsa da brunviTs Tavisuflebis xarisxebs emateba rxeviTi Tavisuflebis xarisxi. amasTan, vinaidan (harmoniulad) merxev wertils

kinetikuri

energiis

garda

gaaCnia

potenciuri

energiac,

romlis

saSualo mniSvneloba iseTivea, rogoric kinetikuri energiisa, amitom rxevis Tavisuflebis xarisxis Sesabamisi energia orjer meti, e.i. kT –s toli iqneba. amrigad, maRal temperaturaze* oratomiani molekulis sruli energia iqneba ara

5 7 kT , aramed kT . 2 2

§64. airis siTbotevadoba. airis universaluri mudmivas fizikuri arsi cdebi gviCvenebs, rom sxvadasxva sxeuli gansxvavebuli siTburi TvisebebiT xasiaTdeba, rac kerZod imiT gamoixateba, rom toli masis sxvadasxva sxeulisaTvis erTi da igive siTbos raodenobis gadacemiT maTi temperatura sxvadasxvanairad icvleba. sxeulebis am Tvisebis dasaxasiaTeblad SemoRebulia siTbotevadobis cneba. s iT b ote v a dob a e w o de b a s iT b o s r ao de no b as , r o m e l ic u n d a ga d av ce T s xe u ls , rom m i s i te m pe r a tu r a g ai z a r d os e rT i gr a du s iT . Tu Q siTbos raodenobis gadacemis Sedegad sxeulis temperatura gaizarda

T sididiT, maSin, ganmartebis Tanaxmad,

siTbotevadoba C

Q . T

(64.1)

igulisxmeba maRali, magram molekulis disocoaciis temperaturaze naklebi temperatura.

198

kuTri siTbotevadoba (c) ewodeba erTeuli masis nivTierebis siTbitevadobas, xolo moluri siTbotevadoba

C  

_ 1 moli nivTierebis siTbotevadobas*.

cxadia, rom 1 Q , m T

c

da C 

(64.2)

 Q

. (64.3) m T (64.3) da (64.3) formulebis SedarebiT miviRebT, rom C   c . erTeulTa SI sistemaSi

siTbotevadobis,

kuTris

siTbotevadobisa

moluri

sitbotevdobis erTeulebi (64.1_64.3) formulebis Tanaxmad Sesabamisad aris j/ K , j/ K  kg, j/ K  mol.; sistemgareSe erTeulebia: siTbotevdobisa _ kal/ K da

kkal/ K ;

kuTri

siTbotevadobisa

kal/ K  kg

kkal/ K ;

moluri

siTbotevadobisa _ kal/ K  mol da kkal/ K  mol. airis siTbotevadoba mniSvnelovnad aris damokidebuli imaze, Tu ra pirobebSi vaTbobT mas _ mudmivi moculobis, Tu mudmivi wnevis dros. amitom arCeven airis orgvar siTbotevadobas: siTbotevdobas mudmivi moculobis

dros CV  da siTbotevadobas mudmivi wnevis dros C   . cdebi gviCvenebs, rom yovelTvis C   CV . idealuri airis

CV

C 

molur siTbotevadobas advilad gamov-

TvliT, Tu visargeblebT wina paragrafSi airis k i ne t ik u r i Teoriis safuZvelze miRebuli SedegiT Sinagani energiis Sesaxeb. a) vTqvaT, 1 moli idealuri airi moTavsebulia daxSul WurWelSi

(sur. 64.1, a) ise, rom mas gafarToebis saSualeba ara aqvs V  const . ganmartebis Tanaxmad, CV aris siTbos raodenoba, romelic unda gadavceT am airs, rom misi temperatura erTi gradusiT gaizardos. (64.3) formulis Tanaxmad *

nivTierebis siTburi Tvisebebis daxasiaTeba ufro xelsayrelia moluri da ara kuTri siTbotevadobiT. marTlac, kuTri siTbotevadobis SemTxvevaSi erTmaneTs vadarebT sxvadasxva nivTierebis erTeulovan masebs. magram sxvadasxva nivTierebis erTeulovani masebi (mag. TiTo grami wyalbadi da Jangbadi) molekulaTa sxvadasxva raodenobebs Seicavs da amitom maTi siTbotevadobebi ar iqneba Sesadari. sul sxvagvaradaa saqme moluri siTbotevadobis ganxilvis dros, vinaidan nebismieri nivTierebis erTi moli Seicavs molekulaTa erTsa da imave raodenobas, maSasadame, gansxvaveba sxvadasxva sxeulis molur siTbotevadobebs Soris ganpirobebuli iqneba am sxeulebisa da maTi molekulebis aRnagobis gansxvavebiT da ara molekulebis sxvadasxva raodenobiT. amitom moluri siTbotevadobis eqsperimentuli Seswavla, cxadia, SesaZleblobas mogvcems gavakeToT garkveuli daskvnebi molekulebis aRnagobis Sesaxeb.

199

ricxobrivad CV  Q

  m, T  1 .

vinaidan airi ar farTovdeba, igi muSaobas

ar asrulebs da amitom gadacemuli siTbo mTlianad ixarjeba misi Sinagani energiis gazrdaze. maSasadame, SeiZleba davweroT:

Q  U  , e.i. CV  U  ,

magram viciT, rom U  

i RT , saidanac 2

U  

i i i RT  1  RT  R , 2 2 2

amitom i R. 2 b) gamovTvaloT C  . vTqvaT, 1 moli CV 

idealuri

airi

moTavsebulia

(64.4)

dguSian

WurWelSi (sur. 64.1 b). dguSze moqmedebs mudmivi

gare

uwonoa

gadaadgileba,

wneva

 p  const  .

dguSi

SeuZlia

xaxunis

gareSe

amitom

airis

wneva

gare

wnevis tolia. imisaTvis, rom gaTbobisas airis wneva ucvleli darCes, igi unda gafarTovdes,

e.i.

dguSi

sur. 64.1

gadaadgildes

raRac h manZilze. gafarToebisas airi Seasrulebs muSaobas gare wnevis dasaZlevad.

maSasadame,

SemTxvevaSi

gadacemuli

siTbos

raodenobis

mxolod nawili moxmardeba Siangani energiis zrdas, nawili ki _ airis gasafarToeblad

Sesrulebul

muSaobas.

Sinagani

energiis

nazrdi

CV –s

tolia.* amrigad, C  metia CV –ze swored am Sesrulebuli muSaobis sididiT, amitom SeiZleba davweroT:

C   CV  A ,

(64.5)

sadac A aris mudmivi wnevis pirobebSi airis gafarToebis dros Sesrulebuli muSaoba misi erTi gradusiT gaTbobisas. gamovTvaloT es muSaoba. dguSis *

idealuri airis Sinagani energia ar aris damokidebuli moculobaze, amitom Sinagani

energiis gazrdaze orive SemTxvevaSi erTi da igive siTbos raodenoba ixarjeba.

200

farTobi aRvniSnoT (aseTive

sididis

S –iT, maSin dguSze moqmedi gare wnevis Zala

wnevis

ZaliT

moqmedebs

airi

gaTbobisas airi farTovdeba da dguSi aiwevs

dguSze).

erTi

F  pS

gradusiT

h sididiT. am dros gare

wnevis Zalis dasaZlevad airi asrulebs muSaobas: A  Fh  pSh  pV ,

(64.6)

sadac V  Sh aris airis moculobis namati. pV –s gansazRvrisaTvis davweroT klapeiron-mendeleevis gantoleba ori mdgomareobisaTvis: gaTbobamde pV  RT da erTi gradusiT gaTbobis Semdeg

(64.7)

p(V  V )  R(T  1) .

(64.8)

Tu (64.8) tolobas gamovaklebT (64.7)-s, miviRebT, rom pV  R ,

anu (64.5)-dan

A  R .

(64.9)

aqedan Cans R –is fizikuri Sinaarsi: ai ri s u niv e rs alu ri m u d m iv a r i c x o b r iv a d im m u S aob is to li a, r om e ls ac as ru le b s 1 m ol i i de a l u r i a ir i i zob a ru l i ga f ar T oe b is as m is i e r T i gr a du s iT g aT b o b is dr os . (64.9) formulis gaTvaliswinebiT miviRebT, rom C p  CV  R .

an (64.4)-is Tanaxmad C p 

(64.10)

gantolebidan

Cans,

rom

(64.10)

i i2 RR  R. 2 2

airis

molur

(64.11) siTbotevadobaTa

sxvaoba

universaluri mudmivas tolia C p  CV  R .

(64.12)

(64.12) Tanafardoba, an, rac igivea, (64.10) Tanafardoba, romelic amyarebs damokidebulebas C p –sa da CV –s Soris, pirvelad miiRo germanelma mecnierma m ai e r m a (1814-1878) da mas m a i e r is g a nt ol e b a ewodeba.

201

siTbotevadobaTa

fardoba aRiniSneba  asoTi. igi siTbotevadobas-

Tan erTad airis erT-erTi mniSvnelovani siTburi maxasiaTebelia. (64.4) da (64.11) formulebis Tanaxmad

  rogorc

vxedavT,

,

iseve

C p



CV

i2 . i

rogorc

(64.13) da

C p

CV ,

damokidebulia

Tavisuflebis xarisxTa ricxvze, e.i. airis gvarobaze. erTatomiani

airisaTvis

i3

SevitanT

R –is

ricxviT

mniSvnelobas (§58), miviRebT:

j kal 3 , R  12,5 3 2 K  moli K  moli j kal 5 , C p  R  20,8 5 2 K  moli K  moli 5    1,67 . 3 oratomiani airisaTvis ( i  5 ) j kal 5 , CV  R  20,8 5 2 K  moli K  moli j kal 7 , C pμ  R  29,2 7 2 K  moli K  moli 7    1,4 . 5 sam- da mravalatomiani airisaTvis i  6 , j kal 6 , CV  R  24,9 6 2 K  moli K  moli j kal 8 , C p  R  33,2 8 2 K  moli K  moli 8    1,33 . 6 Tu ramdenad sworia Cven mier airis kinetikuri Teoriis safuZvelze CV 

miRebuli Sedegebi moluri siTbotevadobis Sesaxeb, amaze pasuxis gacema SeuZlia mxolod eqsperiments. cdebi gviCvenebs, rom erTatomiani airis (He, Ar, Hg da sxv.) SemTxvevaSi Teoriis Sedegebi Zalian kargad Tanxvdeba cdis Sedegebs. eqsperimentis cdomilebis fargelebSi yvela erTatomiani airis moluri siTbotevadoba erTi da 202

igivea da ar icvleba temperaturis cvlilebisas. es gasagebicaa, vinaidan idealurobis pirobas yvelaze ukeT erTatomiani airi akmayofilebs. or-, sam- da mravalatomiani airis SemTxvevaSi Tanxvedra ar aris aseTi kargi, rac imiT gamoixateba, rom jer erTi, cdis Tanaxmad, siTbotevadoba damokidebulia temperaturaze, meorec, sxvadasxva airis siTbotevadoba, romelTa molekulebi Seicaven erTi da imave raodenobis atomebs, sxvadasxvaa. aRniSnuli gadaxrebis axsna xerxdeba mxolod kvanturi Teoriis safuZvelze.

Â§65. barometruli formula 59-e paragrafSi gamoviyvaneT airis kinetikuri Teoriis ZiriTadi gantoleba _ gamovTvaleT idealuri airis wneva, roca molekulebze gare Zala ar moqmedebda. am dros, molekulebis moZraobis qaosurobis gamo, airi WurWlis mTel moculobaSi nawildeba Tanabrad da misi simkvrive (da wneva) yvelgan erTnairia. sxvagvaradaa saqme, roca airi moTavsebulia gare velSi, vTqvaT, simZimis velSi. marTalia, am drosac airis molekulebi ganagrZoben qaosur moZraobas, magram TiToeul molekulaze moqmedebs simZimis Zala, ris gamoc airis simkvrive (da wneva) qveda fenebSi meti iqneba, vidre zedaSi, e.i. adgili eqneba molekulebis garkveul ganawilebas simaRlis mixedviT. marTlac cnobilia, rom atmosferuli wneva simaRlis zrdis mixedviT mcirdeba*. gamovTvaloT airis wnevis cvlileba simaRlis mixedviT. amisaTvis airSi gamovyoT erTeulovani ganikvesur. 65.1

Tis vertikaluri sveti (sur. 65.1) da CavTvaloT, rom temperatura yvelgan erTnairia. davuSvaT, rom airis

wneva

hď&#x20AC;˝0

doneze aris

p 0 , xolo

simaRleze _

p . wnevis cvlileba

atmosferos arseboba ganpirobebulia erTdroulad ori urTierTsapirispiro procesis mimdinareobiT: haeris molekulebis qaosuri (siTburi) moZraobiT, romelic xels uwyobs molekulebis gafantvas da maTze simZimis Zalis moqmedebiT, romelic xels uwyobs molekulebis dedamiwis zedapirisaken moZraobas. siTbur moZraobas rom ar asrulebdnen, molekulebi, simZimis Zalis moqmedebiT `daecemodnen~ dedamiwis zedapirze da warmoiqmneboda haeris metad Txeli fena. simZimis Zala rom ar moqmedebdes, molekulebi, siTburi moZraobis gamo gaifantebodnen msoflio sivrceSi.

203

usasrulod mcire dh simaRleze aRvniSnoT dp –Ti. maSin h  dh

simaRleze

wneva iqneba p  dp . vinaidan wneva ricxobrivad tolia erTeulovan farTobze moqmedi wnevis Zalisa, romelic Cvens SemTxvevaSi airis simZimis Zalaa, amitom wnevis cvlileba (  dp ) toli iqneba im airis simZimis Zalisa, romelic moTavsebulia dh simaRlis cilindrSi.

am cilindris moculoba

dV  1  dh

masSi moTavsebuli airis masa dM  dV  dh sadac  airis simkvrivea, amitom

 dp  gdM  gdV  gdh .

(65.1)

Tu gamoviyenebT airis kinetikuri Teoriis ZiriTad gantolebas p  nkT , sadac

n aris molekulebis raodenoba moculobis erTeulSi, airis  simkvrivisaTvis, romelic tolia n –is namravlisa erTi molekulis masaze, miviRebT:

  mn  m

p , kT

romlis Setana (65.1) formulaSi mogvcems p  mg  dp  dh , kT anu dp mg  dh . p kT vinaidan T  const , (65.2) gantolebis integrebiT miviRebT: p

(65.2)

dp mg p p   kT 0 dh , 0 ln

 h p  e kT p0

p mg  h, p0 kT

da 

(65.3)

formulas

SeiZleba

h

p  p0 e kT . mivceT praqtikuli

gamoyenebisaTvis

(65.3) ufro

mosaxerxebeli saxe. amisaTvis xarisxis maCvenebeli gavamravloT da gavyoT avogadros

N A ricxvze da gaviTvaliswinoT, rom mN A   aris moluri masa,

xolo kN A  R _ airis universaluri mudmiva, maSin miviRebT, rom 

gh



p  p0 e RT . (65.4) (65.3) da misi ekvivalenturi (63.4) formula gamosaxavs airis wnevis damokidebulebas

simaRleze.

mas

b a r om e t ru l i

formula

ewodeba.

igi

gviCvenebs, rom airis wneva simaRlis mixedviT eqsponencialurad mcirdeba. am 204

formuliT SeiZleba atmosferuli wnevis miaxloebiT gamoTvla nebismier h simaRleze, Tu cnobilia wneva nulovan doneze. miaxloebiT, vinaidan formulis gamoyvanisas Cven dauSviT, rom T  const . sinamdvileSi, cxadia, temperatura simaRlesTan erTad icvleba. garda amisa, simaRlis zrdasTan erTad mcirdeba agreTve simZimis Zalis g aCqareba. zemoCamoTvlilis gaTvaliswinebiT (65.3) formulaSi saTanado Sesworebebis Setanis Semdeg, igi saSualebas mogvcems zustad ganvsazRvroT atmosferuli wneva mocemul simaRleze, an piriqiT, atmosferuli wnevis gazomviT gavigoT saTanado simaRle. am principzea agebuli aviaciuri a l tim e tr i

(simaRlemzomi), romlis skalaze wnevebis

nacvlad aRniSnulia Sesabamisi simaRleebi. barometruli formula SeiZleba CavweroT erTeul moculobaSi moTavsebul molekulaTa n ricxvisaTvisac. marTlac, vinaidan airis kinetikuri Teoriis ZiriTadi gantolebis Tanaxmad mudmivi temperaturis pirobebSi airis wneva proporciulia erTeul moculobaSi moTavsebul molekulaTa raodenobisa (59.8 formula), amitom p  nkT da p0  n0 kT , sadac n da n0 Sesabamisad aris

molekulaTa

koncentraciebi

h simaRlesa

h0

doneze.

mniSvnelobebs SevitanT (65.4) formulaSi, miviRebT:

n  n0 e



μgh RT

(65.5)

an 

mgh

n  n0 e kT . (65.5’) (65.5) da (65.5’) gantolebebiT, romelTac agreTve barometrul formulas sur. 65.2

uwodeben,

gamosaxulia

simZimis velSi moTavsebul airSi molekulaTa koncentraciis

cvlileba

simaRlis

mixedviT.

(65.4) an (65.5) formulis grafiki mocemulia 65.2 suraTze. (65.4) an (65.5) formulebi gviCvenebs, rom airis wneva da simkvrive miT ufro swrafad mcirdeba simaRlis mixedviT, rac ufro mZimea airi. amitom, Tu gvaqvs airTa narevi (mag. haeri), simaRlis zrdasTan erTad masSi msubuqi airebis xvedriTi wili unda gaizardos. marTlac, atmosferos maRal fenebSi SeimCneva msubuqi airis, kerZod, wyalbadis siWarbe.

205

§66. bolcmanis kanoni. maqsvel-bolcmanis ganawileba wina paragrafSi gamoyvanil (65.5’) barometrul formulaSi 

mgh kT

(66.1) n  n0 e mgh aris molekulis potenciuri energia dedamiwis gravitaciul velSi h simaRleze. amitom SeiZleba iTqvas, rom (66.1) formula gansazRvravs molekulaTa ricxvis Semcirebas simaRlis mixedviT iseT airSi, romelic simZimis ZalTa velSia moTavsebuli. molekulur-kinetikuri Teoriis safuZvelze bolcmanma daamtkica, rom (66.1) formula zogadia da gamosadegia ara marto simZimis ZalTa velSi, aramed nebismier ZalTa velSi* moTavsebuli airisaTvis. Tu am velSi molekulis potenciur energias aRvniSnavT U –Ti (misi saxe damokidebulia velSi moqmedi Zalebis bunebaze), maSin

n  n0 e



U kT

(66.2)

(66.2) formulas ewodeba bolcmanis formula an bolcmanis ganawilebis kanoni. aq n da n0 aris Sesabamisad nawilakebis raodenoba moculobis erTeulSi sivrcis (velis) im adgilebSi, sadac potenciuri energiebia Sesabamisad U da 0. bolcmanis kanoni gamosaxavs wonasworobis pirobas siTbur moZraobasa, romelic cdilobs nawilakTa Tanabrad ganawilebas mTel moculobaSi da gare

(velis)

ZalTa

moqmedebiT

gamowveul

moZraobas

Soris,

romelic

cdilobs nawilakebis Tavmoyras minimaluri potenciuri energiis areSi. marTalc, moTavsebuli

Zalur

U  0,

maSin

velSi,

n  n0 ,

maSin

e.i.

koncentracia

nawilakTa

sistema

yvelgan

erTnairia

araa da,

maSasadame, nawilakebi mTel moculobaSi Tanabradaa ganawilebuli. Tuki T  0 , maSin n  n0 nulovani potenciuri energiis adgilebSi da n  0 yvelgan,

sadac

U  0 .** e.i. nawilakTa siTburi moZraoba rom ara, yvela nawilaki

daikavebda nulovani potenciuri energiis Sesabamis mdgomareobas.

es veli (simZimis ZalTa velis msgavsad) unda iyos potenciuri, e.i. velis Zalebis mier

Sesrulebuli muSaoba gzis formaze ar unda iyos damokidebuli. **

n  n0e



U kT

 

 n0 eU

1 kT

; Tu U  0, n  n0  11 kT  n0 T -s nebismieri mniSvnelobisaTvis. 206

Tu maqsvelis ganawilebis (60.3) formulaSi mv2

 4  m  2 dn     n υ e 2kT dυ π  2kT  SevitanT n –is mniSvnelobas bolcmanis ganawilebis kanonidan (66.2), miviRebT

maqsvel-bolcmanis ganawilebis kanons:

4  m  dn    π  2kT 

 n0 e



U kT

 υ2e



mv2 2 kT dυ ,

anu 4  m  dn    π  2kT 

sadac E 

 n0 υ e 2



E kT dυ ,

(66.3)

m 2  U aris nawilakis sruli energia. (66.3) formula gamosaxavs 2

molekulaTa im dn raodenobas, romelTa siCqareebi moTavsebulia υ –sa da

υ  dυ –s

Soris da potenciuri energiebi Zalur velSi U –s tolia.

§67. avogadros ricxvis gansazRvra. perenis cda barometruli (65.5’) formulis cdiT Semowmeba, agreTve am formulis saSualebiT avogadros ricxvis gansazRvra moaxdina frangma mecnierma perenma (1871-1942). perenma mravalricxovani cdebis safuZvelze daadgina, rom brounis nawilakebi SeiZleba ganxilul iqnes rogorc giganturi zomis molekulebi, romlebic erTmaneTTan ar urTierTqmedeben. maSasadame, Tu am nawilakebs movaTavsebT simZimis ZalTa velSi, maTi ganawileba simaRlis mixedviT unda iyos airis molekulebis ganawilebis msgavsi, e.i. unda aRiwerebodes (65.5’) formuliT da Tu am ganawilebas cdiT SeviswavliT, amiT saSualeba mogvecema SevamowmoT (65.5’) formulis siswore. perenis cdebma aCvena, rom brounis nawilakebis ganawileba

marTlac

aRiwereba

(65.5’)

barometruli

formuliT,

rac

ukanasknelis sisworis dadasturebaa. idealuri airis molekulaTa nacvlad brounis nawilakTa ganxilvas mTeli rigi upiratesoba aqvs eqsperimentuli TvalsazrisiT: 1.

airis

molekulebisagan

gansxvavebiT

SesaZlebelia

mikroskopiT da maTi daTvla.

207

brounis

nawilakebis

danaxva

xedvis

areSi

moxvedril

brounis

nawilakTa

raodenoba

SedarebiT

mcirea,

rac

aadvilebs maT daTvlas. 3.

brounis nawilakebis masa gacilebiT metia molekulebis masaze, ris gamoc maTi kon-

centracia simaRlis mixedviT gacilebiT ufro swrafad icvleba, vidre molekulebisa. magaliTad, (65.5’) formulis gamoyenebiT miviRebT, rom Jangbadis molekulebis koncentraciis orjer SemcirebisaTvis saWiroa 5 km simaRle ( h1  h2  5 km), xolo brounis nawilakebisaTvis (romelTa masa ~108-jer metia Jangbadis molekulis masaze) es cifri Seadgens mxolod 0,05 mm-s.

cdebisaTvis pereni iyenebda fisis nairsaxeobis _ gumigutis emulsias. specialuri

centrifugis

gamoyenebiT

igi

aRwevda

emulsiis

maqsimalur

erTgvarovnebas, e.i. brounis (gumigutis) nawilakebis masaTa tolobas. emulsias ramodenime

pereni

asxavda

meaTed nawils

kiuvetSi,

romlis

aRemateboda da

simaRle

milimetris

aTavsebda moklefokusiani

mikroskopis obieqtivis qveS. afokusebda mikroskops emulsiis or sxvadasxva

h1 da h2 simaRleze da iTvlida brounis nawilakebis nh1  da nh2  raodenobas Sesabamis sibrtyeebSi. vianidan nh1  da nh2  proporciulia nawilakebis raodenobisa moculobis erTeulSi Sesabamis simaRleebze, amitom maTTvis marTebulia (65.5’) formula:

nh1   n0 e



mgh1 kT

nh2   n0 e aviRoT maTi fardoba



mgh2 kT

 h1  h2  nh1   e kT nh2  mg

gaviTvaliswinoT, rom

k

R , NA

maSin

 nh1  e nh2  tolobis orive mxare gavalogariTmoT:

mgN A h1  h2  RT

nh1  mgN A h2  h1  ,  nh2  RT saidanac ganisazRvreba avogadros ricxvi: ln

208

(67.1)

nh1  nh2  , (67.2) NA  mg h2  h1  am formulaSi R da g mudmivi sidideebia, T -s gazomva Zneli ar aris, nh1  RT ln

nh2 

ricxvebs

uSualod

daviTvliT

mikroskopis

meSveobiT.

h2  h1 ,

romelic warmoadgens mikroskopis obieqtivis gadanacvlebas, didi sizustiT gaizomeba mikrometruli xraxnis saSualebiT. brounis nawilakis

masis

gazomva sakmaod Znelia da daiyvaneba nawilakis r radiusis gazomvaze. saqme isaa, rom brounis nawilaki, zomis simciris gamo, SeiZleba ganvixiloT rogorc r –radiusiani birTvi; maSin masa 4 (67.3) m  ρV  ρ π r 3 . 3 sadac  gumigutis simkvrivea. unda gaviTvaliswinoT isic, rom cdis dros

saqme gvaqvs brounis nawilakis ara namdvil, aramed moCvenebiT masasTan, vinaidan masze moqmedebs wylis amomgdebi Zala. es moCvenebiTi masa tolia sakuTriv brounis nawilakisa da mis mier gamodevnili wylis masaTa sxvaobisa.

m

4 3 π r  ρ  ρ0  , 3

(67.4)

sadac ρ0 wylis simkvrivea.

r -is Sedegebi

gasazomad

pereni

gansxvavdeboda

iyenebda

erTmaneTisagan

sam

sxvadasxva

mxolod

meTods.

1-2%-iT,

rac

miRebuli cdis

did

sizusteze miuTiTebda. erT-erT meTodSi gamoiyeneboda stoqsis formula, romelic gamosaxavs blant garemoSi (Cvens SemTxvevaSi wyalSi) nawilakis Tanabari vardnis siCqares*

υ

2 gr 2  ρ  ρ0  , 9η

r

9ηυ . 2 g  ρ  ρ0 

saidanac (67.5)

g , ρ , ρ0 ,η (wylis siblante) sidideTa ricxviTi mniSvnelobebi cxrilebSia

mocemuli,

υ -s gamoviangariSebT nawilakis Tanabari vardnis manZilisa da

brounis nawilakis vardnis siCqaris uSualo gazomva SeuZlebelia nawilakis qaosuri

moZraobis gamo. amitom pereni zomavda emulsiis Rrublis zeda sazRvris dawevis siCqares.

209

Sesabamisi drois gazomviT, am sidideebis CasmiT (67.5) formulidan gavigebT

r –s, (67.4) formulidan m –s, xolo (67.2) formulidan avogadros N A ricxvs. perenma

pirveli

cdebis

Sedegad

avogadros

ricxvisaTvis

miiRo

N A  6,8  10 23 moli-1 mniSvneloba. SemdegSi sxvadasxva mkvlevaris mier imave

meTodis dazustebiT (mag. WurWlis kedlebis gavlenis gaTvaliswineba da sxva). miRebul iqna

N A –s mniSvnelobebi, romlebic dReisaTvis miRebuli

mniSvnelobisagan mxolod 1%-iT gansxvavdeba. dReisaTvis

avogadros

ricxvis

gazomvis

20-mde

sxvadasxva

meTodi

arsebobs, amitom igi didi sizustiTaa gansazrvruli. amJamad miRebulia, rom avogadros ricxvi N A  6,023  10 23 moli-1

§68. realuri airi. van-der-vaalsis gantoleba cdebi gviCvenebs, rom maRali temperaturisa da dabali wnevis pirobebSi idealuri airis kanonebi sakmao sizustiT asaxavs realuri airis Tvisebebsac. maRali wnevisa da dabali temperaturis pirobebSi ki adgili aqvs rogorc raodenobrivi, aseve Tvisobrivi xasiaTis sagrZnob gadaxrebs. raodenobrivi gadaxra imiT mJRavndeba, rom wnevis TandaTanobiT zrdiT da temperaturis TandaTanobiT SemcirebiT realuri airis Tvisebebi sul ufor scildeba klapeiron-mendeleevis

gantolebidan

gamomdinare

Sedegebs.

Tvisobrivi

gadaxra ufro Rrma xasiaTisaa da mJRavndeba imiT, rom zemoaRniSnuli procesis Sedegad bolos da bolos SesaZlebelia realuri airis gadayvana Txevad an myar mdgomareobaSi (rac SeuZlebeli iqneboda airi rom mkacrad emorCilebodes klapeiron-mendeleevis gantolebas). realuri airis gadaxra idealuri airis kanonebidan ganpirobebulia imiT, rom idealuri airis ganxilvisas ar gagviTvaliswinebia molekulebis mier dakavebuli moculoba (molekulebi miCneuli iyo nivTier wertilebad) da molekulebis urTierTqmedeba (miCneuli iyo, rom urTierTqmedebas adgili aqvs mxolod uSualo drekadi Sejaxebis dros). molekulebs Soris urTierTqmedebis ararseboba niSnavs, rom dajaxebebs Soris SualedSi molekulebze araviTari Zalebi ar moqmedebs, e.i. isini moZ210

raoben Tavisuflad (airis kinetikuri Teoriis ZiriTadi gantolebis gamoyvanisas davuSviT, rom molekulebi kubis mopirdapie waxnagebs Soris moZraoben Tavisflad, maTze araviTari Zalebi ar moqmedebs). sinamdvileSi, molekulebi urTierTqmedeben dajaxebamde gacilebiT adre, ramdenime molekulis diametris tol manZilze miaxloebisas. molekulebis moculobis ararseboba niSnavs, rom WurWlis V moculoba, romelic Sedis gantolebaSi, e.i. is moculoba, romelSic moZraoben molekulebi, mTlianad misawvdomia TiToeuli maTganis moZraobisaTvis. sinamdvileSi, rogorc viciT, molekulebs aqvT Zalian mcire, magram mainc sasruli zomebi. amrigad, idealuri airis TeoriaSi orive daSveba miaxloebiTia. klapeiron-mendeleevis gantolebis vargisianoba realuri airisaTvis imiT ganisazRvreba, Tu ra pirobebSi SeiZleba masze am daSvebis gavrceleba. gavarkvioT

pirobebi.

vTqvaT,

romelime

airs,

magaliTad

haers,

normalur pirobebSi ( p0  1 atm, T0  273K ) ukavia 1 m3 moculoba. molekulebis raodenoba masSi toli iqneba loSmitis ricxvisa n0  2,7  10 25 . vinaidan mole4 kulis radiusi r ~10-10 m, amitom erTi molekulis moculoba   r 3  4  10 30 m3, 3

xolo 1 m3-Si yvela molekulis moculobaTa jami V '  n  4  10 30  2,7  10 25  10 4 m3,

rac Seadgens airis mier dakavebuli saerTo moculobis mxolod 1/10000 nawils da misi ugulebelyofa Tavisuflad SeiZleba. aseT pirobebSi, Tu davuSvebT, rom molekulebi Tanabradaa ganawilebuli mTel moculobaSi (rac marTebulia maTi moZraobis qaosurobis gamo), maT Soris saSualo manZili gacilebiT (ramodenime aTeuljer) meti iqneba maT zomebze, amitom molekulebs Soris urTierTqmedebis ugulebelyofac savsebiT dasaSvebia. amrigad, normalur pirobebSi haeri (da saerTod, realuri airi) sakmaod kargad akmayofilebs idealurobis pirobas, amitom misi Tvisebebi sakmaod zustad aRiwereba klapeiron-mendeleevis gantolebiT. gavzardoT airis wneva p  5000 atm-mde. maSin, Tu kumSva izoTermulia, boil-mariotis kanonis Tanaxmad, moculobac amdenjerve unda Semcirdes da gaxdes 1/5000 m3=210-4 m3. axla ukve airis saerTo moculoba mxolod orjeraa meti molekulebis jamur (sakuTar) moculobaze da cxadia, rom am ukanaskne211

lis ugulebelyofa aRar SeiZleba. garda amisa, vinaidan airis moculobis mkveTri Semcireba moxda molekulebs Soris saSualo manZilis Semcirebis Sedegad, romelic axla ukve molekulis diametris rigisaa, ar SeiZleba molekulebs Soris urTierTqmedebis Zalebis ugulebelyofa. amrigad, wnevis gazrdisas realuri airi aRar akmayofilebs idealurobis pirobebs, amitom misi Tvisebebi aRar aiwereba klapeiron-mendeleevis gantolebiT. magram klapeiron-mendeleevis gantoleba pV  RT

(68.1)

SeiZleba ise gardavqmnaT, rom igi gamosadegi gaxdes realuri airisaTvis. amisaTvis gantolebaSi unda SevitanoT ori Sesworeba, romelTagan erTi iTvaliswinebs molekulebis sakuTar moculobas, xolo meore _ molekulebis urTierTqmedebas. ganvixiloT jer pirveli Sesworeba _ Sesworeba molekulebis sakuTar moculobaze. idealuri airis molekulebs Tavisuflad SeuZliaT moZraoba mTel V  moculobaSi (romelic WurWlis moculobis tolia). realuri airis molekulebis moZraoba ki, imis gamo, rom maT sakuTari moculoba aqvT, ramdenadme SezRudulia: TiToeul molekulas SeuZlia moZraoba ara mTel V moculobaSi, aramed V moculobis im nawilSi, romelic ar ukavia danarCen molekulebs. e.i. molekulebs Tavisufali moZraobisaTvis rCebaT moculoba, romelic naklebia V moculobaze raRac b sididiT, e.i. tolia (V  b) –si. am Sesworebis safuZvelze (68.1) gantoleba miiRebs saxes p(V  b)  RT .

gavarkvioT b

(68.2)

mudmivas fizikuri arsi. (68.1) gantolebidan Cans, rom

roca p   , maSin V  0 , rac realuri airisaTvis, cxadia, swori ar aris. (68.2) gantolebidan viRebT, rom roca p   , maSin V  b , e.i. b aris is moculoba, romliskenac miiswrafvis 1 moli airis moculoba misi wnevis usasrulod gazrdisas. cxadia, rom b ufro metia, vidre 1 molSi Semavali molekulebis sakuTar moculobaTa jami, vinaidan molekulebis mWidro ganlagebisas maT Soris mainc rCeba Tavisufali sivrceebi. gamoTvlebi gviCvenebs, rom

212

tolia 1 moli airis molekulaTa sakuTari moculobis gaoTxkecebuli

sididisa, b  4V ' . unda aRiniSnos, rom molekulebis sakuTari moculobis gaTvaliswineba fizikurad niSnavs maT Soris ganzidvis Zalebis gaTvaliswinebas, vinaidan swored ganzidvis Zalebi uSlis xels molekulebs dauaxlovdnen erTmaneTs garkveul minimalur manZilze ufro axlos (`SeiWran erTmaneTSi~). swored es minimaluri manZili gansazRvravs molkulebis zomebsa da moculobas. maSasadame, b Sesworeba SeiZleba ganvixiloT rogorc parametri, romliTac xdeba molekulebs Soris moqmedi ganzidvis Zalebis gaTvaliswineba da warmoadgens am Zalebis maxasiaTebel sidides. ganvixiloT

axla

meore

Sesworeba,

romelic

dakavSirebulia

molekulebs Soris urTierTmizidvis Zalebis arsebobasTan. (68.1) tolobaSi p aris wneva, romelsac awarmoebs WurWlis kedlebi idealur airze. realuri airis molekulebs Soris mizidvis Zalebis arseboba damatebiT kumSavs airs, ris Sedegadac warmoiqmneba damatebiT p i wneva, romelsac S i g a, anu m o le k u l u r i w ne v a ewodeba. amitom realuri airis faqtiuri wneva Sedgeba

gare da pi Siga wnevebis jamisagan. am Sesworebis gaTvaliswinebiT realuri airis mdgomareobis gantoleba 1 molisaTvis iRebs saxes:

 p  pi (V  b)  RT .

(68.3)

realuri airis mdgomareobis gantoleba am saxiT miiRo holandielma fizikosma van-der-vaalsma (1837-1923) da mis saxels atarebs. van-der-vaalsis gantoleba gacilebiT ukeTes TanxvedraSia eqsperimentTan, vidre klapeironis gantoleba. magaliTad, 1000 atm wnevis dros RT mudmivasagan pV daxra Seadgens 100%-s,

 p  pi (V  b)

sididis ga-

sididisa ki _ mxolod 2%-s.

gamovTvaloT Siga wnevis sidide. urTierTmizidvis Zalebis moqmedeba sxvadasxvanairad iCens Tavs moculobis SigniT da kedlis maxloblad moTavsebuli molekulebisaTvis. vinaidan mizidvis Zala manZilTan erTad Zalian swrafad mcirdeba, amitom airis mocemul molekulas izidavs mxolod mis maxloblad myofi molekulebi. Tu es molekula moculobis SigniTaa, maSin masze moqmedi mizidvis ZalTa tolqmedi nulis tolia, vinaidan mas yoveli mxridan saSualod erTnairi ZaliT izidaven mezobeli molekulebi. Tu moce213

muli molekula kedlis maxlobladaa moTavsebuli, maSin masze moqmedebs airis SigniT myofi molekulebi da amitom mizidvis ZalTa tolqmedi airis SigniTaa mimarTuli.* swored kedlis maxloblad (ganapira fenaSi) myof molekulebze moqmedi es Zalebi ganapirobebs airis Siga

sur. 68.1

wnevas. mizidvis ZalTa swrafi Semcirebis gamo SeiZleba davuSvaT, rom airis ganapira fenaze moqmedebs mxolod misi mezobeli

fenis molekulebi

(sur. 68.1). Zala, romliTac AB fena moqmedebs ganapira fenaze, proporciulia rogorc erT, ise meore fenaSi myofi molekulebis raodenobisa, es ukanaskneli ki proporciulia airis n koncentraciisa, amitom Siga wneva (romelic ganapira fenis erTeulovan farTobze moqmedi mizidvis Zalis tolia) pi   n 2 .

magram

n

NA , V

sadac

(68.4)

avogadros

ricxvia,

xolo

V

molis

moculoba, amitom

pi 

αN A2 V μ2



a V μ2

(68.5)

sadac a   N A2 mocemuli airisaTvis damaxasiaTebeli mudmivaa, romlis meSveobiTac xdeba molekulebs Soris moqmedi mizidvis Zalebis gaTvaliswineba.

p i -s mniSvnelobis Casmis Semdeg, van-der-vaalsis gantoleba 1 moli airisaTvis iRebs saxes:    p  a  V  b  RT . (68.6) 2  μ  V μ   dabali wnevis garda, idealuri airis gantolebis gamoyenebadobis meore





piroba realuri airisaTvis aris maRali temperatura. maRal temperaturaze molekulebis gadataniTi moZraobis saSualo kinetikuri energia bevrad aRemateba maTi urTierTqmedebis potenciur energias. aseT pirobebSi realuri airis Tvisebebi uaxlovdeba idealuri airis Tvisebebs, romlis molekulebsac urTierTqmedebis potenciuri energia saerTod ar gaaCnia. *

kedelTan

myof

airis

molekulaze

moqmedebs

kedlis

molekulebic,

magram

gamoTvlebi gviCvenebs, es moqmedeba umniSvneloa da SeiZleba misi ugulebelyofa.

214

rogorc

van-der-vaalsis gantoleba SeiZleba gardavqmnaT ise, rom igi gamosadegi gaxdes airis nebismieri m masisaTvis. aRvniSnoT m masis airis moculoba V -Ti, maSin (§58)

V sadac  moluri masaa.



V ,

(68.7)

Tu (68.6) formulaSi SevitanT V  –is mniSvnelobas (68.7) formulidan, miviRebT:  m 2 a  m  m  p  2 2 V  b   RT . (68.8)     V   (68.8) gantoleba aris van-der-vaalsis gantoleba airis nebismieri m masisaTvis. amasTan, a da b Sesworebebis mniSvnelobebi igivea. dabolos, SevniSnoT Semdegi. miuxedavad imisa, rom van-der-vaalsis gantoleba gacilebiT ukeT aRwers realuri airis qcevas vidre klapeiron-mendeleevis gantoleba, unda gvaxsovdes, rom igi mainc miaxloebiTi gantolebaa. es kerZod mJRavndeba imiT, rom a da b koeficientebi sinamdvileSi mudmivebs ki ar warmoadgenen, aramed temperaturis cvlilebiT icvlebian. garda amisa, van-der-vaalsis gantolebis izoTermebi gansxvavdeba realuri airis eqsperimentuli izoTermebisagan. ganvixiloT ufro dawvrilebiT realuri airis molekulebs Soris urTierTqmedebis Zalebi.

molekulebi

warmoadgenen

eleqtruli

muxtebisagan

(eleqtronebisa

atombirTvebisagan) Sedgenil rTul sistemas, romelTa Soris moqmedebs rogorc mizidvis, ise ganzidvis Zalebi. vinaidan molekulaSi sruliad kompensirdeba dadebiTi da uaryofiTi muxtebi, igi eleqtroneitraluri sistemaa, amitom mizidvisa da ganzidvis Zalebi molekulebs Soris moqmedebs metad mcire manZilze. Cven ar ganvixilavT molekulebs Soris moqmedi Zalebis bunebas, radgan igi, iseve rogorc TviT atomebisa da molekulebis arseboba, aixsneba mxolod kvanturi meqanikis safuZvelze. moviyvanT mxolod saboloo Sedegs: ori molekulis urTierTqmedebis Zalebi maT Soris manZilis

n –uri xarisxis ukuproporciulia F ~

1 rn

mizidvis ZalebisaTvis n  7 , xolo ganzidvis ZalebisaTvis n  12 . e.i. manZilis gadidebiT ganzidvis Zalebi ufro swrafad mcirdeba, vidre mizidvis Zalebi, amitom SedarebiT did (molekulis ramdenime diametris tol) manZilze sWarbobs mizidva, ufro mcire manZilze ki _

 

ganzidva. 68.2 a suraTze grafikuladaa gamosaxuli mizidvis  F1 ~

215

1  1    , ganzidvis  F2 ~ 12  7 r  r  

 

da marezultirebeli  F ~

1 1    Zalebis damokidebuleba molekulebs Soris manZilze. r 7 r 12 

grafikidan Cans, rom molekulebs Soris garkveul r  r0 manZilze mizidvisa da ganzidvis Zalebi erTmaneTs akompensireben, amitom marezultirebeli Zala nulis tolia. amrigad r0 aris is manZili, romelzec molekulebi erTmaneTis mimarT uZravi iqnebodnen siTburi moZraobis ararsebobis SemTxvevaSi. r  r0 manZilze molekulebi erTmaneTs miizidaven, r  r0 manZilze ki _ erTmaneTs ganizidaven.

u F1 ~

1 r 12

b E2(~T2)

F r2 E1(~T1) E1 0

0 r

u r1 r0

r0 F2 ~

1 r7

b suraTi 68.2

molekulebis urTierTqmedebis Seswavla ufro mosaxerxebelia ara urTierTqmedebis Zalebis, aramed urTierTqmedebis potenciuri energiis ganxilviT *. Tu gamoviyenebT Zalasa da potenciur energias Soris arsebul cnobil damokidebulebas F  

dU , U r  grafiki (urTidR

erTqmedebis potenciuri mrudi) miiRebs 68.2 b suraTze moyvanil saxes. rogorc suraTidan Cans, potenciuri mrudi asimetriulia. roca molekulebs Soris manZili r  r0 (urTierTqmedebis Zala nulis tolia) potenciur mruds aqvs minimumi e.i. xasiaTdeba e.w. potenciuri ormoTi, xolo

r –is gazrdiT asimptoturad uaxlovdeba abscisTa RerZs. mrudis asimetriuloba

gamosaxavs im faqts, rom ganzidvis Sesabamisi potenciuri energia ( ab Sto) icvleba ufro swrafad, vidre mizidvisa ( ac Sto). minimumis arseboba gamosaxavs im faqts, rom r  r0 manZil*

sxeulis Semadgeneli nawilakebis saSualo energia yovelTvis gansxvavdeba nulisagan, am

nawilakebis urTierTqmedebis saSualo Zala ki yovelTvis nulis tolia, vinaidan nawilakTa calkeul

wyvilebs

Soris

moqmedi

Zalebi

urTierTsawinaaRmdegoa.

216

sididiT

toli

mimarTulebiT

ze ori molekulisagan Sedgenili sistema wonasworobis mdgomareobaSia. abscisaTa RerZTan asimptoturad miaxloeba gamosaxavs im faqts, rom didi manZiliT daSorebuli molekulebi ar urTierTqmedeben, urTierTqmedebis potenciuri energia nulis tolia. potenciuri mrudis safuZvelze ganvixiloT molekulebis daaxloebisa da Sejaxebis procesi. simartivisaTvis davuSvaT, rom erTi molekula ( A ) uZravia da misi centri koordinatTa saTaveSia, xolo meore ( B ) molekula usasrulobidan TandaTan uaxlovdeba mas. radgan did manZilze molekulebi ar urTierTqmedebn ( U  o ) da A molekula uZravia, amitom Tavdapirvelad B molekulis kinetikuri energia W gansazRvravs ori ( A da B ) molekulisagan Sedgenili izolirebuli sistemis srul E1 ~ T1  energias:

W  E1 ~ T1  . rodesac B molekula uaxlovdeba A –s, mizidvis Zalis gavleniT r  r0 maZilamde is moZraobs aCqarebulad da misi siCqare da, maSasadame, kinetikuri energiac TandaTan izrdeba. magram, vinaidan, energiis mudmivobis kanonis Tanaxmad izolirebuli sistemis sruli energia mudmivi unda iyos, kinetikuri energiis gazrda iwvevs urTierTqmedebis potenciuri energiis Semcirebas (mrudis ca Sto)*:

W  U r   E1 ~ T1  . rodesac B molekula r0 manZilzea A molekulisagan, potenciuri energia U r0  minimaluria, xolo kinetikuri energia Wr0 _ maqsimaluri. am manZilidan dawyebuli molekulaze moqmedebas iwyebs ganzidvis Zala, igi iwyebs moZraobas Senelebulad da misi siCqare da kinetikuri energiac swrafad mcirdeba, xolo urTierTqmedebis U r  potenciuri energia swrafad izrdeba ( ab

Sto).

U r1   E1 ~ T1  ,

r1 wertilSi potenciuri energia sistemis sruli energiis tolia

amitom

kinetikuri

Wr1

energia

nulis

toli

gaxda

(igi

daixarja

urTierTganzidvis Zalis winaaRmdeg muSaobis Sesrulebaze). nulis tolia am wertilSi B molekulis siCqarec. amis Semdeg B molekula iwyebs ukuqcevas da procesi mimdinareobs Sebrunebuli mimarTulebiT: Tavdapirvelad, ganzidvis Zalis moqmedebiT, B molekula daiwyebs

A –sagan daSorebas sul ufro mzardi siCqariT, ris gamoc misi kinetikuri energia gaizrdeba, potenciuri ki _ Semcirdeba; r  r0 manZilidan dawyebuli masze imoqmedebs mizidvis damamuxruWebeli Zala, ris gamoc kinetikuri energia TandaTan Semcirdeba, potenciuri ki gaizrdeba. bolos, usasrulod didi manZiliT daSorebisas potenciuri energia maqsimaluria (nulis toli gaxda), kinetikuri energia ki iRebs Tavdapirvel mniSvnelobas W  E1 ~ T1  . im minimalur r1 manZils, romelzedac A da

B molekulebis centrebi uaxlovdeba erTmaneTs,

molekulis e f e q t u r i d i a m e t r i ewodeba. *

am dros

potenciuri energiis absoluturi sidide izrdeba, magram igi iRebs uaryofiT

mniSvnelobebs.

217

davuSvaT axla, rom

B molekula uaxlovdeba A molekulas ufro meti sawyisi

E2 ~ T2  energiiT. suraTidan Cans, rom minimaluri molekulebis centrebis daaxloeba, Semcirdeba,

manZili,

romelzedac

moxdeba

r2  r1 . maSasadame, molekulis efeqturi

diametri ar aris mudmivi sidide _ temperaturis (energiis) gazrdiT igi mcirdeba. marTalia, es Semcireba umniSvneloa, vianidan potenciuri mrudis ab Sto TiTqmis vertikaluria. amrigad, potenciuri mrudis safuZvelze Catarebuli analizi gviCvenebs, rom molekulebis dajaxebis procesi ar hgavs myari birTvebis dajaxebas. maT urTierTSexebas saerTod ara aqvs adgili. isini mxolod uaxlovdebian erTmaneTs garkveul minimalur manZilze, romelsac efeqturi diametri ewodeba, ris Sedegadac iwyeba maTi ukuqceva. efeqturi diametri temperaturis cvlilebiT icvleba, magram vianidan es cvlileba umniSvneloa, realuri airis TeoriaSi mas saerTod ar iTvaliswineben _ potenciuri mrudis ab Stos cvlian vertikaluri ab wrfiT, riTac uSveben, rom molekulebi warmoadgenen mudmivi diametris myar birTvebs. amrigad, ganzidvis Zalebis ganxilvis nacvlad molekulebs ganixilaven mudmivi diametris myari birTvebis saxiT, romelTa Soris moqmedebs mxolod mizidvis Zalebi. sabolood,

ganvixiloT,

gansazRvravs

nivTierebis

ama

agregatul

mdgomareobas. winaswar SevniSnoT, rom 68.2, b suraTze gamosaxuli urTierTqmedebis potenciur mruds aseTive saxe aqvs myari sxeulisa da siTxis molekulebisaTvisac. potenciuri ormos siRrme ( U min ) gansazRvravs im muSaobas, romelic unda Sesruldes mizidvis Zalebis sawinaaRmdegod, rom r  r0 manZilze wonasworobaSi myofi molekulebi movwyvitoT erTmaneTs da gadaviyvanoT usasrulobaSi. vinaidan molekulebis siTburi moZraobis saSualo kinetikuri energia kT –s rigisaa, amitom nivTierebis agregatuli mdgomareoba ganisazRvreba U min da kT sidideebis TanafardobiT. a.

Zalian dabal temperaturaze, roca kT  U min , molekulebi miizidaven erTmaneTs

da ganlagdebian mWidrod garkveuli kanonzomierebiT _ nivTiereba iqneba m y a r a g r e g a t u l m dgom areobaSi .

nawilakebis

siTbur

moZraobas

SemTxvevaSi

aqvs

rxevis

xasiaTi

wonasworobis maxloblad. b . Zalian maRal temperaturaze, roca kT  U min , intensiuri siTburi moZraoba xels uSlis molekulebs dauaxlovdnen erTmaneTs r0 manZilze da Seqmnan erTiani, mtkice sistema _ nivTiereba iqneba a i r a d a g r e g a t u l m d g o m a r e o b a S i . g. Sualedur temperaturaze, roca kT  U min , siTburi moZraobis gamo molekulebi ganuwyvetliv gadaadgildebian, magram am dros xdeba molekulebis mier adgilebis gacvla, ise rom maT Soris manZili wonasworul r0 manZilze SesamCnevad meti ar xdeba _ nivTiereba iqneba T x e v a d a g r e g a t u l m d g o m a r e o b a S i .

218

amrigad, nebismieri nivTiereba, imis mixedviT, Tu rogoria misi temperatura, SeiZleba iyos rogorc myar, ise Txevadsa da airad mdgomareobaSi. erTi agregatuli mdgomareobidan meoreSi gadasvlis temperatura damokidebulia potenciuri ormos ( U min ) siRrmeze mocemuli nivTierebisaTvis. inertuli airebisaTvis (heliumi, neoni argoni da sxva) potenciuri ormos siRrme Zalian mcirea. amitom es nivTierebebi Cveulebriv

(oTaxis) temperaturaze

arian airad

mdgomareobaSi da mxolod Zalian dabal (absoluturi nulis maxlobel) temperaturaze gadadian Txevad mdgomareobaSi. liTonebisaTvis piriqiT, potenciuri ormos siRrme didia, amitom oTaxis temperaturaze isini, Cveulebriv, myar mdgomareobaSi arian da SedarebiT maRal temperaturaze gadadian Txevad mdgomareobaSi (dnebian).

§69. van-der-vaalsis gantolebis izoTermebi. kritikuli temperatura viciT

(§57),

rom

idealuri

airis

izoTermebi

hiperbolebia.

vnaxoT,

rogori saxe aqvs van-der-vaalsis (68.6) gantolebis izoTermebs. amisaTvis gantolebaSi gavxsnaT frCxilebi, yvela wevri gavamravloT

V

sidideze da

davalagoT V –s xarisxebad. miviRebT:

 RT  2 a ab V  V  V3   b   0. p  p p 

(69.1)

Tu davuSvebT, rom am gantolebaSi T  const da avagebT

p –sa da V –s Soris

damokidebulebis grafiks, miviRebT realuri airis Teoriul izoTermas. Teoriuls mas imitom uwodeben, rom igi miRebulia (69.1) gantolebis safuZvelze da ara cdiT. (69.1) sur. 69.1

gantoleba

warmoadgens

mesame

xarisxis gantolebas V –s mimarT. algeb-

ridan cnobilia, rom yoveli mesame xarisxis gantolebas aqvs sami amonaxseni, romelTagan an samive namdvilia, an erTi namdvilia da ori kompleqsuri*. es *

namdvil amonaxsenTa raodenoba (moculobis SemTxvevaSi mxolod maT aqvT garkveuli fizikuri azri) damokidebulia gantolebis parametrebis mniSvnelobaze.

219

imas niSnavs, rom V RerZis paraleluri wrfe (izobara) izoTermas gadakveTs an sam wertilSi, an erT wertilSi. marTlac, (69.1) gantolebis safuZvelze agebul izoTermebis

ojaxs 69.1 suraTze mocemuli saxe aqvs, saidanac Cans,

rom garkveul Tk temperaturamde abscisaTa RerZis paraleluri wrfe izoTermas kveTs sam wertilSi (rac gvaZlevs sam V1 , V 2 , V 3 amonaxsens), Tk temperaturaze da zeviT ki _ mxolod erTSi. temperaturis gazrdiT izoTermis talRiseburi BCDE nawili TandaTan mcirdeba da, bolos, Tk temperaturaze, gadadis gadaRunvis K wertilSi. K wertils ewodeba kritikuli wertili, am wertilis Sesabamis Tk , p k , V μk parametrebs _ kritikuli parametrebi, xolo K wertiliT gamosaxul nivTierebis mdgomareobas _ kritikuli mdgomareoba.

kritikuli temperaturis zeviT realuri airis izoTermebi hgavs idealurisas, rac miuTiTebs imas, rom maRal temperaturaze realuri airi iqceva idealuri airis msgavsad. SeviswavloT miRebuli izoTermebi.

amisaTvis

gamovxazoT

erT-erTi maTgani da gavavloT abscisaTa

RerZis

paraleluri

wrfe ise, rom daStrixuli farTobebi

iyos

erTmaneTis

toli

(sur.69.2a). avagoT realuri airis eqsperimentuli izoTerma. amisa-

sur. 69.2

Tvis CavtaroT Semdegi cda: mova-

TavsoT erTi moli romelime realuri airi (vTqvaT CO2 ) dguSian WurWelSi da davuSvaT, rom airis sawyisi mdgomareoba gamoisaxeba A' wertiliT (sur. 69.2 b). davawveT dguSs, e.i. daviwyoT airis moculobis izoTermuli Semcireba. es gamoiwvevs misi wnevis gazrdas da airis mdgomareobis gamomsaxveli wertili gahyveba A' B' mruds. B' wertilidan dawyebuli moculobis Semdgomi Semcireba aRar gamoiwvevs wnevis gazrdas (mdgomareobis gamomsaxveli wertili gahyveba V

RerZis paralelur

B' E '

wrfes). am dros airi gadadis Txevad mdgo-

mareobaSi. es procesi iwyeba B' wertilSi da mTavrdeba E' wertilSi. amis Semdeg, siTxis ukumSvelobis gamo, miiReba TiTqmis vertikaluri E' F' mrudi. 220

a i r s , r om l is w ne v a m o cu l ob i s S e m c ire b iT i zr de b a,

ujeri

o r T q l i e w o de b a, xo lo a irs , ro m l is w ne v a m o cu l ob aze d am o k i d e b u li

ar is

n a je ri

o rT q l i

e w o de b a *. amrigad, eqsperimentul

izoTermaze A' B' ubani gamosaxavs ujeri orTqlis najer orTqlad gadaqcevis process. B' wertilSi orTqli najeria. B' E ' ubani gamosaxavs najeri orTqlis siTxed gadaqcevis process. E' wertilSi orTqli mTlianad gaTxevadebulia. E' F' ubani gamosaxavs siTxis kumSvis process. Tu nivTierebis moculobas, romelic Seesabameba B' E ' horizontaluri monakveTis nebismier wertils, mudmivs davtovebT, maSin nivTierebis nawili darCeba Txevad mdgomareobaSi, nawili ki _ airadSi, amasTan, rogorc siTxis, ise orTqlis raodenoba drois mixedviT ar Seicvleba. es imas niSnavs, rom siTxe

wonasworobaSi

imyofeba

Tavis

orTqlTan,

oRond

wonasworoba

dinamikuria _ drois erTeulSi siTxidan orTqlSi da piriqiT gadasuli molekulebis ricxvi tolia. amitom, Cveulebriv, n a je r orT q ls ga n m ar t a ven

r o gor c

is e T

o rT q ls ,

r om e li c

di n am ik u r

w o n as w o rob a Si a

T av i s s iT xe s T a n. SevadaroT erTmaneTs eqsperimentuli da Teoriuli izoTermebi. 69.2 suraTidan Cans, rom A' B' ubani analogiuria Teoriuli izoTermis AB ubnisa, xolo E ' F ' ubani _ EF

ubnisa. es nisnavs, rom van-der-vaalsis gantoleba

damakmayofileblad aRwers calke airad da calke Txevad mdgomareobas. rac Seexeba

ufro

rTul

sistemas,

romelic

Sedgeba

orTqlisa

siTxis

narevisagan ( B' E ' ubani), aq adgili aqvs sagrZnob gadaxrebs ( B' E ' wrfis nacvlad miiReba BCDE talRiseburi ubani). gavarCioT Teoriuli izoTermis BCDE ubani. SevniSnoT, rom DC ubniT gamosaxul process fizikuri arsi ar gaaCnia, misi praqtikuli ganxorcieleba ar

SeiZleba,

vinaidan

SeiZleba

mudmivi

temperaturis

pirobebSi

erTdroulad izrdebodes airis wnevaca da moculobac. es Teoriis xarvezia. BC da ED ubnebs ki garkveuli fizikuri Sinaarsi aqvT. amasTan, vinaidan BC

warmoadgens airadi mdgomareobis gamomsaxveli ubnis gagrZelebas, xolo ED *

nivTierebis airadi mdgomareobis gamomsaxveli ori termini _ airi da orTqli SemorCenilia im droidan, roca fiqrobdnen, rom orTqlis gaTxevadeba SeiZleba, airisa ki _ ara. dRes, roca miRweulia yvela airis gaTxevadeba, terminis aseTi gaoreba gamarTlebuli ar aris. airi da orTqli erTi da igivea. mxolod CveulebiTaa gamowveuli, rom vambobT wylis orTqls da ara airs, najer orTqls da ara najer airs.

221

_ Txevadi mdgomareobis gamomsaxveli ubnis gagrZelebas, SeiZleba vivaraudoT, rom isini gamosaxaven Sesabamisad airisa da siTxis mdgomareobebs. ra mdgomareobebia es? ganvixiloT jer BC ubani. B wertilSi airi najer mdgomareobaSia. Tu amis Semdeg moculobas SevamcirebT, unda daiwyos misi gaTxevadeba ise, rom wneva darCes mudmivi, e.i. mdgomareobis gamomsaxveli wertili unda gahyves BE wrfes.

magram

airis

gaTxevadeba

(k o n de n s a ci a )

rom

daiwyos,

amisaTvis

saWiroa masSi arsebobdes k o n de n s a ci is c e n tre b i , vTqvaT, mtvris nawilakebi an eleqtruli muxtebi. Tu airSi aseTi centrebi ar arsebobs, e.i. igi Zalian sufTaa, misi siTxed gadaqceva gviandeba da moculobis Semcireba iwvevs wnevis gazrdas _ mdgomareobis gamomsaxveli wertili mihyveba BC

mruds.

i s e T ai r s , r om l is w ne v ac u fr o m e ti a, v i dr e im av e t e m p e r atu r is n a je r i or T q l is w ne v a, ze n a je ri o rT q li e w o de b a. amrigad BC ubani gamosaxavs zenajeri orTqlis mdgomareobas. faqizi eqsperimentis CatarebiT SesaZlebelia

zenajeri

orTqlis

miReba.

zenajeri

orTqis

mdgomareoba

mdgradi ar aris. Tu masSi Seiqmna kondensaciis centri (vTqvaT moxvda mtvris nawilaki), TavisTavad daiwyeba kondensacia da misi wneva gautoldeba najeri orTqlis wnevas. aseT aramdgrad mdgomareobas m e ta s t ab ilu ri mdgomareoba ewodeba. ganvixiloT Teoriuli izoTermis ED ubani. E wertili gamosaxavs nivTierebis Txevad mdgomareobas, am dros dguSiani WurWlis mTeli moculoba siTxes ukavia _ dguSis

zedapri exeba siTxis zedapirs. daviwyoT moculobis

gazrda, e.i. dguSis TandaTanobiT aweva. Cveulebriv pirobebSi am SemTxvevaSi siTxis nawili gadava airad mdgomareobaSi da WurWelSi miviRebT siTxesa da najer orTqls. mdgomareobis gamomsaxveli wertili gahyveba EB wrfes. magram Tu aviRebT Zalian sufTa siTxes, romelic sruliad ar Seicavs gaxsnil airs an sxva raime nawilaks, SeiZleba ganxorcieldes procesi _ WurWelSi gveqneba mxolod siTxe, romlis wneva naklebia, xolo moculoba meti, vidre E mdgomareobaSi. s iT xe s , r om l is w ne v a n ak le b ia, v i dre n a je r orT q lT a n w o n a s w o r o b aS i m y o fi im av e te m pe ra tu r is s iT xi s w ne v a, g aW im u l i s iT xe

e w o de b a.

amrigad, Teoriuli izoTermis

monakveTi gamosaxavs

gaWimuli siTxis mdgomareobas. es mdgomareoba kidev ufro metastabiluria, 222

vidre zenajeri orTlisa da misi miReba did eqsperimentul siZneleebTanaa dakavSirebuli. dabolos, gavarkvioT kritikuli temperaturisa da kritikuli wertilis fizikuri arsi. amisaTvis avagoT eqsperimentuli izoTermebi sxvadasxva temperaturaze. SevamCnevT (sur. 69.3), rom temperaturis gazrdiT B' E ' monakveTis sigrZe TandaTan mcirdeba ( B' da E wertilebi uaxlovdeba erTmaneTs) da, bolos, kritikul Tk temperaturaze nulis toli xdeba _ gadadis gadaRunvis K wertilSi, romelsac kritikuli wertili vuwodeT. es imas niSnavs, rom temperaturis gazrdiT orTqlis siTxed gadaqcevis dro TandaTan mcirdeba da kritikul temperaturaze nulis toli xdeba.

maSasadame,

k ri tik u l i

te m p e r a tu r a

S e i Z le b a g a n v m ar t oT ro go rc is e T i te m p e r a tu r a, r o m e l ze d ac w ne v i s ga d i de b i T a i ri

sur. 69.3

(o r T q l i )

m e y s e u l ad

g a d a dis

T xe v a d m d go m ar e o b a Si. magram, B' wertili gamosaxavs airis mdgomareobas, E ' wertili ki _ siTxis mdgomareobas. amitom maTi urTierTdaaxloeba niSnavs, rom temperaturis gazrdiT TandaTan mcirdeba gansxvaveba siTxisa da airis damaxasiaTebel fizikur sidideeb Soris. wertilSi

nivTiereba

mgdomareobaSi,

magram

wertilSi es gansxvaveba nulis tolia. am

erTdroulad maTi

aris

rogorc

erTmaneTisagan

garCeva

siTxis,

ise

orTqlis

SeuZlebelia,

radgan

mdgomareobis damaxasiaTebeli samive parametri _ wneva, moculoba da temperatura erTnairia*. swored amitom ewodeba K K wertils kritikuli wertili, xolo am wertiliT gamosaxul nivTierebis mdgomareobas _ kritikuli mdgomareoba. amrigad, k r i tik u l i e w o de b a n iv T i e re b is i s e T m dg om are ob as , r o de s ac is p ob a y ov e l gv a ri g a ns xv av e b a s iT xe s a d a m is or T q ls S o r i s **. *

makroskopuli sxeuli, garda wnevisa, moculobisa da temperaturisa xasiaTdeba kidev energiiTa da sxva sidideebiT, magram es sidideebi aRniSnuli parametrebis funqciebia, amitom K wertilSi maTi mniSvnelobebic erTnairia. ** kritikul temperaturaze ufro dabla siTxisa da misi najeri orTqlis wneva da temperatura erTnairia, magram gansxvavdebian simkvriviT, anu kuTri moculobiT, romelic simkvrivis Sebrunebuli sididea.

223

69.3 suraTidan Cans, rom kritikul temperaturaze zemoT realuri airis izoTerma msgavsia idealuri airis izoTermisa, igi ar Seicavs horizontalur B' E ' ubans, romelic gamosaxavs airis siTxed gadaqcevis process. e.i. kriti-

kul temperaturaze zemoT nivTiereba SeiZleba iyos mxolod airad mdgomareobaSi,

wnevis

gazrda

iwvevs

mxolod

moculobis

Semcirebas,

siTxed

gadaqcevis gareSe. amitom, kritikuli temperatura SeiZleba asec ganvmartoT: e s a r is is e T i te m pe r a tu r a, rom l is ze m oT a ir is g aT xe v ad e b a S e u Z l e b e li a, ro g orc ar u n d a g av z a rd oT m is i w ne v a. amrigad, nebismieri airi rom gavaTxevadoT, is jer unda gavacivoT kritikul temperaturamde da Semdeg SevkumSoT. amJamad cnobilia yvela nivTierebis kritikuli mdgomareobis Sesabamisi parametrebi

p k , Vk da Tk . isini mocemuli nivTierebis fundamenturi sidideebia. davamyaroT kavSiri kritikul parametrebsa da van-der-vaalsis a da b mudmivebs Soris. amisaTvis davweroT (69.1) gantoleba kritikuli wertilisaTvis

 RT  a ab (69.2) V3   b  k V2  V  0 pk  pk pk  meore mxriv, Tu (69.1) gantolebis amonaxsens aRvniSnavT V1 –iT, V 2 -iT, V 3 -iT, maSin igi, iseve rogorc nebismieri mesame xarisxis gantoleba, SeiZleba Semdegnairad CavweroT:

Vμ  Vμ1 Vμ  Vμ2 Vμ  Vμ3   0 .

kritikul wertilSi

V1  V 2  V 3  Vk ,

amitom

V

 Vk   0 , 3



anu

V3  3VkV2  3V2kV  V3k  0 .

(69.3)

(69.2) da (69.3) gantolebebi igivuria, amitom V  –is erTnair xarisxebTan mdgomi koeficientebi unda Tanxvdebodnen erTmaneTs. es gavZlevs gantolebaTa sistemas:

RTk a  3V2k ;  3Vk ; pk pk romelTa amoxsna Vk –s, p k -sa da Tk –s mimarT gvaZlevs: b

Vk  3b ;

pk 

a ; 27b 2

Tk 

ab  V3k , pk 8a ; 27bR

miRebuli formulebi saSualebas gvaZlevs gamovTvaloT airis kritikuli parametrebi

a , b da R sidideebis meSveobiT an, piriqiT, Tu cnobilia Vk , 224

p k da Tk , ganvsazRvroT

a , b da R . Cveulebriv, eqsperimentulad sazRvraven airis kritikul parametrebs da maTi meSveobiT igeben van-der-vaalsis gantolebis a da b

mudmivebs.

cxrilSi mocemulia zogierTi nivTierebis kritikuli parametrebi, saidanac cxadia, Tu ratomaa H2O Cveulebriv pirobebSi Txevad mdgomareobaSi, da ratom aris ase Zneli heliumis gaTxevadeba. cx rili

Tk , K

p k (atm)

Vk  10 3 m3/kg

heliumi

5,30

2,26

14,43

wyalbadi

33,2

12,8

32,29

azoti

126,0

33,5

3,22

Jangbadi

154,3

50,6

2,32

naxSirorJangi

304,2

73,0

2,17

wyali

647,2

318,0

2,50

nivTiereba

§70. realuri airis Sinagani energia. joul-tomsonis efeqti viciT, rom idealuri airis Sinagani energia ganisazRvreba misi molekulebis kinetikuri energiebis jamiT da 1 moli airisaTvis ase gamoisaxeba (§63):

U 

i RT . 2

es formula gviCvenebs, rom 1 moli idealuri airis Sinagani energia damokidebulia mxolod airis temperaturaze da ar aris damokidebuli sxva parametrebze, kerZod moculobaze. es niSnavs rom Tu airi izolirebulia (e.i. ar xdeba energiis gacvla-gamocvla garemosTan da garda amisa, moculobis cvlilebisas ar sruldeba muSaoba gare Zalebis winaaRmdeg), misi gafarToebis an kumSvis SemTxvevaSi Sinagani energia ar icvleba ( U  0 ) da, maSasadame, ar icvleba arc temperatura ( T  0 ). realuri airis SemTxvevaSi, imis gamo, rom misi molekulebi urTierTqmedeben da, maSasadame, gaaCniaT urTierTqmedebis potenciuri energia, Sinagani energia ganisazRvreba rogorc molekulebis kinetikuri da potenciuri ener-

225

giebis jami. Tu erTi moli realuri airis molekulebis potenciur energias aRvniSnavT U p –Ti, maSin misi Sinagani energia: U

i RT  U p . 2

(70.1)

magram potenciuri energia ganisazRvreba molekulebis urTierTmdebareobiT da, vinaidan airis moculobis Secvla iwvevs molekulebis urTierTmdebareobis Secvlas, U p damokidebulia moculobaze. amrigad, realuri airis Sinagani energia temperaturis garda damokidebulia moculobaze. izolirebuli airisaTvis Sinagani energia mudmivi sididea _ U  0 , amitom (68.1) formulis safuZvelze gveqneba: O

i RT  U p , 2

(70.2)

saidanac Cans, rom realuri airis moculobis (potenciuri energiis) Secvlam unda gamoiwvios misi temperaturis Secvla. izolirebuli realuri airis gafarToebiT gamowveuli temperaturis cvlileba cdiT Seiswavles ingliselma mecnierebma joulma (1818-1889) da tomsonma (1824-1907) da mas joul-tomsonis efeqti ewodeba. joul-tomsonis cdaSi A WurWelSi moTavsebuli garkveuli wnevis airi forovani C sacobis gavliT

gadadioda

WurWelSi

(sur. 70.1)

farTovdeboda. B WurWelSi Seqmnili iyo vakuumi, amitom es gafarToeba xdeboda muSaobis Sesrulebis gareSe. forovani sacobi airisaTvis aris didi hidrodinamikuli winaRoba, amitom B

Wur-

WelSi airis gadinebis siCqare mcire iyo. sacobis orive mxares moTavsebuli iyo mgrZnobiare Termometrebi, romlebic gviCvenebda airis temperaturas.

sur. 70.1

cdis dros xelsawyo mTlianad izolirebuli iyo. cdam aCvena, rom gafarToebis Sedegad TiTqmis yvela airi civdeba (adgili aqvs joul-tomsobis `dadebiT~ efeqts) da mxolod wyalbadi da heliumi Tbeba (joul-tomsonis `uaryofiTi efeqti~).

226

razea damokidebuli joul-tomsonis efeqtis niSani? viciT, rom realuri airis molekulebs Soris moqmedebs rogorc mizidvis, ise ganzidvis Zalebi. jer davuSvaT, rom molekulebs Soris moqmedebs mxolod mizidvis Zalebi. maSin gafarToebisas

airi muSaobas

Seasrulebda mizidvis

Zalebis

dasa-

Zlevad. vinaidan sistema Tboizolirebulia, am muSaobas airi Seasrulebs Sinagani energiis xarjze da amis gamo igi gacivdeba. amrigad molekulebs Soris mxolod mizidvis Zalebis arseboba ganapirobebs joul-tomsonis dadebiT efeqts. axla davuSvaT, rom molekulebs Soris moqmedebs mxolod ganzidvis Zalebi. airis gafarToebisas am Zalebis moqmedebiT molekulebi aCqardebodnen. rac gamoiwvevda maTi siCqarisa da kinetikuri energiis zrdas da airi gaTbeboda. amrigad, molekulebs Soris mxolod ganzidvis Zalebis arseboba ganapirobebda joul-tomsonis uaryofiT efeqts. realur pirobebSi molekulebs Soris moqmedebs rogorc mizidvis, ise ganzidvis Zalebi, cdis dros Cven vakvirdebiT jamur efeqts, amitom joultomsonis efeqtis niSani damokidebulia imaze, mocemul pirobebSi airis molekulebs Soris mizidva sWarbobs Tu ganzidva. zusti

gamoTvlebi

gvaZlevs

temperaturis

cvlilebis

Semdeg

mniSvnelobas:

T2  T1  T 

 RT1b 1 2a   ,  CV  R  V1  b V1 

(70.3)

sadac V1 airis sawyisi moculobaa, CV _ siTbotevadoba mudmivi moculobis dros, R _ airis universaluri mudmiva, xolo a da b gantolebis mudmivebi. rogorc viciT, Soris mizidvas, xolo

_ van-der-vaalsis

mudmiva axasiaTebs molekulebs

_ ganzidvas. Tu molekulebs Soris arsebobs

mxolod mizidva, e.i. a  0 , b  0 , maSin (70.3)-dan T  0 , e.i. gafarToebisas airi civdeba da adgili aqvs joul-tomsonis dadebiT efeqts. Tu molekulebs Soris mxolod ganzidvis Zalebi moqmedeben e.i. Tu a  0 , b  0 , maSin T  0 , e.i. gafarToebisas airi Tbeba, adgili aqvs joul-

tomsonis uaryofiT efeqts. zogad SemTxvevaSi

a  0,

b  0 , amitom efeqtis niSani damokidebuli

iqneba am sidideebis Tanafardobaze.

227

aRmoCnda, rom garkveuli pirobebis (sawyisi moculobisa da temperaturis) SerCeviT SeiZleba mivaRwioT imas, rom gafarToebisas airis temperatura ar Seicvalos, e.i. adgili hqondes joul-tomsonis nulovan efeqts. nulovani efeqtis Sesabamis maqsimalur temperaturas i nv e rs iis te m pe r at u r a ewodeba. am temperaturis zeviT adgili aqvs joul-tomsonis uaryofiT efeqts, xolo qveviT _ dadebiTs. amrigad, erTi da igive airisaTvis, sawyisi pirobebis SerCeviT SeiZleba miviRoT joul-tomsonis rogorc dadebiTi, ise uaryofiTi efeqti. airebis umravlesobisaTvis (azoti, Jangbadi da sxv.) inversiis temperatura maRalia, amitom cdis dros Cveulebriv vakvirdebiT dadebiT efeqts. wyalbadisa da heliumis inversiis temperatura dabalia (bevrad dabali oTaxis temperaturaze), amitom maTi gafarToebisas, Cveulebriv, adgili aqvs joul-tomsonis uaryofiT efeqts. cxadia, rom Tu am airebze cdas CavatarebT inversiis temperaturaze ufro dabal temperaturaze, maSin gafarToebisas airi gacivdeba. joul-tomsonis dadebiT efeqtze dafuZnebulia airis gasaTxevadebeli manqanis muSaobis principi, romelic Seqmna 1895 wels germanelma inJiner-fizikosma lindem.

X Tavi

T e r m od i n a mi k i s s a fu Zv l e b i

ยง71. ZiriTadi cnebebi Termodinamika aris fizikis nawili romelic Seiswavlis Termodinamikur sistemebs da Termodinamikur procesebs energetikuli gardaqmnebis TvalsazrisiT. Termodinamikuri sistema ewodeba makroskopul sxeuls (an sxeulTa jgufs), romelSic mimdinareobs siTbos sxva saxis energiad gardaqmnasTan dakavSirebuli procesi da piriqiT. Termodinamikuri sistemis magaliTia airi, siTxe, myari sxeuli, eleqtruli

wredi, 228

adamianis organizmi da sxva. yoveli

Termodinamikuri

sistemis

damaxasiaTebeli

niSani

isaa,

rom

igi

Sedgeba

kolosaluri raodenobis nawilakebisagan. Termodinamikis zogierT sakiTxs (airis cdiseuli kanonebi, airis siTbotevadoba) ukve gavecaniT wina TavSi. Termodinamikuri sistemis mdgomareoba xasiaTdeba mTeli rigi fizikuri sidideebiT, romlebsac Termodinamikuri parametrebi ewodeba. umartivesi Termodinamikuri sistemis _ idealuri airis SemTxvevaSi aseTi parametrebia wneva, moculoba (kuTri moculoba) da temperatura. * gantolebas, romelic amyarebs damokidebulebas Termodinamikur parametrebs Soris, mdgomareobis gantoleba ewodeba. idealuri airisaTvis aseT gantolebas warmoadgens klapeiron-mendeleevis gantoleba, realuri airisa da (nawilobriv) siTxisaTvis _ van-dervaalsis

gantoleba.

myari

sxeulis

mdgomareobis

gantoleba

jerjerobiT

dadgenili ar aris. klapeiron-mendeleevis gantoleba saSualebas gvaZlevs ori cnobili parametris mixedviT ganvsazRvroT mesame, amitom idealuri airisaTvis

damoukidebel

Termodinamikur

parametrTa

ricxvi,

romlebic

gansazRvravs mis mdgomareobas, oria ( p, V ; p, T an T ,V ). T e r m o d in am ik u r i (anu siTburi) w on a s w or ob a ewodeba sistemis iseT mdgomareobas,

romlis

drosac

yvela

Termodinamikuri

parametri

ragind

didxans rCeba ucvleli. Termodinakuri wonasworobis dros sistemis yvela nawilSi wneva, temperatura da sxva parametrebi erTnairia**, amitom, am dros sistemaSi ar SeiZleba adgili hqondes iseT movlenebs, rogoricaa difuzia, Tbogamtaroba, qimiuri reaqciebi, nivTierebis erTi agregatuli mdgomareobidan meoreSi gadasvla (fazuri gardaqmnebi), da sxv. T e r m o d in am ik u r i

pr oc e s i

ewodeba sistemis erTi mdgomareobidan

meoreSi gadasvlas. Tu procesi dakavSirebulia sistemis temperaturis an agregatuli mdgomareobis SecvlasTan, mas s iT b u ri pr oce s i ewodeba. Ter*

Termodinamikuri parametrebi makroskopuli parametrebia, romlebic axasiaTebs sistemas,

rogorc mTlians. yoveli sistemis mdgomareoba ganisazRvreba misi Semadgeneli nawilakebis damaxasiaTebeli

fizikuri

sidideebis

(e.w.

mikroskopuli

parametrebis)

mniSvnelobebiT.

idealuri airis SemTxvevaSi aseTi mikroskopuli maxasiaTeblebia molekulebis koordinatebi da siCqareebi. rogorc wignis meore nawilis SesavalSia aRniSnuli, Termodinamika ganixilavs sxeuls rogorc mTlians, misi molekulur-kinetikuri aRnagobis gaTvaliswinebis gareSe. **

simZimis velSi moTavsebul airSi wonasworobis dros wnevis mniSvneloba wertilidan wertilamde icvleba.

229

modinamikur process, romelic mimdinareobs erT-erTi parametris mudmivobis pirobebSi, iz o pro ce s i ewodeba. aseTi izoprocesebi ganvixileT 57-e paragrafSi airis eqsperimentuli kanonebis Seswavlisas. sistemis a raw on a s w oru li m d gom ar e ob a ewodeba iseT mdgomareobas, rodesac Termodinamikuri parametrebis mniSvnelobebi mis sxvadasxva nawilSi sxvadasxvaa. sistemis gadasvla wonasworuli mdgomareobidan arawonasworulSi xdeba gare zemoqmedebiT. gare zemoqmedebis Sewyvetis Semdeg sistema garkveuli drois ganmavlobaSi aucileblad T a v i s T av a d gadava Termodinamikuri wonasworobis mdgomareobaSi. magaliTad, Tu dguSian WurWelSi moTavsebul airs swrafad SevkumSavT, sistema aRmoCndeba arawonaworul mdgomareobaSi _ dguSis mimdebare fenebSi airis wneva da simkvrive meti iqneba, vidre WurWlis danarCen nawilSi. magram garkveuli drois Semdeg simkvrive da wneva gaTanabrdeba, sistema aRmoCndeba Termodinakuri wonasworobis (axal) mdgomareobaSi. sistemis arawonasworuli mdgomareobidan wonasworulSi gadasvlis process r e laq s aci is p ro ce s i ewodeba, xolo am gadasvlisaTvis saWiro dros _ r e l aq s a ci is d ro ( ď ´ ). amrigad, wonasworuli mdgomareoba aris is mdgomareoba, romliskenac miiswrafvis yoveli Termodinamikuri sistema gare zemoqmedebis ararsebobis SemTxvevaSi. wonasworuli mdgomareobidan arawonasworulSi gadasvla xdeba mxolod gare zemoqmedebiT*, xolo arawonasworuli mdgomareobidan wonasworulSi gadasvla xdeba TavisTavad. Tu koordinatTa RerZebze gadavzomavT romelime or Termodinamikur parametrs, vTqvaT, wnevasa da moculobas, maSin sistemis wonasworuli mdgomareoba SeiZleba gamoisaxos wertiliT koordinatTa sibrtyeze. arawonasworuli mdgomareobis aseTnairad gamosaxva ar SeiZleba, vinaidan am dros erT-erT parametrs mainc ara aqvs garkveuli mniSvneloba. amrigad, grafikulad SeiZleba gamoisaxos mxolod wonasworuli procesi, e.i. iseTi procesi, romelic Sedgeba wonasworuli mdgomareobebis uwyveti Tanamimdevrobisagan**. *

aq ar viTvaliswinebT wonaworuli mdgomareobidan TviTnebur gadaxrebs _ fluqtuaciebs.

wonasworul process kidev uwodeben kvazistatikurs (warmosdgeba laTinuri sityvisagan

`kvazi~

TiTqos).

arawonasworuli

procesis

ganvixilavT.

230

grafikuli

gamosaxvis

meTods

Cven

cxadia, rom wonasworuli procesi garkveuli gaidealebaa. yvela realuri procesi arawonasworulia, e.i. Sedgeba arawonasworul mdgomareobaTa Tanamimdevrobisagan, vinaidan, rogorc zemoT aRvniSneT, sistemis erTi mdgomareobidan meoreSi gadasvlas aucileblad Tan sdevs wonasworobis darRveva. miuxedavad amisa, aseTi gaidealeba, iseve rogorc materialuri wertilis, idealuri airisa da sxv. cnebebis Semotana sasargebloa, vinaidan gvexmareba realuri procesebis garCevaSi da saSualebas gvaZlevs movnaxoT realuri arawonasworuli procesebis maqsimaluri miaxloebis gzebi idealur wonasworul procesebTan. rogorc qvemoT vnaxavT, idealuri siTburi manqanis mq koeficienti udidesia nebismieri realuri manqanis mq koeficientTan SedarebiT, romelic muSaobs saxureblisa da macivris temperaturebis imave mniSvnelobebze. procesis arawonasworulobis xarisxi damokidebulia misi mimdinareobis siswrafeze _ rac metia siswrafe, miT metia arawonasworuloba, e.i. miT metia gansxvaveba Termodinakur parametrebs Soris sistemis sxvadasxva nawilSi. marTlac, zemoaRniSnul cdaSi, airis swrafad SekumSvisas, gansxvaveba sxvadasxva nawilSi wnevebs, simkvriveebsa da temperaturebs Soris didia, neli SekumSvisas ki _ mcire. Tu SekumSvis procesi mimdinareobs Zalian nela, ise rom misi warmarTvisaTvis saWiro t dro gacilebiT metia relaqsaciis

ď ´

droze, maSin mTeli

procesi SeiZleba warmovidginoT rogorc didi raodenobis Zalian mcire SekumSvebis (mikroSekumSvebis) jami, amasTan, ise, rom TiToeuli mikroSekumSvisaTvis saWiro dro mainc meti iqneba

relaqsaciis

droze.

maSin,

TiToeuli

mikroSekumSvis

ganmavlobaSi

sistemaSi

moeswreba wonasworobis damyareba da mTeli procesi SeiZleba ganxilul iqnas rogorc erTi wonasworuli mdgomareobidan meoreze gadasvlebis erToblioba.

vinaidan wonasworuli procesi SeiZleba iyos mxolod usasrulod nela mimdinare procesi, amitom, airis gafarToebis an SekumSvis procesi rom wonasworuli iyos, saWiroa, misi wneva usasrulod mcired gansxvavdebodes gare wnevisagan (zRvarSi es gansxvaveba SeiZleba CaiTvalos nulis tolad). airis moculobis arawonasworuli cvlilebisas gansxvaveba airis wnevasa da gare wnevas Soris sasruli sididea _ gafarToebisas gare wneva sasruli sididiT naklebia airis wnevaze (amitom arawonasworuli gafarToebis wiri unda mdebareobdes ufro dabla, vidre wonasworulis), xolo kumSvisas ki piriqiT _ gare wneva sasruli sididiT sWarbobs airis wnevas. 71.1 suraTze mocemulia airis wonasworuli gafarToebis ( acb ) da kumSvis ( bc' a ) da arawonaswosur.71.1

ruli gafarToebisa ( adb ) da kumSvis ( bd' a ) grafike231

bi. arawonasworuli procesebis Sesabamisi grafikebi wyvetili wiriTaa warmodgenili, vinaidan, rogorc aRvniSneT, arawonasworuli procesis grafikuli gamosaxva ar SeiZleba da Tu mainc gamovsaxavT, mxolod im azriT, rom gviCvenos am procesis fardobiTi svla wonasworulTan SedarebiT.

§72. siTbo da muSaoba Termodinamikis erT-erTi ZiriTadi cneba aris Sinagani energiis cneba. 63-e paragrafSi Cven ganvmarteT Sinagani energia da gamovTvaleT misi sidide idealuri airisaTvis. sxeulis sruli energia aris misi meqanikuri da Sinagani energiebis jami. viciT, rom sxeulis (sistemis) meqanikuri energia icvleba muSaobis Sesrulebis procesSi, rom am cvlilebis raodenobrivi zoma aris Sesrulebuli muSaobis sidide (`muSaobis raodenoba~). Tu sxeulze moqmedi Zalis

muSaobas

A –Ti,

aRvniSnavT

xolo

sxeulis

meqanikuri

energiis

cvlilebas _ E -Ti, maSin E  A . (72.1) (72.1) formula gamosaxavs meqanikuri energiis mudmivobis kanons. vnaxoT,

procesebis

asrulebs

Sedegad

muSaobas

icvleba

xaxunis

sxeulis

Zalis

Sinagani

winaaRmdeg,

energia. igi

Tbeba,

sxeuli izrdeba

molekulebis saSualo kinetikuri energia da, maSasadame, sxeulis Sinagani energiac. imave SemTxvevasTan gvaqvs, rodesac gareSe sxeulebTan meqanikuri urTierTqmedebis

Sedegad

sxeuli

deformirdeba

(am

dros

icvleba

nawilakebis potenciuri energia) an icvleba misi agregatuli mdgomareoba (icvleba nawilakebis rogorc potenciuri, ise kinetikuri energia) da a.S. amrigad, Sinagani energiis cvlilebis erT-erTi gza aris muSaobis Sesruleba. Sesrulebuli muSaobis sidide warmoadgens Sinagani energiis cvlilebis zomas, amitom (72.1) formula, romelic gamosaxavs meqanikuri energiis mudmivobis kanons, marTebulia im SemTxvevaSic, rodesac igulisxmeba rom E aris sxeulis sruli energia (da es energia icvleba mocemuli sxeulis sxva sxeulebTan

meqanikuri

urTierTqmedebis

Sedegad,

muSaobis

Sesrulebis

procesSi). magram Sinagani energia sxva gziTac SeiZleba Seicvalos. Tu, magaliTad, mocemul A sxeuls SevaxebT B sxeuls, romlis temperaturac ufro maRalia, 232

maSin

A sxeulis temperatura da, maSasadame, Sinagani energiac gaizrdeba,

Tumca am dros araviTari muSaoba ar sruldeba. s xe u l is S i an g an i e ne r gi i s c v li le b i s p ro ce s s m u S a ob is Se s ru le b i s g a re Se T b o g a da c e m a ( T b o c v l a ) e w o de b a. Tbogadacemis dros erTi sxeulis nawilakebis energia gadaecema meore sxeulis nawilakebs, ganxilul magaliTSi es gadacema xdeboda molekuluri dajaxebebis Sedegad; Semxeb sazRvarze adgili aqvs A da B sxeulebis molekulebis dajaxebebs. swrafad moZravi molekulebi, Sejaxebisas gadascemen energiis nawils nela moZrav molekulebs, ris Sedegadac A sxeulis molekulebis saSualo kinetikuri energia izrdeba,

mcirdeba, e.i. adgili aqvs energiis gadacemas

sxeulis molekulebisa ki B

sxeulidan

sxeulSi.

amrigad, T b og a d ace m a ar is m ik r op r oce s e b i s e rT ob l iob a, r om e lT a m e Sv e ob i T a c e ne r g ia e rT i s xe u li d a n g ad ae ce m a m e ore s . Tbogadacemis ganxilul process, rodesac sxeulebi uSualod exebian erTmaneTs T b o g a m t a r o b a an konduqtiuri Tbogadacema ewodeba. sxeulebs Soris Tbocvlas maSinac aqvs adgili, rodesac isini uSualod ar exebian erTmaneTs. movaTavsoT A sxeuli B sxeulis zemoT garkveul manZilze, airis (an siTxis) fena, romelic exeba B sxeuls, gaTbeba (airis molekulebis B sxeulis molekulebTa dajaxebisa da energiebis gadanawilebis gamo) da imoZravebs zemoT (vinaidan misi simkvrive Semcirda), mis adgils ki daikavebs ufro civi fenebi. haeris Tbili fenebi A sxeulTan Sexebisas gamoiwvevs mis gaTbobas. T b o g a d a c e m a s ,

romelic

xorcielde ba

airis

(an siTxis) W a v l i s

me Sve obiT, konv eq ci uri Tbogadacema ewode ba.

B sxeulidan A sxeuls energia maSinac gadaecema, roca gamoricxulia konveqcia, magaliTad, roca A sxeuli moTavsebulia B sxeulis qvemoT garkveul manZilze. am dros konveqcia gamoricxulia, vinaidan Tbili haeris Wavli zeviTken moZraobs. energiis gadacema arc TbogamtarobiT xdeba, vinaidan airis, kerZod haeris Tbogamtaroba Zalian mcirea (Tanac energiis aseT gadacemas maSinac aqvs adgili, roca A da B sxeulebi vakuumSia moTavsebuli da gamoricxulia rogorc konveqciis, aseve Tbogamtarobis fizikuri SesaZlebloba). energiis gadacema am dros xdeba eleqtromagnituri talRebis meSveobiT. T b o g a d a c e m a s , r o m e l i c xorcielde ba

ele qtrom agn ituri

talRe biT,

ewode ba

Tbogadacem a

g amosxiv ebis saSuale biT .

Tbogadacemis procesSi sxeulis mier miRebuli an gacemuli energiis zomad miRebulia gansakuTrebuli sidide, romelsac s iT b os 233

r a ode no b a

ewodeba ( Q ). sxvanairad, siTbos raodenoba aris energiis is raodenoba, romlis gadacema xdeba TbocvliT. siTbos

raodenoba

SeiZleba

ganvixiloT

rogorc

raRac

saxis

energia, romelsac Seicavs sxeuli. imis Tqma, rom sxeuli Seicavs siTbos raodenobas, iseTive uazrobaa, rogorc imisa, rom sxeuli Seicavs muSaobis raodenobas.

siTbo

iseTive

dinamikuri

cnebaa,

rogoric

muSaoba

igi

vlindeba mxolod procesSi. iseve, rogorc muSaoba aris energiis gadacemis (da gardaqmnis) forma, xolo Sesrulebuli muSaobis sidide _ am gadacemis zoma, siTbo aris energiis gadacemis forma, xolo siTbos raodenoba _ am gadacemis zoma. magram muSaobisagan gansxvavebiT, romelic aris energiis gadacemis makroskopuli* forma, s iT b o a ri s e ne r gi is g a d ace m is m ik r o f i zi k u r i f orm a, e . i. is e T i fo rm a, r om lis d ros a c x or cie ld e b a e ne r gi i s

u Su al o

g acv l a

u rT ie rT q m e d i

s xe u le b is

q aos u r a d

m oZ r av n aw i l ak e b s Sor is . vinaidan mdgomareobas

muSaoba (Tvisebas),

siTbo

isini

axasiaTebs

warmoadgenen

process procesis

ara

funqciebs

sxeulis da

ara

mdgomareobisas. maTematikurad es imiT gamoisaxeba, rom muSaobisa da siTbos usasrulod mcire raodenoba srul diferencials ar warmoadgens da amitom maTi aRniSvna dA –Ti da dQ –Ti, sazogadod swori ar aris. d diferencialis aRniSvnaa mxolod usasrulod mcire nazrdisaTvis da amitom misi dawera SeiZleba mxolod iseTi sididis win, romelic sxeulis Tvisebas axasiaTebs, magaliTad,

energia,

moculoba,

temperatura

sxv.

cxadia

uazrobaa

vilaparakoT magaliTad muSaobis nazrdze. amrigad, sxeulis energiis cvlilebis zoma meqanikur procesSi aris gare Zalebis mier Sesrulebuli muSaoba, xolo siTbur procesSi _ siTbos raodenoba. amitom energiis mudmivobis kanoni siTburi procesebis gaTvaliswinebiT Semdegnairad Caiwereba:

E  A  Q .

(72.2)

terminiT `makroskopuli~ aq xazgasmulia is faqti, rom muSaoba yovelTvis dakavSirebulia

sxeulis an misi nawilebis makroskopul gadaadgilebasTan.

234

vianidan siTbos raodenoba aris energiis cvlilebis zoma, igi izomeba imave erTeulebiT, riTac energia da muSaoba. erTeulTa SI sistemaSi siTbos raodenoba izomeba joulebiT, CGS sistemaSi ki _ ergobiT. arsebobs siTbos raodenobis sistemgareSe erTeuli, romelsac kaloria ewodeba. kaloria aris siTbos raodenoba, romelic saWiroa 1 g. gamoxdili wylis gasaTbobad erTi gradusiT, kerZod, 19,5 oC-dan 20,5 oC-mde.

aTasjer met

erTeuls

gamoTvlebiT

kilokaloria

ewodeba.

cdebiT

Teoriuli

dadgenilia, rom 1 kaloria=4,1868 j. kaloriis

SemoReba

dakavSirebulia

istoriul

ganixilavdnen ara rogorc nawilakebis Sinagan moZraobas,

periodTan,

roca

siTbos

aramed rogorc nivTierebis

gansakuTrebul saxes. jer kidev galileim (1564-1642) gamoTqva mosazreba imis Sesaxeb, rom siTbo Taviseburi saxis nivTierebaa, romelic met-naklebi raodenobiT aris yvela sxeulSi da SeuZlia erTi sxeulidan meoreSi gadasvla. Semdgom am nivTierebas siTbombadi ewoda. siTbombadis Teoriis Tanaxmad, misi Sesvla sxeulSi iwvevs sxeulis temperaturis gazrdas da piriqiT. Tu sxeulSi sruliad ar aris siTbombadi, misi temperatura iqneba absoluturi nuli. vinaidan cdebi aCvenebda, rom temperaturis Secvla ar iwvevs sxeulis wonis Secvlas, iTvleboda, rom siTbombadi uwonoa. garda amisa, vinaidan siTbo miCneuli iyo nivTierebis raodenobad, iTvleboda, rom SeuZlebeli iyo misi gaqroba an warmoqmna SesaZlebeli iyo mxolod misi gadanawileba sxeulebs Soris. es Tvalsazrisi siTbos nivTieri bunebis Sesaxeb gabatonebuli iyo meTvramete saukunis ganmavlobaSi da mecxramete saukunis dasawyisSi. am droidan modis gamoTqmebi s i T b o s r a o d e n o b a , s i T b o t e v a d o b a , f a r u l i s i T b o d a a . S . siTbombadis Teoriis safuZvelze ganviTarda kalorimetruli gazomvebis meTodebi da ganisazRvra siTbos raodenoba rogorc sidide, romelic unda gadavceT an wavarTvaT sxeuls, rom misi temperatura gavzardoT an SevamciroT

T sididiT. dadginda, rom es siTbos

raodenoba, garda T –si, damokidebulia sxeulis m masaze da nivTierebis gvarobaze:

proporciulobis

koeficienti

(c )

Q  cmT .

(72.3)

gamosaxavs

Q –s damokidebulebas nivTierebis

gvarobaze da mas kuTri siTbotevadoba ewodeba. (70.3)-dan Cans, rom kuTri siTbotevadoba ricxobrivad im siTbos raodenobis tolia, romelic saWiroa erTeulovani masis sxeulis temperaturis Sesacvlelad erTi gradusiT. kaloriis ganmartebis Tanaxmad, wylis kuTri siTbotevadobaa 1 kal/K.g. sapirispiro Tvalsazrisi siTbos bunebis Sesaxeb gamoTqva galileis Tanamedrovem, ingliselma filosofosma f.bekonma (1561-1626), romelic Tvlida, rom siTbo sxeulSi myofi siTbombadi ki ara, sxeulis Semadgeneli mcire uxilavi nawilakebis Sinagani moZraobaa; rom sxeulis temperaturas gansazRvravs nawilakebis moZraobis siCqare da ara siTbombadis

235

raodenoba masSi. am Tvalsazriss miemxro bevri cnobili mecnieri (niutoni, boili da sxv.), maT Soris lomonosovic, romelmac Seqmna siTbos pirveli mecnieruli Teoria _ siTbos meqanikuri

Teoria.

TeoriaSi

lomonosovi

siTbos

ganixilavda,

rogorc

sxeulis

molekulebis Sinagan (meqanikur) moZraobas da asabuTebda absoluturi nulis arsebobas, rogorc

mdgomareobisa,

umravlesoba

mainc

rodesac

siTbombadis

Sewyvetilia Teorias

yovelgvari

emxroboda,

moZraoba.

vinaidan

igi

magram ufro

mkvlevarTa martivi

TvalsaCino iyo, siTbos meqanikuri Teoria ki eyrdnoboda nivTierebis molekulur-kinetikur aRnagobas, rac Znelad gasagebi da dasajerebeli iyo molekulebisa da atomebis Sesaxeb konkretuli cnobebis uqonlobis gamo. TviT sadi karnoc ki, romelic Termodinamikis erTerTi fuZemdebelia, Tavdapirvelad siTbombadis Teorias iziarebda da mxolod SemdegSi miemxro siTbos meqanikur Teorias. siTbos meqanikuri Teoriis gamarjvebaSi gadamwyveti roli iTamaSa XVIII saukunis bolos aRmoCenilma movlenebma, romlebic ewinaaRmdegebodnen siTbos raodenobis mudmivobis kanons. 1798 wels ingliselma mecnierma rumfordma (1753-1814) SeamCnia, rom qvemexis lulis burRvisas gamoiyofoda didi siTbos raodenoba, romelic iwvevda lulaze dadgmul qvabSi wylis aduRebas. ris xarjze gamoiyo siTbo? siTbombadis Teoriis momxreebi cdilobdnen es movlena aexsnaT daSvebiT, rom rkinis burbuSelas siTbotevadoba ufro naklebia, vidre masiuri rkinisa da siTbotevadobaTa am sxvaobis xarjze xdeba siTbos gamoyofa. magram gazomvebiT dadginda, rom rkinis burbuSelas siTbotevadoba zustad iseTivea, rogorc masiuri rkinisa. 1899 wels ingliselma mecnierma devim (1778-1829) aRmoaCina, rom Tbogaumtar WurWelSi moTavsebuli yinulis ori naWris erTmaneTze xaxunis Sedegad gamoiyofa siTbo, ris Sedegadac yinulis naWrebi dneba. siTbombadis TeoriiT am movlenis axsna SeuZlebeli iyo, vinaidan wylis siTbotevadoba ara Tu naklebi, aramed TiTqmis orjer metia yinulis siTbotevadobaze. cxadia, rom orive SemTxvevaSi siTbos raodenobis gamoyofa dakavSirebulia muSaobis SesrulebasTan, siTbo gamoiyofa Sesrulebuli muSaobis xarjze. amrigad, siTbos raodenoba aris sidide, romelic garkveuli gagebiT muSaobis msgavsia. sxeulis temperaturis erTnairi sididiT aweva SeiZleba gamoiwvios rogorc raime

Q siTbos raodenobis gadacemam, ise

garkveuli A muSaobis Sesrulebam. siTbombadis Teoria sabolood iqna uaryofili siTbos raodenobisa da muSaobis sididis ekvivalentobis dadgenis Semdeg. XIX saukunis Sua wlebSi maierma Teoriulad, xolo joulma eqsperimentulad daadgines, rom 4,1868 j m u S a o b a e k v i v a l e n t u r i a 1 k a l o r i a siTbos raoden obisa, e .i. 4,18 68 j m uSaobis Se srule ba iwvev s tempe raturi s iseTsave

m ate bas,

rog orsac

sxeulisaTv is

k alori a

siTbos

raodenobis

m i n i W e b a . amrigad, kaloria energiis siTburi erTeulia, da sxva araferi. ricxvi 4,1868 j/kal aris siTburi erTeulebidan meqanikurSi gadasayvani koeficienti da mas ewodeba s i T b o s meq anikuri ekviv ale nti .

236

§73. Termodinamikis I kanoni wina paragrafSi miRebuli (72.2) formula, romelic gamosaxavs

energiis

mudmivobis kanons siTburi procesebis gaTvaliswinebiT, gamosaxavs Termodinamikis I kanons. magram TermodinamikaSi, rogorc wesi, ganixileba uZravi sistemebi, romelTa meqanikuri energia ar icvleba, amitom (72.2) formulaSi, Cveulebriv, sistemis srul E energias cvlian misi Sinagani U energiiT: (73.1) U  A  Q , sadac U aris sxeulis Sinagani energiis cvlileba, A aris gare Zalebis mier Sesrulebuli muSaoba, xolo Q _ sxeulze gadacemuli (sxeulis mier miRebuli an gacemuli) siTbos raodenoba. niutonis III kanonis Tanaxmad A  A' , sadac A' aris TviT sxeulis mier Sesrulebuli muSaoba, e.i. im Zalebis muSaoba, romliTac mocemuli A

sxeuli moqmedebs gare sxeulebze. Tu (73.1) formulaSi gare Zalebis muSaobis nacvlad CavsvamT TviT sxeulis A' muSaobas, miviRebT: da

U  Q  A'

(73.2)

Q  U  A' .

(73.3)

am saxiT Cawerili Termodinamikis I kanoni sityvierad ase gamoiTqmis: s xe u lze g ad a ce m u l i s iT b o s rao de nob a xm ar de b a am s xe u li s Si n ag a n i e ne rg ii s Se cv l as d a m is m ie r m u S aob is Se s ru le b as .

Q dadebiTia, Tu sxeuli iRebs siTbos raodenobas da uaryofiTia, Tu igi gascems mas. aseve, A' dadebiTia Tu muSaobas asrulebs TviTon sxeuli da uaryofiTia, Tu muSaobas asrulebs gare sxeulebi. Tu (73.3) formulas CavwerT Tbogadacemis elementaruli procesisaTvis, anu usasrulod mcire sidideebiT, miviRebT:

 Q  dU   A ,

(73.4) sadac  Q sistemaze gadacemuli usasrulod mcire siTbos raodenobaa, dU sistemis Sinagani energiis usasrulod mcire cvlileba, xolo  A sistemis mier Sesrulebuli usasrulod mcire muSaoba. am CanawerSi gaTvaliswinebulia, rom Sinagani energia aris mdgomareobis funqcia da misi cvlileba warmoadgens srul diferencials, siTbos raodenoba da Sesrulebuli muSaoba ki procesis funqciebia da  Q da  A ar warmoadgens srul diferencialebs. 237

cxadia,

rom

(73.3)

(73.4)

gantolebebSi

Semavali

samive

sidide

gazomilia erTi da imave erTeulebiT, kerZod SI sistemaSi joulobiT, xolo CGS sistemaSi ergobiT. istoriulad

Termodinamikis

kanonis

Camoyalibeba

dakavSirebulia

iseTi manqanis Seqmnis cdebis warumateblobasTan, romlis moqmedebis erTaderTi Sedegi iqneboda muSaobis Sesruleba. aseT manqanas, romelic Seasrulebda muSaobas garedan siTbos raodenobis miuReblad da Sinagani energiis Seucvlelad, I gvaris pe rpe tu u m m ob il e ewodeba (laTonurad perpetuum mobile mudmivad moZravs

niSnavs).

Termodinamikis

I kanonidan gamomdinare

aseTi

manqanis arseboba SeuZlebelia. marTlac, vinaidan energia mdgomareobis funqciaa, sistemis sawyis mdgomareobaSi dabrunebisas energia iRebs pirvandel mniSvnelobas. amitom periodulad moqmedi manqanisaTvis yoveli periodis bolos dU  0 da manqanis nulisagan gansxvavebuli muSaobis

 A  0

Sesru-

leba SeuZlia mxolod garedan miwodebuli energiis xarjze

 Q  0 .

amitom

Termodinamikis I kanons zogjer asec ayalibeben: S e u Z le b e li a pi rv e l i gv ar is pe r pe tu u m m o b il e s g a nx or c i e leba.

§74. Termodinamikis I kanoni idealuri airisaTvis davweroT Termodinamikis I kanoni idealuri airisaTvis. viciT (§64), rom airis muSaoba izobaruli gafarToebisas sadac

V

aris

airis

A  pV , moculobis namati.

airis

sawyis

(74.1) moculobas

aRvniSnavT V1 –iT, saboloss _ V2 -Ti, maSin

V  V2  V1 da

A  pV2  V1  . (74.2) abscisaTa RerZze gadavzomoT moculoba, ordinatTa RerZze _ wneva.

vianidan ( p,V ) koordinatebiT gamosaxuli izobara aris abscisaTa RerZis paraleluri wrfe, amitom airis muSaoba ricxobrivad tolia 74.1 a suraTze gamosaxuli daStrixuli farTobisa. 238

Tu procesi ar aris izobaruli, maSin (74.1) formula marTebulia mxolod moculobis usasrulo mcire dV

cvlilebisaTvis, roca dasaSvebia

wnevis cvlilebis ugulebelyofa:

 A  pdV .

(74.3) (74.3) formuliT gamosaxuli elementaruli muSaoba ricxobrivad tolia 74.1 b suraTze xSirad daStrixuli svetis farTobisa. sasruli muSaoba, romlis drosac moculoba icvleba V1 –dan V2 –mde, miiReba (74.3) gantolebis integrebiT

A

 pdV ,

rac gamoisaxeba 74.1 b suraTze mocemuli daStrixuli farTobiT. p

p2 p1

p1 pdV

V2 a

b suraTi 74.1.

amrigad, nebismieri procesis dros Sesrulebuli muSaoba gamoisaxeba im nakvTis farTobiT, romelic Semosazrvrulia procesis

p  pV 

grafikiT,

abscisaTa RerZiTa da sawyisi da saboloo ordinatebiT (sur. 74.1 a,b). 1 moli idealuri airis Sinagani energia (§63)

i RT  CV T , 2 aris moluri siTbotevadoba mudmivi moculobis dros. CV  const U 

sadac CV

idealuri airisaTvis , amitom dU   CV dT ,

(74.4)

Tu  A –s da dU  –s miRebul mniSvnelobebs SevitanT (73.4) formulaSi, Termodinamikis I kanoni 1 moli idealuri airisaTvis miiRebs saxes:

 Q  CV dT  pdV .

239

(74.5)

gavaanalizoT miRebuli formula. davuSvaT, airi farTovdeba pdV

muSaoba

dadebiTia.

muSaobis

miwodebuli siTbos raodenobis xarjze xarjze

dV



C

V

Sesruleba

dQ  0 ,

airs

dV



 0, maSin

SeuZlia

garedan

an Sinagani energiis klebis

dT  0 , an orive procesis gamo erTdroulad. Tu airi ikumSeba

 0, Sesrulebuli muSaoba

siTbos gare sxeulebs

dQ  0 ,

pdV

uaryofiTia. am dros airi gadascems

an izrdeba misi Sinagani energia CV dT  0 , an

orive procesi erTdroulad mimdinareobs. izoqoruli procesis dros pdV  0 da

 Q  CV dT , e.i. izoqoruli procesis dros gadacemuli siTbos raodenoba mTlianad xmardeba airis temperaturis (Sinagani energiis) gazrdas. izoTermuli procesis dros dT  0 da

 Q  pdV , e.i. izoTermuli procesis dros gadacemuli siTbos raodenoba mTlianad xmardeba muSaobis Sesrulebas.

§75. adiabaturi procesi. puasonis gantoleba a d i ab atu ri pro ce s i e w o de b a is e T pro ce s s , rom e l ic m im d i n a r e ob s T b o cv li s ga re Se m oce m u l s i s te m as a d a ga re m om cv e l s xe u l e b s So ri s , e.i. am dros sistema (airi) ar iRebs siTbos raodenobas garedan da arc gascems mas  Q  0 . procesi rom adiabaturi iyos, airi unda movaTavsoT absoluturad Tbogaumtari masalisagan damzadebul WurWelSi, e.w. a di ab at u r ga r s S i * . magram absoluturad Tbogaumtari masala bunebaSi ar arsebobs, amitom procesi rom adiabaturi iyos, igi unda warimarTos Zalian swrafad, vinaidan swrafi procesis dros sistema ver aswrebs siTbos raodenobis miRebas an gacemas da adiabaturobis piroba sruldeba.

aseT WurWels yvelaze ufro emsgavseba e.w. d i u a r i s WurWeli, romelic ormagkedliani

minis WurWelia movercxlili kedlebiT, romelTa Soris Seqmnilia maRali vakuumi.

240

amrigad, p r a q t ik u l a d y ov e l i m e ta d s w r af i p roc e s i a di ab at u r i p r oc e s i a. SevadaroT erTmaneTs adiabaturi da izoTermuli procesebi. davuSvaT, airi farTovdeba. am dros igi asrulebs dadebiT muSaobas. es muSaoba SeiZleba Sesruldes garedan siTbos raodenobis miwodebis xarjze, an Sinagani energiis Semcirebis xarjze, an orive procesis gamo erTdroulad. Sinagani energia (e.i. temperatura) rom ar Semcirdes da, maSasadame, procesi iyos izoTermuli, miwodeba

uzrunvelyofili

garedan.

amisaTvis

unda airi

iyos

siTbos

moTavsebuli

raodenobis unda

iyos

idealuri idealurad

Tbogamtari masalisagan damzadebul WurWelSi. magram aseTi masala bunebaSi ar

arsebobs.

amitom

g a f arT oe b is as

a i ris

t e m p e r atu r a

r om

S e m c ir de s , pr oce s i u nd a w a rim a rT os im de n a d n e l a , rom moeswros yovel momentSi garedan siTbos saWiro raodenobis miReba. analogiurad, airis kumSvisas rom misi temperatura ar gaizardos, e.i. procesi iyos izoTermuli, man unda moaswros yovel momentSi zedmeti siTbos raodenobis gadacema, rac realurad SesaZlebeli iqneba mxolod maSin, Tu procesi mimdinareobs Zalian nela. Catarebuli msjelobidan cxadia, rom adiabaturi gafarToebisas airis Sinagani energia unda Semcirdes, e.i. igi unda gacivdes da piriqiT _ adiabaturi kumSvisas airi unda gaTbes. es Sedegi uSualod gamomdinareobs Termodinamikis I kanonidan. marTlac, vinaidan adiabaturi procesis dros  Q  0 , Termodinamikis I kanoni 1 moli idealuri airisaTvis miiRebs saxes: CV dT  pdV  0 .

(75.1)

es formula gviCvenebs, rom Tu airi adiabaturad farTovdeba igi civdeba

dT  0

da piriqiT _ adiabaturi SekumSvisas

dV



dV  0, maSin  0 airi Tbeba 

dT  0 . airis temperaturis cvlileba adiabaturi procesis dros farTod gamoiyeneba teqnikaSi: magaliTad, Tu gavxsniT balons, romelSic moTavsebulia naxSirorJangi, romlis wneva

p  40 atmosferos, airi adiabaturad gafarTovdeba. am dros misi temperatura Semcirdeba - 80 o C -mde

airis

nawili

gardaiqmneba

Tovlis

SegrovebiT da dawnexviT miiReba e.w. `mSrali yinuli~.

241

msgavs

kristalur

masad.

masis

dizelis tipis Sigawvis ZravaSi haeri, adiabaturad SekumSvis gamo xurdeba 600 C – mde, rac iqvevs masSi Sefrqveuli Txevadi sawvavis TviTaalebas. es ki saSualebas iZleva damzaddes am tipis Zrava anTebis sistemis gareSe.

amrigad, adiabaturi procesis dros airis temperatura damokidebulia moculobaze. gamovsaxoT es damokidebuleba maTematikurad. amisaTvis ganvsazRvroT wneva klapeiron-mendeleevis gantolebidan

 pV



 RT  da SevitanoT

(75.1) formulaSi:

CV dT  anu

RT dV  0 , V

dT R dV   0. T CV V

(75.2)

maieris gantolebis Tanaxmad R  C p  CV , amitom

C p  CV C p R    1    1. CV CV CV maSin (75.2) gantoleba iRebs saxes dV dT    1  0, T V romlis integrebiT miviRebT: ln T    1ln V  const , anu ln TV 1  const . Tu raime sididis logariTmi mudmivia, maSin TviT es sididec, cxadia, mudmivi iqneba. amitom sabolood miviRebT: TV 1  const .

(75.3)

an, sazogadod, TV γ 1  const . (75.3) gantoleba gamosaxavs damokidebulebas temperaturasa da moculobas Soris adiabaturi procesis dros. igi SeiZleba Semdegnairad CavweroT: T1V1 1  T2V21 . (75.3’) davamyaroT axla damokidebuleba wnevasa da moculobas Soris adiaba-

turi procesis dros. amisaTvis (75.3) gantolebaSi SevitanoT T –s mniSvneloba pV  klapeiron-mendeleevis gantolebidan  T  R 

242

  miviRebT: 

pV

 V 1  const ,

R anu R

pV  const  R . airis universaluri mudmivaa. misi namravli mudmiv sidideze agereTve

mudmivi inqbea, amitom saboloos pV  const .

(75.4)

an, sazogadod, pV γ  const . (75.4) gantoleba, romelic amyarebs damokidebulebas idealuri airis wnevasa da moculobas Soris adiabaturi procesis dros, aris a di ab ati s gantoleba. am gantolebis grafiks ad i ab at a ewodeba. adiabatis gantoleba pirvelad miiRo frangma mecnierma p u as onm a (1781-1840), amitom mas puasonis gantolebas uwodeben. adiabatis

pV γ  const

gantolebis

Sedareba

izoTermis pV  const gantolebasTan gviCvenebs, rom (radgan

  1)

moculobis

mocemuli

cvlilebisas

wnevis cvlileba adiabaturi procesis dros ufro swrafia, vidre izoTermuli procesis dros (marsur. 75.1

Tlac, Tu izoTermuli procesis dros moculobis orjer

gazrda

iwvevs

wnevis

orjer

Semcirebas,

adiabaturi procesis dros wneva 2  –jer unda Semcirdes, rac 2-ze metia).

gasagebicaa, vianidan izoTermuli procesis dros airis wnevis cvlilebas iwvevs mxolod misi moculobis cvlileba, adiabaturi procesis dros ki wnevis

cvlileba gamowveulia airis

rogorc

moculobis,

ise

temperaturis

cvlilebiT. adiabaturi gafarToebisas airi civdeba da wneva ufro swrafad ecema, adiabturi kumSvisas ki piriqiT _ Tbeba da wneva ufro swrafad izrdeba. maSasadame, adiabata ufro metad unda iyos daxrili abscisaTa RerZisadmi, vidre izoTerma. 75.1 suraTze 1-2 mrudi izoTermaa, 1-2’ mrudi ki _ adiabata.

vinaidan

nebismieri

procesis

dros

Sesrulebuli

muSaoba

ricxobrivad tolia im farTobisa, romelic SemosazRvrulia am procesis gamomsaxveli mrudiT, abscisaTa RerZiT da sawyisi da saboloo ordinatebiT, amitom cxadia, rom izoTermuli gafarToebisas airi ufro met muSaobas Seasrulebs, vidre adiabaturi gafarToebisas. 243

§76. politropuli procesi Cven mier

ganxiluli airis

mdgomareobis

Secvlis

izoTermuli

adiabaturi procesebi gaidealebuli procesebia, vinaidan pirvelis ganxorcielebisaTvis saWiroa airis idealuri siTburi kontaqti garemosTan. meoresaTvis ki _ aseTive idealuri izolacia. orive es procesi zRvruli SemTxvevaa ufro zogadi, e.w. po li tro pu li pr o ce s i s a . p o li tro pu li p ro c e s i e w o de b a y ov e lgv a r pro ce s s , ro m l is m im di na re ob is d ros a c s i T C

b ote v a dob a

Q dT

m u dm i v i s i d ide a.

gamoviyvanoT politropuli procesis gantoleba _ politropis gantoleba; es gamoyvana savsebiT analogiuria adiabatis gantolebis gamoyvanisa. Termodinamikis I kanonis Tanaxmad sistemaze gadacemuli siTbos raodenoba

 Q  CV dT  pdV , magram politropuli procesisaTvis  Q  C  dT , amitom C  dT  CV dT  pdV , anu CV  C dT  pdV  0 . gamovricxoT am gantolebidan

(76.1) dT . amisaTvis visargebloT klapeiron-

mendeleevis gantolebiT pV  RT , romlis gadiferencialeba gvaZlevs: pdV  V dp  RdT ,

saidanac maieris C p  CV  R gantolebis gamoyenebiT miviRebT:

dT 

pdV  V dp



. R C p  CV dT -s miRebuli mniSvneloba SevitanoT (74.1) gantolebaSi:

C

V

anu

 C 

pdV  V dp C p  CV

 pdV  0 ,

CVμ  C μ pdVμ  CVμ  C μ Vμ dp  C pμ  CVμ pdVμ  0 ,

aqedan miRebuli tolobis

C pμ  C μ pdVμ  CVμ  C μ Vμ dp  0 . yvela wevri gavyoT C  C  pV –ze: V

244



C p  C  dV dp    0. CV  C  V p aRvniSnoT

C p  C  CV  C 

 n,

(76.2)

maSin

dV

dp (76.3)  0. V p n -s ewodeba politropis maCvenebeli. mocemuli procesisaTvis igi mudmivia, n



vinaidan, ganmartebis Tanaxmad, C  const . amis gamo (76.3) gantolebis integreba gvaZlevs: n ln V  ln p  const ,

anu ln pVn  const ,

e.i. pVn  const ,

an, sazogadod,

pV n  const . (76.4) gantolebas

(76.4)

ewodeba politropis gantoleba.

adiabaturi procesis dros C   0 , (vinaidan  Q  0 ), amitom

C p

 , CV da (76.4) gantoleba gardaiqmneba adiabatis gantolebad. n

izoTermuli procesis dros

C    , (vinaidan dT  0 ), amitom C p

1 C n   1, CV CV  C  1 C da (76.4) gantoleba gardaiqmneba izoTermis gantolebad. C p  C 

Tu davuSvebT, rom C   C p

politropis

maCvenebeli xdeba nulis

toli da (76.4) gantolebidan, rogorc unda yofiliyo, viRebT p  const .

1   Tu C   CV , maSin n   da (76.4) gantolebidan  misi xarisxSi ayvanis Semdeg n   viRebT

V  const .

adiabaturi

procesis

245

msgavsad,

politropuli

procesic

SeiZleba

Caiweros

T ,V ,

parametrebiT.

amisaTvis

klapeiron-mendeleevis

gantolebidan ganvsazRvroT p da SevitanoT (76.4) formulaSi:

m RT  V n  const ,  V anu

TV n1  const .

§77. muSaoba sxvadasxva Termodinamikur procesSi viciT, rom gafarToebisas airis mier Sesrulebuli elementaruli muSaoba

 A  pdV , xolo sruli muSaoba

A  δ A 

 pdV .

(77.1)

gamovTvaloT

A –s

sidide

moli

idealuri

airisaTvis

sxvadasxva

Termodinamikur procesSi. 1. i zoq oru li pro ce s i . am dros V  const ,

amitom dV  0 da

V 2

 pdV  0 ,

A

V 1

e.i. izoqoruli procesis dros airi muSaobas ar asrulebs, am procesis ganxorcielebisaTvis airi unda movaTavsoT daxurul WurWelSi ise, rom mas gafarToebis

 Q  C

v

saSualeba

hqondes.

Termodinamikis

kanonis

Tanaxmad

dT  pdV  am dros gadacemuli siTbos raodenoba mTlianad xmardeba

airis Siangani energiis gazrdas. 2. i zob ar u l i pr oce s i , p  const V 2

A

 pdV .

V 1

vinaidan wneva mudmivia, igi SeiZleba gavitanoT integralis niSnis gareT, amitom miviRebT: V 2

A  p  dV  pV 2  V1  . V 1

an, sazogadod,

A  pV2  V1  . 246

(77.2)

izobaruli

procesis

gansaxorcieleblad

airi

unda

movaTavsoT

idealurad moZravi dguSiT daxurul WurWelSi. Termodinamikis I kanonis Tanaxmad am dros airze gadacemuli siTbos nawili xmardeba muSaobis Sesrulebas (dguSis gadaadgilebas), nawili ki misi Sinagani energiis gazrdas. 3 . i zo T e rm u li pro ce s i , T  const V 2

A

 pdV .

V 1

am procesis dros wneva ganuwyvetliv icvleba, amitom misi gatana integralis niSnis gareT ar SeiZleba.

SevcvaloT

misi

toli

sididiT

 RT  klapeiron-mendeleevis gantolebidan  p  .   V    maSin sur. 77.1

V 2



A

V 1

dV V

V 2



 RT

V 1

dV V

 RT ln

V 2 V1

(77.3)

zogad SemTxvevaSi, roca farTovdeba nebismieri m masis airi

V2 .  V1 grafikulad es muSaoba gamoixateba 77.1 suraTze mocemuli daStrixuli A

RT ln

farTobiT, sadac ab procesis grafikia (izoTerma). izoTermuli procesis gansaxorcieleblad airi unda movaTavsoT idealurad Tbogamtar WurWelSi, an, vinaidan iseTi WurWeli ar arsebobs, igi unda warvmarToT Zalian nela. vinaidan izoTermuli procesis dros T  const da CV dT  0 , Termodinamikis I kanonis Tanaxmad gadacemuli siTbos raodenoba mTlianad xmardeba muSaobis Sesrulebas. 4. a d i ab atu ri p ro ce s i ,  Q  0 V 2

 pdV .

A

V 1

radgan am procesis dros airi garedan siTbos raodenobas ar iRebs, amitom Termodinamikis I kanonis Tanaxmad igi muSaobas asrulebs Sinagani energiis xarjze 247

O  CV dT  pdV , saidanac pdV  CV dT . pdV -s miRebuli mniSvneloba SevitanoT muSaobis formulaSi T2

A    CV dT  CV  dT  CV T1  T2  , anu  T  A  CV T1 1  2  .  T1  magram adiabaturi procesis dros kavSiri moculobasa da

(77.4)

temperaturas Soris (75.3) formulis Tanaxmad aseTia: T1V1 1  T2V21 ,

saidanac T2  V1   T1  V 2  garda amisa, maieris gantolebis Tanaxmad,

 1

(77.5)

 C p  R  C p  CV  CV   1  CV   1 , C   V  saidanac

CV  Tu

R .  1

(77.6)

T2 -is da CV –s mniSvnelobebs (77.5) da (77.6) formulidan SevitanT (77.4) T1

formulaSi, gveqneba:  1 RT1   V1   . A 1   1   V 2     zogad SemTxvevaSi, roca farTovdeba nebismieri m masis airi

(77.7)

 1 m RT1   V1   1     , A    1   V2    

an, Tu SevcvliT

RT1 –s misi toli



p1V1

sididiT klaperon_mendeleevis

gantolebidan, miviRebT: p V  V A  1 1 1   1   1   V2 

248

  

 1

 . 

(77.8)

p izoTerma

adiabata

izoTerma

b V1

suraTi 77.2. grafikulad

gamoisaxeba

77.2 a

suraTze

mocemuli

adStrixuli

nakvTis farTobiT, sadac ab adiabataa. 77.2 b da 77..2 g suraTze gamosaxulia Sesabamisad izoTermuli da adiabaturi gafarToebisa da kumSvis grafikebi, saidanac Cans, rom adiabaturi gafarToebisas airi nakleb muSaobas asrulebs, vidre izoTermuli gafarToebisas, xolo adiabaturi kumSvisas gare Zaleba ufro met muSaobas asruleben, vidre izoTermuli kumSvisas.

ยง 78. wriuli procesi (cikli). siTburi manqanis mq koeficienti istoriulad Termodinamika viTardeboda teqnikis moTxovnebis safuZvelze, rac iTvaliswinebda siTburi manqanis, gansakuTrebiT orTqlis manqanis, srulyofas, anu margi qmedebis koeficientis gazrdas. miuxedavad konstruqciuli Taviseburebebisa, yovel siTbur manqanaSi, gadaiWreba erTi da igive amocana _ Sinagani energia gardaiqmneba meqanikur muSaobad.

amisaTvis sawvavis

wvis an birTvuli reaqciis Sedegad gamoyofili energia Tbocvlis meSveobiT gadaecema airs (m u S a s xe u ls ),

romelic farTovdeba da asrulebs muSaobas

gare Zalebis winaaRmdeg, mohyavs ra moZraobaSi esa Tu is meqanizmi. cxadia, rom siTbur manqanaSi, misi sasruli zomebis gamo, airs ar SeuZlia usasrulod gafarToeba. maSasadame, gafarToebis Semdeg airi unda SevkumSoT ise, rom igi da manqanis yoveli detali daubrundes sawyis mdgomareobas, ris Semdegac es airi kvlav gafarTovdeba da a.S. amrigad, mivdivarT daskvnamde, rom yoveli siTburi manqana unda muSaobdes periodulad, anu, rogorc am SemTxvevaSi amboben, c ik lu r ad .

TiToeuli ciklis ganmavlobaSi muSa sxeulis

gafarToebas unda mohyves misi SekumSva sawyis mdgomareobamde. 249

maSasadame, yovel siTbur manqanaSi muSa sxeuli asrulebs cikls, anu w r iu l p roc e s s , e.i. iseT process, romlis Sesrulebis Semdegac igi ubrundeba sawyis mdgomareobas, amitom, sxvadasxva wriuli procesis Seswavlas didi praqtikuli da Teoriuli mniSvneloba aqvs. realuri

siTburi

manqana,

Cveulebriv,

muSaobs

e.w.

g ax sni li

cikliT ,

roca

gafarToebis Semdeg `namuSevari~ airi gareT gamoityorcneba da ikumSeba airis axali porcia. magram es arsebiTad ar moqmedebs procesis Termodinamikaze, amitom Cven ganvixilavT daxSul cikls, roca farTovdeba da ikumSeba airis erTi da igive porcia.

cxadia, rom grafikulad wriuli procesis Sekruli mrudiT gamoisaxeba. marTlac,

davuSvaT,

rom

airi

mdgomareobidan

(romlis

parametrebia

p1 , V1 , T1 ) gafarToebis Sedegad gadavida b mdgomareobaSi romelic

( p2 , V2 , T2 ) acb

mrudiT

procesis

meSveobiT,

gamoisaxeba

(sur. 76.1),

Semdeg ki SekumSvis Sedegad dabrunda sawyis mdgomareobaSi procesiT.

bc' a

acbc' a

mrudiT

Sekruli

mrudi

gamosaxuli gamosaxavs

wriul process, anu cikls. gamovTvaloT Sesrulebuli V1

sur. 78.1

wriuli

muSaoba.

procesis

dros

gafarToebisas

airis

mier Sesrulebuli muSaoba A1 dadebiTia da igi

V1acbV2 farTobiT gamoisaxeba (§ 74). A1  farT. V1acbV2 .

kumSvisas airis mier Sesrulebuli muSaoba uaryofiTia (am dros dadebiTia gare Zalebis mier Sesrulebuli muSaoba) da igi V1ac'bV2 farTobiT gamoisaxeba A2  farT. V1ac' bV2 ,

amitom ciklis dros Sesrulebuli sruli muSaoba A  A1  A2  farT. V1acbV2 _farT. V1ac' bV2  farT. acbc' a .

maSasadame, w r iu l i pr oce s i s dros S e s r u le b u li m u S aob a am p r oc e s is g am om s a xv e l i m ru diT Se m o s az R v ru li f a rT ob iT g am ois a xe b a. yoveli

siTburi

manqana

Sedgeba

sami

ZiriTadi

nawilisagan:

muSa

sxeulisagan (airi an orTqli, romelic asrulebs wriul process), saxureb250

lisa (romlisganac muSa sxeuli iRebs siTbos raodenobas gafarToebisas) da macivrisgan (romelsac muSa sxeuli gadascems siTbos raodenobas kumSvisas). gamovTvaloT siTburi manqanis margi qmedebis koeficienti. saxureblidan muSa sxeuli iRebs raRac Q1 siTbos raodenobas, ris Sedegadac farTovdeba da asrulebs dadebiT A1 A muSaobas ( acb procesi). Semdeg iwyeba muSa sxeulis SekumSva. am dros igi asrulebs uaryofiT A2 muSaobas da gadascems macivars

Q2 siTbos raodenobas (e.i. iRebs macivridan _ Q2 siTbos raodenobas, bc' a procesi). aRvniSnoT muSa sxeulis Sinagani energia a mdgomareobaSi U 1 –iT, xolo b mdgomareobaSi U 2 –Ti. maSin Termodinamikis I kanonis Tanaxmad acb da bc' a

procesebisaTvis Sesabamisad daiwereba:

Q1  U 2  U1  A1 ,  Q2  U1  U 2  A2 . wevr-wevrad Sekrebis Semdeg miviRebT: Q1  Q2  A1  A2  A . amrigad erTi ciklis ganmavlobaSi moxda Semdegi: muSa sxeulma saxureblisagan miiRo Q1 siTbos raodenoba, misi nawili _ Q2 siTbos raodenoba gadasca macivars, xolo danarCeni Q1  Q2 siTbos raodenobis xarjze Seasrula sasargeblo A muSaoba. s iT b u ri m a nq an is m ie r Se s ru le b u li m u S a ob i s fa r dob as s a x u r e b li s g a n m i R e b u l Q1 s i T b os r a od e n ob a s T an e w o de b a s i T b u r i m anq a ni s m ar g i q m e de b is k oe f i c ie n t i (m q k oe f ici e n ti ):

Q A Q1  Q2 (78.1)   1 2 . Q1 Q1 Q1 (78.1) formulidan Cans, rom siTburi manqanis mq koeficienti miT metia, rac



metia muSa sxeulis mier saxuleblisagan miRebuli siTbos raodenoba da rac mcirea mis mier macivarze gadacemuli siTbos raodenoba. Cveni msjelobis safuZvelze (§71, sur. 71.1) cxadia, rom ciklis dros Sesrulebuli muSaoba da, maSasadame siTburi manqanis mq koeficienti maSin aris maqsimaluri, rodesac wriuli procesi wonasworulia (kvazistatikuria). ismis kiTxva: ratom mTlianad ar gardaiqmneba siTbur manqanaSi muSa sxeulis mier miRebuli siTbos raodenoba meqanikur muSaobad, ratom aris aucilebeli 251

macivrisaTvis siTbos gadacema? saqme isaa, rom siTbur manqanaSi muSa sxeuli asrulebs wriul process* da maSasadame, igi, saxureblisgan miRebuli siTbos raodenobis xarjze gafarToebisa da muSaobis Sesrulebis Semdeg, kvlav unda SevkumSoT da davabrunoT sawyis mdgomareobaSi, e.i. masze unda SevasruloT muSaoba. am muSaobas asrulebs gare Zalebi. cxadia, rom es muSaoba naklebi unda iyos gafarToebis muSaobaze, winaaRmdeg SemTxvevaSi manqana dadebiT muSaobas ver Seasrulebs da ciklis warmoebas azri ar eqneba. imisaTvis ki, rom kumSvis dros Sesrulebuli muSaoba naklebi iyos gafarToebis muSaobaze, kumSvis mrudi pV diagramaze gafarToebis mrudze qvemoT unda mdebareobdes. magram, diagramaze ufro qvemoT mdebare mruds Seesabameba ufro dabali temperatura. maSasadame, muSa sxeuli SekumSvis dawyebamde unda gavacivoT, _ unda wavarTvaT garkveuli siTbos raodenoba da gadavceT macivars **. amitomaa, rom verc erTi periodulad momuSave siTburi manqana ver imuSavebs siTbos erTi wyaros (saxureblis) xarjze, macivris gareSe. vinaidan ciklis dros Sesrulebuli muSaoba am ciklis gamomsaxveli mrudiT SemosazRvruli farTobiT ganisazRvreba, amitom cxadia, rom es muSaoba miT meti iqneba, rac ufro maRal temperaturaze xdeba airis gafarToeba da dabal temperaturaze misi kumSva. Cven mier ganxilul acbc' a wriul process, romelic gamosaxavs siTburi manqanis muSaobis princips, p i r d a p i r i wriuli procesi an pirdapiri cikli ewodeba. wriuli procesi SeiZleba warimarTos Sebrunebuli ac' bca mimarTulebiTac. aseTi Sebrunebuli wriuli procesi, anu u k u c i k l i gamosaxavs m a c i v a r i m a n q a n i s muSaobis princips. ukuciklis dros gafarToeba ( ac' b ) mimdinareobs ufro dabali wnevis pirobebSi, vidre kumSva ( bca ). kumSvis A2 muSaoba metia gafarToebis A1 muSaobaze amitom aseTi ciklis dros muSaobas asrulebs ara muSa sxeuli, aramed gare Zalebi, romlebic kumSavs muSa sxeuls. es muSaoba gardaiqmneba siTbod. amrigad, macivari manqanis ciklis ganxorcielebisas muSa sxeuli arTmevs macivars raRac Q2 siTbos raodenobas da gadscems saxurebels ufro

laparakia swored wriul procesze, vinaidan siTbos gadacemis e r T j e r a d i aqtis dros,

izoTermuli

gafarToebisas,

(  A   Q  dU ; dU  0,

rogorc

 A Q)

gamomdinareobs

Termodinamikis

kanonidan

airis mier Sesrulebuli muSaoba saxureblisgan miRebuli

siTbos raodenobis tolia. **

Sigawvis Zravas SemTxvevaSi macivris rols asrulebs atmosfero.

252

met, Q1 siTbos raodenobas*, damatebiTi Q1  Q2 siTbos raodenoba, romelic gadaeca saxurebels, miiReba im gare Zalebis muSaobis xarjze, romelmac SekumSa airi. amrigad, naklebad Tbili sxeulidan (macivari) metad Tbils (saxurebeli) siTbos raodenoba gadaecema gare Zalis mier

Sesrulebuli

muSaobis

xarjze.

imisaTvis

rom

ukeT

gavarkvioT, Tu rogor xdeba es, warmovidginoT macivari manqanis Semdegi gamartivebuli sqema. Tbogamtari masalisagan damzadebuli dguSiani cilindri, romelSic moTavsebulia airi, CavuSvaT garemoSi (vTqvaT siTxeSi), romlis gacivebac gvsurs (sur. 78.2 a). Semdeg SevamciroT wneva dguSze, e.i. mivceT airs gafarToebis saSualeba. airi gafarTovdeba, Seasrulebs muSaobas Sinagani energiis xarjze da gacivdeba. roca airis tempe-

sur. 78.2

ratura gaxdeba garemos temperaturaze naklebi, es ukanaskneli daiwyebs airze siTbos gadacemas, da TviTon gacivdeba.

Semdeg cilindri gadaviatanoT meore garemoSi, romlis temperaturac ufro maRalia, vidre pirvelisa (sur. 78.2 b) da dguSis qvemoT daweviT SevkumSoT airi sawyis mdgomareobamde ise, rom

misi temperatura gaxdes am

garemos temperaturaze

maRali. maSin airi mTlianad

gadascems garemos im siTbos raodenobas, romelic man pirvel WurWelSi moTavsebul garemos waarTva. amrigad, Cveni mowyobiloba moqmedebs rogorc siTburi tumbo _ arTmevs siTbos raodenobas ufro civ sxeuls da gadascems ufro Tbils, magram yvelaferi es xdeba gare Zalebis muSaobis xarjze.

§ 79. Seqcevadi da Seuqcevadi procesebi procesi SeiZleba mimdinareobdes rogorc piradapiri, ise Sebrunebuli mimarTulebiT. magaliTad, dnobis Sebrunebuli procesia gamyareba, sxeulis gaTbobisa _ misi gaciveba, gafarToebisa _ kumSva dasxv. S e q c e v a d i p roc e s i e w od e b a is e T pr oce s s , ro m e l ic m i m d i n a r e ob s

r o go rc pi rd a p iri , is e S e b r u ne b u li m im arT u le b iT ,

am as -

T an , Se b r u ne b u li p ro ce s is dr os s i s te m a T an am im de v ru l a d g a d i s y v e la im Su ale du r m d gom a re ob as , rom e l Si c m a n g a i ar a p ir d a p ir i

muSa

sxeuli

Q1  Q2  A

rom

siTbos

daubrundes raodenoba,

sawyis sadac

mdgomareobas,

A  A2  A1

Sesrulebuli muSaobaa.

253

man

ciklis

saxurebels dros

gare

unda

gadasces

Zalebis

mier

p r oc e s is dro s d a u b ru nde b a s aw y i s m d gom a re ob as is e , r om g ar e m om c v e l s xe le b Si ar av i T a ri cv l il e b a ar x de b a. process, romelic ar akmayofilebs zemoCamoTvlil pirobebs S e u q c e v i procesi ewodeba. y ov e l i

p ro ce s i,

ro m e ls a c

T an

a xl av s

x a xu n i,

Se u q c e v i a .

marTlac xaxunis Sedegad muSaobis nawili gardaiqmneba siTbod. moxaxune zedapirebi Tbeba, gamoyofili siTbos raodenoba gaibneva sivrceSi da ganawildeba garemomcvel sxeulebSi. ar arsebobs procesi, romelic am siTbos raodenobas gare Zalebis mier muSaobis Seusruleblad daubrunebs moxaxune zedapirebs da kvlav mTlianad gardaqmnis muSaobad. maSasadame, garemomcvel sxeulSi rCeba procesis kvali _ gaTboba; procesi iqneba Seuqcevi. nebismier r e a lu r

meqanikur process Tan axlavs xaxuni da amitom igi Seuqcevia.

Seqcevadi SeiZleba iyos mxolod id e alu ri meqanikuri procesi, romelic mimdinareobs wminda konservatul sistemaSi, xaxunis gareSe. Seqcevadi procesis magaliTi iqneboda absoluturad drekadi birTvis vardna h simaRlidan vakuumSi absolurad drekad horizontalur zedapirze. aseT idealur pirobebSi zedapirze dacemisas birTvi iZens kinetikur energias, romelic zustad tolia misi potenciuri energiisa h

simaRleze, ris

Sedegadac igi iwyebs vertikalurad zeviT moZraobas, gaivlis Sebrunebuli TanamimdevrobiT imave Sualedur mdgomareobebs, rac man gaiara vardnisas (amasTan, traeqtoriis mocemul wertilSi mas eqneba erTi da igive siCqare da energia ganurCevlad imisa, Tu ra mimarTulebiT moZraobs igi) da isev ava imave

simaRleze. es procesi SeiZleba ganmeordes usasrulod bevrjer.

magram vinaidan ar arsebobs arc absoluturi vakuumi, arc absoluturad drekadi sxeulebi, amitom birTvis meqanikuri energiis nawili ixarjeba haerTan xaxunis gamo, nawili ki zedapirze dacemisas gardaiqmneba zedapirisa da birTvis

Sinagan

energiad.

amitom

birTvi

imave

simaRleze

ver

ava,

garemomcvel sxeulSi darCeba procesis kvali _ gaTboba, e.i. procesi iqneba Seuqcevi. cxadia, rom rac ufro maRalia vakuumi da birTvisa da zedapiris drekadi Tvisebebi, miT ufro didi sizustiT SeiZleba ganxiluli procesi CaiTvalos Seqcevadad.

254

Seqcevadi iqneboda maTematikuri qanqaras rxevis procesi, rom ar arsebobdes haeris winaRoba da xaxuni sakidTan. am SemTxvevaSi saqanis potenciuri energia mTlianad gardaiqmneba kinetikur energiad da piriqiT; saqanis rxeva iwarmoebs ragind didxans amplitudis Seumcireblad. aqac, realur pirobebSi, mTlianad

verc

haeris

winaRobas

gamovricxavT

verc

xaxuns,

amitom

realurad saqanis rxeva mxolod miaxloebiT SeiZleba CaiTvalos Seqcevad procesad. ganxiluli magaliTebidan Cans, rom Seqcevadi procesi meqanikuri sistemebis SemTxvevaSic ki realuri procesis garkveuli gaidealebaa, magram aRsaniSnavia, rom meqanikuri moZraoba, p r in ci pu l a d mainc, Seqcevadi procesia. sxvagvaradaa saqme molekulur sistemaSi, sadac p r i nc ipu l ad a a Se u Z l e b e l i Seqcevadi procesis warmarTva, Tumca rig SemTxvevebSi, aqac, procesi garkveuli sizustiT SeiZleba CaiTvalos Seqcevad procesad. naTqvamis naTelsayofad ganvixiloT ramdenime magaliTi. 1. Tu sxvadasxva temperaturis or sxeuls SevaxebT erTmaneTs, maSin siTbos raodenoba TavisTavad gadava metad Tbili sxeulidan naklebad TbilSi da temperaturebi gaTanabrdeba. es procesi mimdinareobs TavisTavad. magram Sebrunebuli, anu ukuprocesi _ Tanabari temperaturis sxeulebidan siTbos raodenobis gadasvla erTi sxeulidan meoreze da temperaturaTa sxvaobis warmoqmna

TavisTavad

arasodes

moxdeba.

saerTod,

yoveli

procesi,

romelsac Tan axlavs Tbogadacema Tbili sxeulidan civSi, Seuqcevia. 2. Tu erTmaneTTan miliT SevaerTebT or WurWels, romlebSic moTavsebulia sxvadasxva airi (vTqvaT, argoni da heliumi) maSin difuziis Sedegad moxdeba airebis urTierTSereva da garkveuli drois Semdeg orive WurWelSi miviRebT heliumisa da argonis Tanabari simkvrivis narevs. es procesi mimdinareobs TavisTavad. Sebrunebuli procesi _ narevis dayofa calkeul komponentebad,

TavisTavad

arasodes

moxdeba.

amrigad,

difuziis

procesi

(iseve

rogorc sxva gadatanis movlenebi, ยง102) Seuqcevia. 3. inerciiT moZravi sxeulis gaCerebis mizezi aris am sxeulis xaxuni Semxeb sxeulTan da haeris winaRoba. am dros sxeulis meqanikuri energia ixarjeba TviT am sxeulis, Semxebi sxeulisa da garemos Sinagani energiis gazrdaze. es procesi mimdinareobs TavisTavad. Sebrunebuli procesi _ sxeulis

255

amoZraveba Sinagani energiis xarjze, rogorc amas yoveldRiuri dakvirvebebi gviCvenebs, TavisTavad arasdros ar xdeba. 4. Seuqcevia airis (idealurisac ki) gafarToebis procesi sicarieleSi, vinaidan am dros airs ar uxdeba winaRobis daZleva da maSasadame, igi muSaobas ar asrulebs. es procesi mimdinareobs TavisTavad. xolo imisaTvis, rom airi davabrunoT sawyis mdgomareobaSi (warvmarToT Sebrunebuli procesi), e.i. movuyaroT erTad Tavi molekulebs, sxva sxeulis mier garkveuli muSaobis Sesrulebaa saWiro, rac iwvevs cvlilebebs am sxeulSi (rCeba procesis kvali). ganvixiloT airis gafarToebis SemTxveva, rodesac airi asrulebs muSaobas. vTqvaT, airi moTavsebulia dguSian WurWelSi. dguSze devs tvirTi, romelic awonasworebs airis wnevas. Tu tvirTs aviRebT, airi gafarTovdeba, amasTan jer gafarTovdeba airis is fena, romelic dguSTan axlosaa, Semdeg ki danarCeni fenebi. maSasadame airis gafarToebisas wonasworoba dairRveva: dguSis maxloblad wneva da temperatura ufro naklebi gaxdeba, vidre dguSisagan moSorebiT. saWiro iqneba garkveuli dro darRveuli wonasworobis aRsadgenad, e.i. wnevisa da temperaturis gasaTanabreblad. isev

davdoT

tvirTi

dguSze.

airi

daubrundeba

pirvandel

mdgomareobas (Sesruldeba imave sididis muSaoba), magram axla ukve dguSis maxloblad wneva da temperatura iqneba ufro meti, vidre airis danarCen fenebSi, e.i. wonasworoba isev dairRveva, magram sapirispiro mimarTulebiT, procesi iqneba Seuqcevi, vinaidan Sebrunebuli procesis (kumSvis) dros airi ver gaivlis

saSualedo

mdgomareobebs,

romelbic

man

gaiara

pirdapiri

procesis (gafarToebis) dros. cxadia, rom rac ufro nela vafarToebT an vkumSavT airs, miT ufro mcire iqneba gadaxrebi wonasworobis mdgomareobebidan da miT ufro dauaxlovdeba ganxiluli procesi Seqcevad process. zRvarSi, roca procesis siCqares usasrulod SevamcirebT, wonaworobis mdgomareobebidan gadaxrebic nulis toli (usasrulod mcire) iqneba, e.i. procesi Sedgenili iqneba mxolod wonasworuli

mdgomareobebisagan.

amitom

Sebrunebuli

procesis

(kumSvis)

dros airi Tanamimdevrulad gaivlis yvela im Sualedur mdgomareobas, rom-

256

lebic

man

gaiara

pirdapiri

procesis

(gafarToebis)

dros,

e.i.

procesi

Seqcevadi iqneba. wonasworuli mdgomareobebisagan Sedgenil process wonasworuli, anu kvazistatikuri

procesi

vuwodeT.

amrigad,

a i ri s

ga f ar T oe b is a

k u m Sv is k v a zi s t at ik u r i pr oce s i Se q ce v a d pr oce s s w a rm o a dg e ns . ganxiluli magaliTebi gviCvenebs, rom Seqcevadi procesi garkveuli gaidealebaa, rom yvela realuri procesi Seuqcevia, vinaidan ar arsebobs moZraoba xaxunis gareSe da arc erTi realuri procesi ar mimdinareobs usasrulod nela. magram praqtikaSi, rig SemTxvevaSi, SesaZlebelia, rom realuri procesi sakmaod dauaxlovdes Seqcevad process. amitom Seqcevadi procesis ganxilvisas miRebul Sedegebs aqvs garkveuli praqtikuli mniSvneloba, rogorc zRvrul Sedegebs. marTlac, Tu romelime sakiTxis gadawyvetisas mniSvneloba ara aqvs procesis mimdinareobis siCqares da arsebiTia mxolod saboloo

Sedegi

(vTqvaT,

miRebuli

siTbos

raodenoba

Sesrulebuli

muSaoba), maSin msjelobisas SeiZleba visargebloT warmodgenebiT Seqcevadi procesis Sesaxeb. rogorc aRvniSneT,

pV diagramaze grafikulad SeiZleba mxolod wonasworuli

procesis gamosaxva. vinaidan Seuqcevi procesi ar warmoadgens wonasworul process, amitom misi gamosaxva

pV (an sxva) diagramaze uwyveti wiriT ar SeiZleba, rac metad arTulebs

aseTi procesisa da ciklis gamokvlevas. magram, praqtikuli sakiTxebisaTvis gvWirdeba Seuqcevi procesis saboloo Sedegi, anu integraluri maxasiaTeblebi. magaliTad, Tu sistema Seuqcevi gziT gadadis a mdgomareobidan b –Si, maSin CvenTvis mniSvnelovania ara procesis mimdinareobis calkeuli detalebi, aramed sistemis mier miRebuli da mis mier Sesrulebuli

Q ' siTbos raodenobisa

A' muSaobis sididis codna. amitom Seuqcevi procesi SeiZleba

Seicvalos misi `tolfasi~ Seqcevadi procesiT, romelic mimdinareobs imave

mdgomareobebs Soris ise, rom sistema iRebs Q  Q' siTbos raodenobas da asrulebs A  A' muSaobas. aseTi Secvlis mniSvneloba isaa,

rom es `tolfasi~ Seqcevadi procesi

diagramaze uwyveti wiriT gamoisaxeba. Semdgom swored aseTi azriT gamovsaxavT Seuqcev process uwyveti wiriT, amasTan, igulisxmeba, rom sinamdvileSi muSa sxeuli ar gadis im Sualedur mdgomareobebs, romlebsac wirze aRebuli wertilebi Seesabameba.

257

Seuqceobis fizikuri bunebis gageba yvelaze ufro iolia Cvens mier ganxiluli magaliTis safuZvelze ori sxvadasxva airis (heliumisa da argonis) urTierTdifuziis Sesaxeb. heliumisa da argonis molekulebi moZraobisas ejaxebian erTmaneTs da TandaTan iWrebian erTmaneTis mier dakavebul moculobaSi. molekulebis es moZraoba da Sejaxeba xdeba mkacrad, meqanikis kanonebis mixedviT; maSasadame calkeuli molekulebis moZraobisa da urTierTqmedebis aqti Seqcevad process warmoadgens, magram uamravi molekulis mravalgzis Sejaxebis Sedegad sistemaSi Cndeba Seuqcevadi cvlilebebi. cdis dawyebamde sistemaSi gvqonda garkveuli wesrigi _ or sxvadasxva WurWelSi gvqonda sxvadasxvagvari airi. molekuluri Sejaxebebis Sedegad es Tavdapirveli wesrigi dairRva _ moxda maTi sruli urTierTSereva. u fr o m ow e s r i ge b u l i m gd om are ob i d an na k lebad

m ow e s r ige b u l Si

g a d as v l a

a ris

S e u q c e ob i s

f i zik u r i a rs i . Seuqceoba aris statistikuri kanonzomierebis gamovlena da igi damaxasiaTebelia sistemisaTvis, romelic Sedgeba Zalian didi raodenobis nawilakebisagan.

§ 80. karnos cikli 78-e paragrafSi miviReT, rom siTburi manqanis margi qmedebis koeficienti

A Q1  Q2 . (80.1)  Q1 Q1 am formulis gamoyvanisas Catarebul msjelobaSi igulisxmeboda, rom wriuli



procesis yvela etapi kvazistatikuri, e.i. Seqcevadia da, maSasadame, ciklis dros Sesrulebuli muSaoba maqsimaluria; amitom (80.1) gamosaxulebaSi  warmoadgens siTburi manqanis maqsimaluri margi qmedebis koeficients. realuri manqanis mq koeficienti yovelTvis ufro naklebia, vinaidan, garda imisa, rom manqanaSi mimdinare procesebi (an maTi nawili) Seuqcevadia, raRac garkveuli

siTbos

raodenoba

usargeblod

ixarjeba

xaxunis

daZlevaze,

dabali

temperaturis garemomcvel sxeulebTan Tbocvlaze da sxv. amrigad,

siTburi

manqanis

maqsimaluri

koeficientis

misaRebad

saWiroa, rom masSi mimdinare procesebi (airis gafarToebisa da kumSvis) iyos 258

Seqcevadi. aseTi Seqcevadi wriuli procesi Teoriulad ganixila 1824 wels frangma inJinerma sadi karnom (1796-1832) da igi k a r nos c ik l is saxelwodebiTaa cnobili. karnos cikli Sedgeba ori izoTermuli da ori adiabaturi procesisagan. mis gansaxorcieleblad saWiroa gvqondes saxurebeli, muSa sxeuli da macivari. amasTan, saxureblisa da macivris siTbotevadoba unda iyos Zalian didi, rom siTbos raodenobis gacemisa da miRebisas maTi temperatura ar Seicvalos. ganvixiloT ufro dawvrilebiT, rogor SeiZleba ganxorcieldes Seqcevadi wriuli procesi, rodesac siTburi manqana Seicavs mxolod erT saxurebelsa da erT macivars, romelTa T1 da T2 temperatura procesis ganmavlobaSi ar icvleba. muSa sxeulis saxurebelTan an macivarTan Tbocvlis procesi mxolod im SemTxvevaSi iqneba Seqcevadi, Tu muSa sxeulis temperatura tolia T1 –isa, roca igi saxureblisagan iRebs Q1 siTbos raodenobas da tolia T2 –isa, roca igi gadascems macivars Q2 siTbos raodenobas* (ufro zustad gansxvaveba muSa sxeulis temperaturasa da T1 –s Soris siTbos raodenobis miRebis procesSi, aseve muSa sxeulis temperaturasa da T2 –s Soris siTbos raodenobis gacemis procesSi unda iyos usasrulod mcire, zRvarSi _ nulis toli. aseT pirobebSi Tbocvlis procesic usasrulod nela warimarTeba, e.i. igi iqneba kvazistatikuri). marTlac, Tu davuSvebT, rom muSa sxeulis temperatura naklebia saxureblis temperaturaze, maSin igi siTbos raodenobas miiRebs pirdapiri procesis dros, magram ver gascems Sebrunebuli procesis dros (vinaidan siTbos raodenoba TavisTavad ar gadadis civi sxeulidan Tbilze), e.i. procesi iqneba Seuqcevadi. radgan muSa sxeulis Tbocvlis procesi saxurebelTan da macivarTan mimdinareobs mudmiv temperaturaze, wriuli procesi e.i. cikli unda Seicavdes or Seqcevad izoTermul process, romelTagan erTi mimdinareobs T1 , xolo meore T2 temperaturaze. am or process Soris muSa sxeulis temperatura Seqcevadad unda icvlebodes T1 –dan T2 –mde da piriqiT da vinaidan, pirobis Tanaxmad, siTbos

mxolod ori wyaro gvaqvs (romelTa temperaturebia T1

da T2 ) izoTermul process Soris mimdinare procesebi SeiZleba iyos mxolod adiabaturi. amrigad, SemTxvev aSi

SegviZlia

davaskvnaT,

Seqce vadi

w ri uli

rom

erTi

proce si

sax ure blis

SeiZle ba

erTi

Se dge bode s

m acivri s

mxolod

ori

i zoTe rm uli da ori adiabaturi proce si sagan .

mxolod

maSinaa

SesaZlebeli,

rodesac

usasrulod didia.

259

saxureblisa

macivris

siTbotevadoba

gamovTvaloT karnos ciklis margi qmedebis koeficienti. davuSvaT, rom muSa

sxeuli

cilindrSi,

aris

romlis

moli

idealuri

kedlebi

airi.

idealurad

airi

moTavsebulia,

Tbogaumtaria,

xolo

dguSian

fskeri

idealurad Tbogamtari. davuSvaT, rom ciklis dasawyisSi airi aris

mdgomareobaSi,

romelic

xasiaTdeba

p1 , V1 , T1 parametrebiT (sur. 80.1). airi saxureb-

lidan iRebs Q1 siTbos raodenobas, izoTermulad farTovdeba,

asrulebs

A12 muSaobas da

gadadis II mdgomareobaSi, romelic xasiaTdeba p2 , V2 , T1 parametrebiT. amis ganxorcieleba Sesur. 80.1

iZleba, Tu cilindris idealurad Tbogamtar fskers SevaxebT saxurebels, romlis temperaturaa T1

(ufro zustad, usasrulo mcire sididiT aRemateba airis T1

temperaturas). amis Semdeg fskeri davxuroT Tboizolirebuli firfitiT da airi

adiabaturi

gafarToebiT

gadaviyvanoT

III

mdgomareobaSi,

romlis

parametrebia p3 , V3 , T2 . T2 < T1 , vinaidan airi adiabaturad gafarTovda Sinagani energiis xarjze. Semdeg fskers movaSoroT izolacia da cilindri movaTavsoT macivarze, romlis temperaturaa T2

(ufro zustad, usasrulo mcire

sididiT naklebia airis T2 temperaturaze). maSin airi gadascems macivars Q2 siTbos raodenobas, izoTermulad SeikumSeba, Seasrulebs uaryofiT A34 muSaobas da gadava IV mdgomareobaSi, romlis parametrebia cilindris

fskeri

isev

davxuroT

Tboizolirebuli

p4 , V4 , T2 . bolos,

firfitiT

airi

adiabaturi SekumSviT davabrunoT sawyis (I) mdgomareobaSi. airi

muSaobas

gafarToebisa

asrulebs

kumSvisas,

rogorc

magram

advili

izoTermuli, misaxvedria,

ise

adiabaturi

rom

adiabaturi

procesis dros Sesrulebuli muSaoba ciklis energetikul balansSi ar Seva, vinaidan II→III procesis dros Sesrulebuli muSaoba A23  CV T1  T2  toli da sapirispiroa IV→I procesis dros Sesrulebuli muSaobisa A41  CV T2  T1  .

260

karnos Seqcevadi ciklis, rogorc idealuri siTburi manqanis mq koeficienti gamoiTvleba (80.1) formuliT. vipovoT Q1 –sa da Q2 –s mniSvnelobebi. I→II procesis dros airi saxureblisagan iRebs Q1 siTbos raodenobas, izoTermulad

farTovdeba

V1 –dan

V 2 –mde

asrulebs

A12

muSaobas,

romelic gamoiTvleba formuliT (§75): Q1  A12  RT1 ln

V 2 V1

III→IV procesis dros airi izoTermulad ikumSeba, gadascems macivars

siTbos

raodenobas

asrulebs

uaryofiT

A34

muSaobas

(dadebiT

muSaobas asrulebs gare Zalebi, romlebic kumSavs airs), romelic tolia:  Q2  A34  RT2 ln

V 4 V 3

  RT2 ln

V 3 V 4

Q1 -isa da _ Q2 -is mniSvnelobebi SevitanoT (78.1) formulaSi:



vaCvenoT, rom

V 2 V1



V3 V 4

Q1  Q2  Q1

RT1 ln

V 2 V1

 RT2 ln

RT1 ln

V3 V 4

V 2

(80.2)

V1

, amisaTvis II→III da IV→I adiabaturi procesebisaTvis

gamoviyenoT puasonis gantoleba (§75). gveqneba: T1V21  T2V3 1

da T2V41  T1V1 1 saidanac, tolobaTa gayofiT miviRebT: V 2 V3 .  V1 V 4

(80.3)

(80.3)-is gamoyenebiT da R –ze SekveciT (80.2) toloba iRebs saxes: Q1  Q2 T1  T2 . (80.4)  Q1 T1 amrigad, karnos ciklis mq koeficienti damokidebulia mxolod saxureb-



lisa da macivris absolutur temperaturaze. es Sedegi, romelic miviReT idealuri

airisaTvis,

marTebulia,

rodesac

muSa

sxeulad

gamoyenebulia

nebismieri drekadi nivTiereba (magaliTad, realuri airi an airTa narevi, wylis orTqli da sxv.) karnom daamtkica, rom k a rn os Se q ce v ad i cik liT 261

m om u S av e ( i de alu r i ) s iT b u r i m an q a ni s m q k oe f ici e n ti ar a a d am o k ide b u l i m as Si g am oy e n e b u l i m u S a s xe u lis n iv T i e re b az e da g a ni s az R v r e b a m xol o d s a xu r e b l is a d a m a civ ri s t e m p e r atu ri T * . muSa sxeulad idealuri airi avirCieT mxolod imitom, rom zustadaa cnobili misi mdgomareobis gantoleba, ramac saSualeba mogvca iolad gamogveTvala karnos ciklis mq koeficienti. karnos cikli ise CavatareT, rom airis gafarToebisa da kumSvis procesi Seqcevadi iyo, rac, rogorc viciT, uzrunvelyofs maqsimaluri muSaobis miRebas. garda amisa, Seqcevadi procesis Catareba gamoricxavda xaxuns, sxvadasxva temperaturis sxeulebis Sexebas da sxva Seuqcev procesebs, romlebic siTbur danakargebs iwvevs. es imas niSnavs, rom ciklis Sesrulebisas uzrunvelyofili iyo siTburi energiis gamoyenebis idealuri pirobebi **. amitom ufro didi margi qmedebis koeficientis miReba, vidre es (80.4) formuliTaa gamosaxuli, principulad SeuZlebelia. karnom daamtkica, rom a r c e rT re a lu r s iT b u r m anq a na s , r o m e l ic m u Sa ob s s axu re b lis a d a m ac iv r is m oc e m u l (T1 , T2 ) t e m pe r a t u r az e ,

S e iZ le b a

h q on de s

u fr o

di di

k o e f ic ie n ti ,

v i dr e

i m av e te m pe ra tu r u l Su ale d Si k a r no s Se q ce v a di c ik l iT m om u S av e s iT b u r m anq a n as * * * . realuri manqanis margi qmedebis koeficienti gamoiTvleba formuliT: Q1  Q2 , Q1 aris ara marto macivarze gadacemuli, aramed



oRond aq igulisxmeba, rom Q2

xaxunis daZlevaze da garemomcvel sxeulebTan Tbocvlis Sedegad dakarguli siTbos raodenobac. (80.4)

formula,

romelic

idealuri

siTburi

manqanisaTvis

miviReT

zogadad ase SeiZleba Caiweros:

 *

Q1  Q2 T1  T2 .  Q1 T1

(80.5)

am daskvnas zogjer karnos meore Teoremas uwodeben.

amitomaa, rom karnos Seqcevadi cikliT momuSave siTbur manqanas idealur siTbur manqanas

uwodeben. ***

am daskvnas zogjer karnos pirvel Teoremas uwodeben.

262

tolobis

niSani

marTebulia

idealuri

siTburi

manqanisTvis

(karnos

Seqcevadi cikli), utolobis niSani ki _ realuri siTburi manqanisaTvis (Seuqcevi cikli). rogorc

(80.4)

formula

gviCvenebs,

karnos

idealuri

manqanis

koeficienti yovelTvis naklebia erTze da miT metia, rac metia saxureblis temperatura da naklebia macivris temperatura*. realuri manqanis mq koeficientic, miT umetes, erTze naklebi iqneba. gansakuTrebuli yuradReba unda mieqces imas, rom siTburi manqana principulad gansxvavdeba meqanikuri da eleqtruli

manqanisagan,

srulyofiT

maTi

marTlac,

meqanikuri

koeficientis

mniSvneloba

eleqtruli

SeiZleba

manqanis

dauaxlovdes

Teoriul zRvrul mniSvnelobas, e.i. (100%-s). siTburi manqanis srulyofisas ki Cven maT mq koeficients vuaxlovebT ara 100%-s, aramed karnos idealuri manqanis koeficients. aqedan gamomdinareobs, rom siTburi Zravas mq koeficientis

gazrdis

gadamwyveti faqtoria saxureblis temperaturis

gazrda da

macivris temperaturis Semcireba. magram macivars, Cveulebriv, warmoadgens atmosfero,

saxureblis

temperaturis

gazrda

ki,

cnobili

mizezis

gamo

(dguSis masalis Tbomedegoba), garkevulad SezRudulia, amitom inJinerTa da konstruqtorTa amocanaa siTburi Zravas mq koeficienti gazardon xaxunis SemcirebiT, garemomcvel sxeulebze siTbos raodenobis gadacemis SemcirebiT, saTbobis arasrul wvasTan dakavSirebuli danakargebis SemcirebiT da a.S. realuri siTburi manqanis mq koeficientis gazrdis SesaZlebloba am mxriv jer kidev didia. magaliTad,

orTqlis

qvabSi

orTqlis

temperatura

T1  500K ,

xolo

macivris

temperatura T2  300K . am monacemebiT, aRniSnuli manqanis mq koeficienti



500  300  0,4 , 500

anu 40%.

praqtikulad ki Tanamedrove orTqlis manqanis mq koeficienti ar aRemateba 20%-s. orTqlis manqanis mq koeficientis gasazrdelad unda gaizardos orTqlis temperatura (rac xorcieldeba e.w. maRali wnevis qvabSi) da Semcirdes siTburi danakargebi. Sigawvis Zravas mq koeficienti gacilebiT metia (45%), rac ganpirobebulia imiT, rom, jer erTi, muSa sxeulis temperatura gafarToebis dawyebamde ufro maRalia, meorec, sawvavi iwvis TviT *

mq koeficienti erTis toli gaxdeba or SemTxvevaSi: roca T1   an, roca T2  0 . rogorc

erTi, ise meore piroba praqtikulad ganuxorcielebelia.

263

cilindrSi da amiT gamoricxulia energiis danakargi, romelic aucileblad Tan sdevs saxureblisagan wylisaTvis siTbos raodenobis gadacemas. gansakuTrebiT maRalia dizelis Zravas mq koeficienti, romelSic ikumSeba ara benzinisa da haeris narevi (rogorc benzinis ZravaSi), romlis maRal wnevamde SekumSva gaZnelebulia narevis TviTaalebis gamo, aramed sufTa haeri, romelic swrafad ikumSeba 30-35 atmosferomde da amis gamo aqvs maRali temperatura

T1 . amis Semdeg cilindrSi afrqveven sawvavs, xdeba misi TviTaaleba da

gamoyofili siTbos xarjze sruldeba muSaoba.

mocemul paragrafSi miRebuli Sedegebis safuZvelze SeiZleba davaskvnaT, rom Sinagani energia (siTburi energia) yvelaze ufro naklebfasovani energiaa yvela saxis energiebs Soris, vinaidan meqanikuri, eleqtruli da sxva saxis energiebi mTlianad gadadian erTimeoreSi da Sinagan energiaSi, siTburi energia ki mxolod nawilobriv gardaiqmneba meqanikur energiad.

§ 81. Termodinamikis II kanoni rogorc viciT, Termodinamikis I kanoni aris energiis mudmivobis kanoni siTbur movlenebSi. magram es kanoni ar miuTiTebs realuri (bunebaSi TavisTavad mimdinare) procesebis mimarTulebas. meore mxriv, SeiZleba davasaxeloT mTeli rigi procesebisa, romlebic energiis mudmivobis kanons ar ewinaaRmdegeba, magram bunebaSi T av is T av a d praqtikulad arasodes ganxorcieldeba. es miuTiTebs, rom Termodinamikis I kanoni arasakmarisia bunebaSi mimdinare procesebis calsaxad gansazRvrisaTvis, amitom am kanonis paralelurad unda arsebobdes damatebiTi piroba, romelic miuTiTebs bunebrivi procesebis mimarTulebas. swored aseTi damatebiTi pirobis Sinaars gamosaxavs Termodinamikis II kanoni. arsebobs Termodinamikis II kanonis ramdenime formulireba*, romlebic erTmaneTis savsebiT ekvivalenturia _ erTidan miiReba meore. gavecnoT maT konkretuli magaliTebiT: 1. davuSvaT, mocemuli gvaqvs sxvadasxva

da T2

temperaturis ori

sxeuli. T1  T2 . Tu am sxeulebs SevaxebT erTmaneTs (sur. 81.1). garkveuli Q *

maTgan yvelaze ufro zogadi ganxilulia momdevno paragrafSi entropiis cnebis Semotanis

Semdeg.

264

siTbos

raodenoba

TavisTavd

gadava

metad

Tbili

sxeulidan

naklebad

TbilSi da procesi gagrZeldeba manam, sanam temperaturebi ar gaTanabrdeba. energiis mudmivobis kanoni ar dairRveoda Sebrunebuli procesi rom warmarTuliyo _ siTbos raodenoba gadasuliyo naklebad Tbili sxeulidan metad TbilSi, magram cxadia, rom aseTi gadasvla T av i s T av ad arasodes moxdeba*. amitom Termodinamikis II kanonis erT-erTi (istoriulad pirveli) formulireba _ klauziusis formulireba _ (1850 w.) aseTia: ` s iT b o * * T a v is T av a d ar as d ros a r g a dava

n ak le b a d

T b il i

s xe u lid a n

m e t ad

T b il S i ~ . 2. viciT, rom erTi saxis energia SeiZleba gardaiqmnas meore saxis energiad. mag. meqanikuri _ siTburad da piriqiT. amasTan, Termodinamikis I kasur. 81.1

nonis mixedviT araviTari gansxvaveba ar arsebobs am gardaqmnis mimarTulebebs Soris.

meore mxriv, cdebi gviCvenebs, rom eleqtruli, meqanikuri da sxva saxis energiebi mTlianad gardaiqmneba siTbur energiad da amisaTvis araviTari damatebiTi pirobis Sesruleba ar aris saWiro, siTburi energia ki mxolod nawilobriv gardaiqmneba meqanikurad da isic garkveuli damatebiTi pirobis dacviT. naTqvamis naTelsayofad davweroT siTburi manqanis mq koeficientis gamomsaxveli formula:

Q1  Q2 . Q1 am formulis Sinaarsi aseTia: muSa sxeulma saxureblisagan miiRo Q1



siTbos raodenoba, ufro naklebi Q2

siTbos raodenoba gadasca macivars,

xolo danarCeni Q1  Q2 siTbos raodenobis xarjze Seasrula muSaoba. yovelgvari cda _ Sesrulebuliyo muSaoba siTbos erTi wyaros (saxureblis) gacivebis xarjze, macivris gareSe _ marcxiT damTavrda. aq macivris arseboba *

macivar manqanaSi siTbos raodenoba gadaecema naklebad Tbili sxeulidan metad Tbils,

magram iq es procesi mimdinareobs ara TavisTavad, aramed gare Zalebis muSaobis Sesrulebis Sedegad. **

igulisxmeba Sinagani energia

265

is damatebiTi pirobaa, romelic saWiroa siTburi energiis meqanikur energiad gardasaqmnelad.

amitom

plankma

kelvinma

Termodinamikis

kanoni

Semdegnairad Camoayalibes: S e u Z le b e li a i s e T i pe r io du l ad m oq m e d i m a nq an is age b a, r om l is

e r T a de rT i

erTi

w y a r os

Se de gi

g a civ e b is

iq ne b od a

m u S aob is

Se s ru le b a

siTbos

xa rj ze (e.i. saxureblisagan miRebuli siTbos

raodenobis xarjze macivris gareSe). iseTi manqanis arseboba, romelic Seasrulebda muSaobas siTbos erTi wyaros gacivebis xarjze, metad xelsayreli iqneboda kacobriobisaTvis, vinaidan siTbos am wyarod SeiZleboda agveRo dedamiwis qerqi, zRva, okeane, atmosfero da sxv., e.i. gveqneboda energiis ufaso da praqtikulad gamouleveli wyaro. aseTi manqanis mq koeficientis zRvruli mniSveneloba iqneboda erTis toli,

vinaidan

Q2  0

(gavixsenoT, rom karnos

idealuri manqanis

margi

qmedebis koeficientic ki, romelic maqsimaluria yvela SesaZlo siTburi manqanis mq koeficients Soris, naklebia erTze). manqana konstruqciuladac ufro martivi iqneboda, vinaidan misTvis saWiro iqneboda mxolod saxurebeli da muSa sxeuli. aseTi manqana Tavisi praqtikuli RirsebiT mogvagonebs mudmiv Zravas, amitom germaneli mecnieris ostvaldis (1858-1932) winadadebiT mas meore gvaris perpetuum mobile uwodes. aqedan gamomdinare, Termodinamikis II kanoni mokled ase gamoiTqmis: S e u Z le b e li a m e o re gv ar is pe r pe t u u m m ob i le s age b a . wylis moculoba msoflio okeaneSi daaxloebiT 1300 milioni kuburi kilometria. misi masaa 1,31018 tona, anu 1,31021 kg. zedapiruli fenebis temperatura weliwadis droisa da geografiuli ganedis mixedviT icvleba 32



C –dan _ 1,9  C –mde, magram ukve 100 metrze da ufro 

Rrmad wylis temperatura mcired icvleba da saSualod 3-4 C –is tolia. rom SegveZlos iseTi manqanis ageba, romelic msoflio okeanes gamoiyenebda saxureblad da misi 0,01 gradusiT gacivebis Sedegad miRebuli siTbos raodenobas gardaqmnida muSaobad, es muSaoba sakmarisi iqneboda dedamiwaze arsebuli manqanebisa da Carxebis asamoZraveblad 150 wlis ganmavlobaSi (!). Tu gaviTvaliswinebT, rom wyali kvlav gaTbeboda mzis energiis xarjze, aseTi manqana mudmivi Zravas ekvivalenturi iqneboda. amitom mas daarqves m e o r e

gvaris

mudmivi Zrava, riTac ganasxvaves p i r v e l i g v a r i s mudmivi Zravasagan, romelic muSaobas asrulebs arafrisagan, raime energiis daxarjvis gareSe.

266

Termodinamikis II kanonis plankisa da kelvinis formulireba tolfasia klauziusis formulirebisa, vianidan plank-kelvinis debulebis darRvevis SemTxvevaSi SesaZlebeli gaxdeboda naklebad Tbili sxeulidan metad Tbil sxeulze siTbos raodenobis gadasvlac. marTlac, am debulebis darRvevis SemTxvevaSi SesaZlebeli iqneboda naklebad Tbili sxeulis gacivebis Sedegad meqanikuri muSaobis miReba, romelsac gamoviyenebdiT metad Tbili sxeulis kidev ufro gasaTbobad, amasTan, yovelgvari muSaobis daxarjvis gareSe, rac tolfasia Termodinamikis II kanonis klauziusiseuli formulirebis darRvevisa. SeiZleba kidev mravali magaliTis moyvana, romelic gviCvenebs, rom mTeli rigi procesebi, romlebic `nebadarTulia~ Termodinamikis I kanoniT, arasodes xorcieldeba, vinaidan maT krZalavs Termodinamikis II kanoni. magaliTad, rodesac sxeuli raime h simaRlidan vardeba, dedamiwaze dacemisas misi meqanikuri energia mTlianad gardaiqmneba am sxeulisa da dedamiwis Sinagan energiad. Sebrunebuli procesi _ Sinagani energiis xarjze (dedamiwis

TviT

sxeulis

gacivebis

xarjze)

sxeulis

asvla

imave

simaRleze arasodes xorcieldeba. marTalia, aseTi procesi nebadarTulia Termodinamikis I kanoniT, magram igi ewinaaRmdegeba Termodinamikis II kanons. marTlac, aseTi procesi rom SesaZlebeli yofiliyo, miviRebdiT II gvaris perpemtuum mobiles _ manqanas, romelic muSaobas Seasrulebda siTbos erTi wyaros gacivebis xarjze, macivris gareSe. Tu airi moTavsebulia daxSuli WurWlis

naxevarSi, B -Si ki sicarielea (sur. 81.2), sarqvlis gaxsnis

Semdeg

molekulebis

nawili

TavisTavad

gadava A â&#x20AC;&#x201C;dan B â&#x20AC;&#x201C;Si ise, rom orive naxevarSi airis simkvrive gaTanabrdeba. Termodinamikis I kanoni ar

sur. 81.2

dairRveoda, Sebrunebuli procesi rom warmarTuliyo _ molekulebs emoZravaT ise, rom maT Tavi moeyaraT WurWlis mxolod A naxevarSi. aseTi procesi arasdros ganxorcieldeba, vinaidan mas krZalavs Termodinamikis II kanoni (marTlac, gafarToebuli airis TavisTavad SekumSvis SesaZlebloba niSnavs II gvaris perpetuum mobiles SesaZleblobas). ganxiluli magaliTebis safuZvelze SeiZleba gavakeToT zogadi daskvna, rom b u ne b a S i m im di na re y v e l a T av is T av ad i pr oc e s i Se u q c e v i a . 267

§ 82. klauziusis utoloba. entropia wina

paragrafSi

miviReT,

rom

siTburi

manqanis

koeficienti

ganisazRvreba formuliT:

Q1  Q2 T1  T2 ,  Q1 T1 marTebulia idealuri



sadac

tolobis

niSani

(82.1) siTburi

manqanisaTvis

(Seqcevadi cikli), utolobisa ki _ realuri siTburi manqanisaTvis (Seuqcevi cikli). (82.1) formulidan

1

Q2 T  1 2 ; Q1 T1



Q2 T  2 . Q1 T1

an, Tu SevucvliT niSans,

Q2 T2 * ,  Q1 T1

(82.2)

anu

Q2 Q1 ,  T2 T1 saidanac miviRebT:

aris

siTbos

Q1 Q2   0. T1 T2 raodenoba, romelsac

muSa

sxeuli

gadascems

macivars. Tu ukanasknel utolobaSi niSan minuss mivakuTvnebT TviT Q2 –s, anu ganvixilavT mas rogorc algebrul sidides

Q2  0 ,

maSin igi Semdegnairad

Caiwereba:

Seqcevadi ciklis SemTxvevaSi (80.2) utoloba gadadis tolobaSi:

Q1 T1  . Q2 T2 rogorc viciT, karnos ciklis mq koeficienti da, maSasadame, Q1 / Q2 Sefardebac ar aris damokidebuli nivTierebis gvarobaze. Tu gavzomavT

Q1 da Q2 siTbos raodenobas da

gvecodineba erTi temperatura, vTqvaT T2 , maSin am tolobidan gavigebT T1 temperaturas. maSasadame, es toloba saSualebas gvaZlevs SemoviRoT iseTi temperaturuli skala, romlis Cvenebac ar aris damokidebuli Termometruli sxeulis gvarobaze da saWiroebs mxolod erT eqsperimentul mudmiv wertils (e.i. reperul wertils). aseT skalas kelvinis winadadebiT ewoda T e r m o d i n a m i k u r i s k a l a . Termodinamikuri skalis reperul wertilad zoma-wonis X konferenciaze miRebuli iyo wylis sammagi wertili (273,16K).

268

Q1 Q2 (82.3)  0. T1 T2 (82.3) Tanafardobas ewodeba k l au z iu s is u to lob a . sxeulis mier miRebuli siTbos raodenobis Sefardebas im temperaturasTan, romelzedac igi sxeulma miiRo, d ay v a ni li s iT b o s ra o de no b a ewodeba (cxadia, rom sxeulis gaTbobisas igi dadebiTia, gacivebisas ki _ uaryofiTi). maSasadame, klauziusis utolobis

Tanaxmad,

s x e u l is

m ie r

m iR e b u li

d ay v an il

s iT b o s

r a od e n ob a T a jam i a r Se iZ l e b a iy o s nu l ze m e t i . es Sedegi, romelic miviReT karnos ciklis konkretuli magaliTiT, marTebulia nebismieri ciklisaTvis. marTlac, SeiZleba imis Cveneba, rom nebismieri wriuli procesi SeiZleba ganvixiloT rogorc karnos elementaruli ciklebis jami.* erT-erTi aseTi elementaruli i –uri ciklisaTvis klauziusis utoloba Semdegnairad Caiwereba:

Qi Q 'i   0, Ti T 'i romlis ajamva yvela elementaruli ciklis mixedviT mogvcems: Qi (82.4)  0. Ti i amrigad, nebismieri ciklis dros sxeulis mier miRebuli dayvanil siTbos



raodenobaTa jami ar SeiZleba iyos nulze meti. zRvarSi, Tu karnos elementaruli ciklebis raodenobas usasrulod gavzrdiT, ise rom TiToeuli Qi  0 , maSin miviRebT:

∮ δTQ  0 .

(82.5)

rgoli integralis niSanze imis maCvenebelia, rom integreba warmoebs mTel Caketil wirze. Seqcevadi ciklis SemTxvevaSi (80.5) formulidan miviRebT:

∮ δTQ  0 .

(82.6)

maTematikidan cnobilia, rom Tu romelime sididis integrali Caketil wirze nulis

tolia,

funqciisa, *

karnos

maSin es sidide

romelic

cikli

aris

damokidebulia

erTaderTi

Seqcevadi

sruli diferenciali romelime

sistemis

wriuli

SemTxvevaSi.

269

mdgomareobaze

procesia

mxolod

siTbos

ar ori

aris wyaros

damokidebuli

gzaze,

romliTac

sistema

movida

mdgomareobaSi.

maSasadame,

Q

(82.7)  dS . T S funqcia, romelsac e nt ro p ia * ewodeba, pirvelad SemoiRo klauziusma 1855 wels. magram misi fizikuri arsis gaxsna SesaZlebeli gaxda mxolod mas Semdeg (1896 w), rac bolcmanma daakavSira igi mdgomareobis albaTobasTan. energiasTan

erTad

entropia

warmoadgens

sistemis

mdgomareobis

metad

mniSvnelovan maxasiaTebels. igi, iseve rogorc energia, aditiuri sididea. entropiis gansazRvridan Cans, rom SI sistemaSi igi (iseve rogorc siTbotevadoba) izomeba j/K-obiT, an erTi molisaTvis, j/K.mol-obiT. entropiis cvlilebis gamoTvla xSirad metad martivia. magaliTad, gamovTvaloT entropiis cvlileba 200g wylis, 180C-dan 00C-mde gacivebis SemTxvevaSi:

S 



Q T



j T cmdT 273  cm ln 2  4190  0,2 ln  54 . T T1 291 K T1



e.i. wylis gacivebisas misi entropia mcirdeba.

(82.7) toloba marTebulia Seqcevadi procesisaTvis. SeiZleba vaCvenoT, rom Seuqcevi procesisaTvis dS 

amitom, sazogadod: dS 

Q T

(82.8)

Q

. (82.9) T aq, iseve rogorc wina SemTxvevaSi, tolobis niSani marTebulia Seqcevadi procesisaTvis, utolobisa ki _ SeuqcevisaTvis. (82.9) Tanafardoba Seesabameba im SemTxvevas, roca sistemis mdgomareoba usasrulod mcired icvleba. mdgomareobis sasruli cvlilebisaTvis, roca sistema gadadis A mdgomareobidan B –Si (82.9) formulis Tanaxmad gveqneba: B

 dS  

Q T

warmodgeba berZnuli sityvebisagan. misi azrobrivi Targmania S e m d g o m i g a r d a q m n e b i saTv is mi uwv dome li .

270

S B  S A  S   A B

toloba

SB  S A   A

Q T

saSualebas

Q T

(82.10)

gvaZlevs

ganvsazRvroT

ara

mocemuli

mdgomareobis Sesabamisi entropiis absoluturi mniSveneloba, aramed misi cvlileba erTi mdgomareobidan

meoreSi

gadasvlisas.

magram,

rogorc

yovelTvis

xdeba

xolme

aseT

SemTxvevaSi, SeiZleba SevarCioT iseTi mdgomareoba, romlis entropias miviCnebT nulis tolad

integrali

yvela



Q T

sxva

mdgoamreobas

ganvixilavT

mis

mimarT.

amitom

miRebulia,

rom

gansazRvravs entropiis mniSvnelobas.

S

Q T

praqtikaSi yovelTvis saWiroa entropiis cvlilebis da ara TviT entropiis codna, amitom sulerTia, Tu ra mdgomareobas SevusabamebT entropiis nulovan mniSvnelobas. germanelma

mecnierma

nernstma

(1864-1941)

dabal

temperaturaze

Catarebuli

mravali

cdis

safuZvelze daadgina (1906), rom t e m p e r a t u r i s a b s o l u t u r n u l z e y v e l a s x e u l i s e n t r o p i a n u l i s t o l i a . es daskvna, romelic cnobilia nernstis Teoremis an Termodinamikis III kanonis saxelwodebiT, miuTiTebs, rom absolutur nulze mdgomareobis yoveli cvlileba xdeba entropiis Seucvlelad.

Tu sistema adiabaturad izolirebulia, maSin  Q  0 da

S B  S A  S  0 an, usasrulod mcire cvlilebisaTvis e . i.

i zo lir e b u l

s is t e m aS i

dS  0 , pr oce s i

(82.11)

ise

m im di n are ob s ,

(82.12) rom

s is te m i s e nt ro pi a ar m ci r de b a (mudmivi rCeba an izrdeba). viciT (§81), rom bunebaSi yvela realuri procesi, anu procesi, romelic mimdinareobs TavisTavad, Seuqcevia. amitom SeiZleba vTqvaT, rom i zo l ir e b u l s i s te m a S i m im di n are y v e l a r e a lu r i pro ce s i iw v e v s e n t r o pi i s zr d as . es daskvna, romelic cnobilia e nt ro p iis z rd is p ri nc ipi s saxelwodebiT, aris Termodinamikis II kanonis yvelaze ufro zogadi formulireba: i zo li re b u l s is te m a Si Se s a Z le b e li a m xol o d is e T i pro c e s i, r om e li c iw v e v s s i s te m is e n tr opi is z rd as . aseTnairad Camoyalibebuli Termodinamikis II kanoni saSualebas gvaZlevs davadginoT izolirebul sistemaSi mimdinare procesis mimarTuleba: igi 271

SeiZleba iyos mxolod iseTi, romelic Seesabameba entropiis zrdis mimarTulebas. SesaZlebelia Tu ara rom sistema gadavides pirveli mdgomareobidan meoreSi? am kiTxvaze pasuxis gasacemad unda gamovTvaloT sistemis entropiis cvlileba am gadasvlis dros. Tu aRmoCnda, rom

S  0 , aseTi gadasvla

SesaZlebelia da piriqiT. entropia izrdeba manam, sanam sistemaSi damyardeba Termodinamikuri wonasworoba. am mdgomareobas Seesabameba sistemis entropiis maqsimumi. aseT sistemaSi yovelgvari makroskopuli procesi Sewyvetilia. entropiis zrdis principi ganvixiloT iseTi mniSvnelovani procesebis magaliTiT, rogoricaa Tbogadacema da airis gafarToeba vakuumSi. 1. T b o g a da ce m a . Tu adiabatur garsSi moTavsebul or sxeuls, romelTa temperaturaa T1 da T2 , SevaxebT erTmaneTs (sur. 81.1), maSin siTbos raodenoba gadaecema metad Tbili sxeulidan naklebad Tbils, ris Sedegadac orive sxeulis temperatura gaTanabrdeba. gamovTvaloT entropiis cvlileba am procesSi. rogorc viciT (§ 79), Tbogadacemis aRniSnuli procesi, romelic iwvevs orive sxeulis mdgomareobis Secvals, Seuqcevia, amitom misTvis S 

Q forT

mula, sazogadod, umarTebuloa (vinaidan entropiis cvlileba tolia dayvanili siTbos raodenobisa mxolod Seqcevadi procesis dros); am SemTxvevaSi iqcevian Semdegnairad: vinaidan entropiis cvlileba ganisazRvreba sistemis sawyisi da saboloo mdgomareobiT, aRniSnul Seuqcev process cvlian tolfasi Seqcevadi procesiT, e.i. iseTi Seqcevadi procesiT, romelic iwvevs sistemis mdgomareobis iseTsave cvlilebas* da angariSoben misTvis S 

Q sidides T

(ra Tqma unda, es Secvla xdeba azrobrivad; sistemis entropia srulebiT ar aris damokidebuli imaze, sinamdvileSi moxdeba Tu ara es Seqcevadi procesi). davuSvaT,

T1  T2 , maSin pirveli sxeuli gascems siTbos raodenobas,

amitom misi entropia Seicvleba (Semcirdeba)

es SeiZleba ganxorcieldes, magaliTad, mesame, m u S a s x e u l i s meSveobiT, rogorc es iyo

karnos ciklis ganxilvisas.

272

Q T1 sididiT, meore sxeuli ki miiRebs Q siTbos raodenobas, e.i. misi entropia S1  

gaizrdeba S 2 

Q sididiT. T2

sistemis entropiis cvlileba toli iqneba orive sxeulis entropiis cvlilebaTa algebruli jamisa: 1 1 S  S1  S 2  Q    0 ,  T2 T1  anu entropia izrdeba, vianidan, pirobis Tanaxmad, T1  T2 .

entropiis es cvlileba, miRebuli idealuri (Seqcevadi) procesisaTvis, tolia misi cvlilebisa Tbogadacemis realur (Seuqcev) procesSi. amrigad, Tumca Tbogadacemis Sedegad sistemis energia ar Seicvala (Termodinamikis I kanonis Sesabamisad), misi entropia gaizarda (Termodinamikis II kanonis Sesabamisad). aseve advilia vaCvenoT, rom meore sxeulidan pirvels siTbos raodenoba ar gadaecema, vinaidan es gamoiwvevs sistemis entropiis Semcirebas, rac akrZalulia Termodinamikis II kanoniT. 2. i de alu ri a i ris g a f ar T oe b a v a k u u m S i . davuSvaT, rom adiabaturad izolirebuli WurWeli gayofilia or tol A da B nawilebad. A nawilSi moTavsebulia 1 moli idealuri airi, romlis temperaturaa T , B -Si ki sicarielea. rodesac K sarqvels gavxsniT, airi adiabaturad gafarTovdeba da daikavebs WurWlis mTel V A B moculobas. vinaidan airi farTovdeba vakuumSi, misi temperatura ar icvleba (§76). erTi SexedviT SeiZleba mogveCvenos, rom am dros airis entropia ar unda Seicvalos. vinaidan igi arc iRebs siTbos raodenobas da arc gascems, magram es ase ar aris. gamovTvaloT entropiis cvlileba. airis gafarToeba vakuumSi Seuqcevi procesia (§79), amitom am cvlilebis gamosaTvlelad unda gamovTvaloT entropiis cvlileba nebismieri Seqcevadi procesisaTvis, romelic iwvevs sistemis mdgomareobis iseTsave cvlilebas. aseT Seqcevad process, kerZod, warmoadgens airis izoTermuli (e.i. usasrulod neli da, maSasadame, kvazistatikuri) gafarToeba imave T temperaturaze. radgan procesis dros T  const , amitom 273

2 

1 Q S    Q  . T 1 T

Q siTbos

raodenoba

tolia

izoTermuli

procesis

dros

Sesrulebuli

muSaobisa, romelic (77.3) formulis Tanaxmad tolia

Q  RT ln amitom

V A B  RT ln 2 , VA

S  R ln 2  0 ,

e.i. entropia izrdeba. aseve martivad vaCvenebT, rom airis urTierTdifuziis dros entropia izrdeba. marTlac, davuSvaT, rom A naxevarSi moravsebulia 1 moli romelime idealuri airi,

B -Si ki igive raodenobis sxva airi. gavxsnaT sarqveli. maSin airebi erTmaneTs Seereva da miRebuli narevi daikavebs

V A  VB  V A B

moculobas. es procesi SeiZleba ganvixiloT

rogorc TiToeuli airis gafarToebis procesi: pirveli gafarTovda V A -dan VA B –mde, xolo meore V B –dan VA B –mde. magram am dros, rogorc vnaxeT, izrdeba rogorc pirveli, aseve meore airis entropia, amitom sistemis entropiac, cxadia, gaizrdeba:

 V V S  S1  S 2  R ln A B  ln A B VB  VA

   2 R ln 2  0 

ganvixiloT kidev erTi magaliTi. davuSvaT, mocemulia izolirebuli sistema, romelic Sedgeba adiabatur garsSi moTavsebuli erTi moli airisa da inerciiT mbrunavi m masis diskosagan. airis molekulebTan xaxunis gamo gaTbeba rogorc airi, ise disko, e.i. diskos meqanikuri energia gardaiqmneba TviT diskosa da airis siTbur enrgiad da sistemis temperatura gaizrdeba

T1 –dan

T2 –mde. vinaidan sistema izolirebulia, cxadia, rom misi sruli

energia mudmivi rCeba. gamovTvaloT sistemis entropiis cvlileba. airis moluri siTbotevadoba mudmivi moculobis dros aRvniSnoT C V –Ti. xolo diskos masalis kuTri siTbotevadoba c –Ti. maSin  Q' da  Q' ' siTbos raodenoba, romlebsac miiRebs Sesabamisad airi da disko temperaturis dT sididiT cvlilebisas, toli iqneba:

 Q'  C V dT da

 Q' '  cmdT . 274

airis entropiis cvlileba aRvniSnoT S ' –iT, xolo diskosi S ' ' –iT, maSin gveqneba:

S ' 

 Q'



S ' ' 



 C V

 Q' ' T

T2 dT  C ln  0,  V T T T 1 1 T2

T dT  cm ln 2  0 . T T1 T1

 cm 

sistemis entropiis cvlileba S  S 'S ' '  0

e.i. sistemis entropia gaizarda. amrigad, am SemTxvevaSic miviReT, rom Tumca sistemis sruli energia ar Secvlila, misi entropia gaizarda. saWiroa xazgasmiT aRiniSnos, rom sistemis energia ar Seicvala r a ode n ob r iv ad ,

magram Seicvala x a r is x ob riv a d . kerZod, meqanikuri energia

(diskos mowesrigebuli moZraobis energia), gardaiqmna siTbur energiad (molekulebis qaosuri moZraobis energiad). moxda energiis xarisxis Semcireba an, rogorc amboben, e ne rg ii s ga u f a s u r e b a . energiis gaufasurebaSi igulisxmeba misi sxva saxis energiad TavisTavad gardaqmnis unaris Semcireba. marTlac meqanikuri, eleqtruli da sxva saxis energiebi TavisTavad da mTlianad gardaiqmneba Sinagan energiad, maSin roca Siangani energia mxolod nawilobriv gardaiqmneba sxva saxis energiad da isic garkveuli pirobebis dacviT. maSasadame, izolirebuli sistemis entropiis zrda miuTiTebs, rom sistemaSi xdeba energiis gaufasureba, amitom e n tr op i a Se i Z le b a g a nv ix i l oT r og or c e ne rg ii s g au f as u r e b is zom a . zogjer ixmareba aseTi gamoTqmac, rom e ntropia aris s istemis siTburi muSaobis uunarobis zoma. SeiZleba imis Cveneba, rom entropiis zrdis kanonidan gamomdinareobs II gvaris mudmivi Zravas SeuZlebloba. marTlac, meore gvaris mudmivi Zrava Sedgeba saxureblisa da muSa sxeulisagan. muSa sxeuli asrulebs wriul process, amitom misi entropia ar icvleba, saxu-

 

reblis entropia ki mcirdeba, vinaidan is gascems siTbos  dS 

Q

 ,  Q  0, amitom dS  0  . T 

maSasadame, mTliand sistemis entropia unda Semcirdes, rac ewinaaRmdegeba entropiis zrdis kanons.

275

§ 83. Termodinamikis II kanonis statistikuri arsi gavecaniT Termodinamikis II kanonis ramdenime formulirebas. yoveli maTgani erTmaneTis tolfasia da gamosaxavs bunebaSi mimdinare realuri procesis Seuqceobas, magram arc erTi maTgani ar xsnis am kanonis fizikur Sinaarss. saWiroa aixsnas, ratomaa, rom realuri procesi, romelic ganpirobebulia molekulaTa erTobliobis moZraobiT (magaliTad airis gafarToeba, difuzia da sxv.) Seuqcevia, maSin roca calke aRebuli TiToeuli molekulis moZraoba

warmoebs

mkacrad

meqanikis

kanonebis

mixedviT

da,

maSasadame,

Seqcevadia? am kiTxvaze pasuxis gasacemad sakiTxi unda ganvixiloT molekulurkinetikuri Teoriis TvalsazrisiT da gavecnoT a lb aT ob is cnebas, romelic aris statistikuri fizikis ZiriTadi cneba. r a im e Se m T xv e v iT i m ov le n is a lb aT o b a e w o de b a am m ov le ni s x e l Se m w y ob

S e m T xv e v aT a

ri cxv i s

f a rd ob as

T an ab r a d

S e s aZ l o

S e m T xv e v aT a s ae rT o ri cxv T a n. Tu A movlenis moxdenis xelSemwyob SemTxvevaTa ricxvi aris n A , xolo SemTxvevaTa saerTo ricxvia N , maSin A movlenis moxdenis albaToba nA . (83.1) N Tu magaliTad, daxurul yuTSi moTavsebulia sami TeTri da Svidi Savi WA 

burTula, maSin albaToba imisa, rom amoRebuli burTula aRmoCndeba TeTri (83.1) formulis Tanaxmad iqneba W1 

3 , xolo albaToba imisa, rom amoRebuli 10

burTula aRmoCndeba Savi, iqneba W2 

7 . 10

albaToba imisa, rom amoRebuli burTula iqneba TeTri an Savi, tolia

W  W1  W2  1, xolo albaToba imisa, rom burTula iqneba wiTeli W  0 . am magaliTebiT Cans, rom a. albaToba moTavsebulia nulsa da erTs Soris, 0  W  1 , b. aucilebeli movlenis albaToba erTis tolia, g.

SeuZlebeli movlenis albaToba nulis tolia.

276

albaTobis TeoriaSi ganixileba albaTobaTa Sekrebisa da gamravlebis kanoni. Tu

movlena Sedgeba

kerZo SemTxvevebisagan, maSin am

movlenis albaToba tolia B da C movlenebis albaTobaTa jamisa:

W  A  W B   W C  . Tu ori A da B movlena damoukidebelia*, maSin orive movlenis erTdroulad

moxdenis

albaToba

tolia

calkeul

movlenaTa

albaTobebis

namravlisa:

W  AB   W  A  W B gamoviyenoT albaTobis Teoriis meTodebi da ganvixiloT umartivesi Termodinamikuri procesi _ airis gafarToeba (sur. 83.1). simartivisaTvis jer warmovidginoT, rom WurWlis A naxevarSi gvaqvs mxolod

erTi

molekula.

maSin,

sarqvels

imoZravebs qaosurad da aRmoCndeba xan

A , xan

SemTxveva erTnairad albaTuria da es albaToba W  davuSvaT axla, rom ori

molekula

(1 da 2).

gavxsniT,

molekula

naxevarSi, Tanac, orive 1 –s (sur. 83.1. a). 2

A naxevarSi moTavsebulia winaswar danomrili

sarqvels

gavxsniT,

molekulebi

daiwyeben

moZraobas WurWlis mTel moculobaSi _ `airi~ gafarTovdeba. molekulebis moZraobis

qaosurobis

gamo orive molekula SeiZleba isev

WurWlis

naxevarSi aRmoCndes, e.i. `airi~ TavisTavad SeikumSos. vnaxoT, rogoria imis albaToba,

rom

amisaTvis

unda

orive

molekula

ganvixiloT

ori

isev

WurWlis

molekulis

yvela

naxevarSi

aRmoCndes.

SesaZlo

ganawileba

WurWlis orive naxevars Soris. es ganawilebani mocemulia 83.1 b suraTze, saidanac Cans, rom SesaZlo ganawilebaTa ricxvi aris 4=2 2, xolo CvenTvis saintereso SemTxvevis albaToba W 

1 1  4 22

**,

es albaToba sakmaod didia, ami-

tom didxans ar mogvixdeba lodini imisaTvis, rom orive molekula A naxevarSi aRmoCndes e.i. `airi~ TavisTavad SeikumSos. am SemTxvevaSi gafarToebis procesi Seqcevadi iqneba. molekulaTa ricxvis gazrdiT airis TavisTavad *

or movlenas ewodeba damoukidebeli, Tu erT-erTis Sesruleba ar cvlis meoris albaTobas.

igulisxmeba rom TiToeuli ganawilebis albaToba erTnairia da tolia ¼1/4-is.

277

SekumSvis albaToba TandaTan mcirdeba. 83.1 g, d suraTebidan Cans, rom sami molekulis SemTxvevaSi es albaToba W  TxvevaSi W  SekumSvis W

1 1  4 . zogadad, N 16 2

(e.i.

yvela

1 1 . xolo oTxi molekulis Sem 8 23

molekulis SemTxvevaSi airis TavisTavad

molekulis

naxevarSi

dabrunebis)

albaToba

1 ; airSi molekulebis ricxvi Zalian didia, amitom es albaToba Zalian 2N

mcirea. magaliTad, roca N  2,7  1019  3  1019 (loSmitis ricxvi), molekulebis WurWlis A naxevarSi Tavmoyris albaToba imdenad mcirea, rom praqtikulad igi arasdros ar ganxorcieldeba, e.i. airis gafarToebis procesi Seuqcevi iqneba. davubrundeT isev 83.1 suraTs. ori, sami da oTxi molekulis SemTxvevaSi

sur.83.1 278

SesaZlo ganawilebaTa ricxvi aris Sesabamisad 4, 8 da 16. TiToeuli es ganawileba gansazRvravs sistemis e.w. m ik r om dg om are ob as . maSasadame, mikromdgomareoba ganisazRvreba imiT, Tu romeli molekula romel naxevarSia. magram erTgvarovani airis molekulebi savsebiT erTnairia, amitom cxadia, rom sistemis mdgomareobas, e.w. m ak ro m d go m a re ob a s , gansazRvravs ara is, Tu r om e li molekulebia WurWlis A naxevarSi da romeli WurWlis B naxevarSi, aramed is, Tu r a m de ni molekulaa

A –Si da ramdeni

B –Si. amitom

mdgomareobebi, romelsac Seesabameba molekulebis erTi da igive ricxvi WurWlis TiToeul naxevarSi, savsebiT igivuria, miuxedavad imisa, Tu konkretulad romeli molekulebia A –Si da romeli B –Si da gamosaxavs erTsa da imave makromdgomareobas. ase, magaliTad, 83.1 d suraTze naCvenebi II, III, IV da V ganawilebebi gamosaxavs erTsa da imave makromdgomareobas (e.i. sistemis makroskopuli

parametrebis

mniSvneloba

oTxive

SemTxvevaSi

erTnairia),

romelic xasiaTdeba A –Si sami da B –Si erTi molekulis yofniT. am mdgomareobis albaToba, cxadia 4-jer meti iqneba TiToeuli mikromdgomareobis albaTobaze*. VI-XI ganawilebebi gamosaxavs sxva makromdgomareobas, romelic xasiaTdeba

molekulebis

Tanabari

ganawilebiT

molekula). am makromdgomareobis albaToba W 

(TiTo

naxevarSi

or-ori

6 da a.S. 16

m ik r om d go m a re o b aT a r ic xv s , rom liT ac xo rc ie l de b a m o c e m u l i m ak r om d gom a re ob a, T e rm o di na m ik u ri alb aT ob a (  ) e w od e b a . maTematikuri

albaTobisagan

gansxvavebiT,

romelic

yovelTvis

≤1. Termodinamikuri albaToba   1**. magaliTad, Termodinamikuri albaToba imisa, rom

A –Si iyos 4 molekula,

B -Si ki arc erTi, an piriqiT, erTis

tolia,  4,0   0,4  1 , aseve  3,1   1,3  4 , xolo  2,2  6 . am magaliTiT Cans, rom maqsimaluri Termodinamikuri albaToba Seesabameba molekulaTa

makromdgomareobis

SemTxvevaSi

albaToba

roca

Tanxvdeba

mocemuli

mikromdgomareobis

makromdgomareoba

albaTobas

xorcieldeba

mxolod mxolod

erT erTi

mikromdgomareobiT. **

cxadia, rom maTematikur da Termodinamikur albaTobas Soris arsebobs kavSiri. kerZod,

matematikuri albaToba miiReba Termodinamikuri albaTobis gayofiT mikromdgomareobaTa saerTo ricxvze.

279

Tanabar ganawilebas, minimaluri ki _ yvelaze ufro araTanabars. gansxvaveba am albaTobebs Soris didi ar aris, rac gamowveulia molekulaTa mcire ricxviT. molekulaTa ricxvis zrdiT es gansxvaveba swrafad izrdeba da Zalian didi ricxvis SemTxvevaSi metismetad didia. vTqvaT, mocemulia N molekula. SeiZleba vaCvenoT, rom Termodinamikuri albaToba imisa, rom molekulaTa saerTo

ricxvidan WurWlis

naxevarSi moTavsebulia n , xolo B –Si ( N  n ) molekula, tolia

N! . n!N  n ! formulis analizi gviCvenebs, rom  udidesia, roca molekulebi orive



naxevarSi Tanabradaa ganawilebuli e.i. n 

N ; magaliTad, Tu N  3 1019 , maSin 2

 N ,0   0, N   1 , xolo

N! N N 1020 , ,   2 2  2 2  N   ! 2 

 rac Zalian didi ricxvia.

amrigad, statistikuri ganxilvis safuZvelze cxadi xdeba, Tu ratom gadadis airi TavisTavad WurWlis A naxevridan B –Si da ratom aris airis gafarToebis es procesi (iseve rogorc yvela realuri procesi) Seuqcevi. airi TavisTavad imitom gadadis WurWlis A naxevridan B –Si, rom am gadasvlas

Seesabameba

Termodinamikuri

albaTobis

zrda

piriqiT,

airi

TavisTavad ar ikumSeba imitom, rom am process Seesabameba Termodinamikuri albaTobis Semcireba. maSasadame, SeiZleba davaskvnaT, rom izolirebul sistemaSi mimdinare realuri procesis mimarTuleba aris albaTobis zrdis mimarTuleba. Termodinamikuri albaToba maqsimaluria, roca moculobaSi molekulebi Tanabradaa ganawilebuli. izolirebul sistemaSi molekulebis Tanabari ganawilebis

mdgomareoba

Seesabameba

sistemis

Termodinamikuri

wonaworobis

mdgomareobas. maSasadame, SeiZleba vTqvaT (da es mkacrad mtkicdeba), rom sistemis maqsimaluri albaTobis mdgomareoba warmoadgens misi wonasworobis mdgomareobas.

280

magram, moculobaSi molekulebis Tanabari ganawileba amave dros aris maTi yvelaze ufro mouwesrigebeli ganawileba. amitom sistemis yvelaze ufro albaTuri mdgomareoba amave dros aris misi yvelaze ufro mouwesrigebeli

mdgomareoba.

gagebiT

SeiZleba

iTqvas,

rom

T e rm o di na m ik u r i

a lb aT ob a ar is s i s te m is m ou w e s r i ge b l ob i s zom a . cxadia, rom rac ufro didia mdgomareobis albaToba, miT ufro xSirad ganxorcieldeba igi da miT ufro met dros dahyofs sistema am mdgomareobaSi. amitomaa, rom izolirebuli sistema masSi Termodinamikuri wonasworobis damyarebis Semdeg TiTqmis mTel dros (praqtikulad ragind didxans) am mdgomareobaSia*. maSasadame, Termodinamikuri albaToba saSualebas gvaZlevs davadginoT izolirebul sistemaSi mimdinare procesis mimarTuleba _ igi Seesabameba albTobis zrdis mimarTulebas. magram aseTive TvisebiT xasiaTdeba wina paragrafSi ganxiluli fizikuri sidide _entropia. amrigad, arsebobs ori fizikuri sidide _ entropia ( S ) da Termodinamikuri albaToba (  ), romlebic izrdeba izolirebul sistemaSi mimdinare realuri (Seuqcevi) procesis dros. gasagebia, rom maT Soris unda arsebobdes garkveuli kavSiri. es kavSiri 1896 wels Teoriulad daadgina bolcmanma: S  k ln  ,

(83.1)

sadac k bolcmanis mudmivaa. maSasadame, sistemis entropia Termodinamikuri albaTobis logariTmis proporciulia. amitom, iseve rogorc albaToba, e nt ro pi a s is te m i s

m o u w e s r i ge b l ob i s

zo m a s .

aseTive

w a rm o a dg e n s

daskvnamde,

ramdenadme

gansxvavebuli msjelobis safuZvelze, mivediT wina paragrafSi. kavSiri entropiasa da albaTobas Soris saSualebas gvaZlevs gavxsnaT Termodinamikis II kanonis statistikuri arsi. igi miuTiTebs, rom Termodinamikis II kanoni _ entropiis zrdis kanoni, aris statistikuri kanoni da gamosaxavs bunebis fundamentur Tvisebas, romelsac Cven ukve gavecaniT airis gafarToebis kerZo magaliTiT: *

sistemis

wonasworuli

mdgomareobis

albaToba

didad

gansxvavdeba

maxlobeli

mdgomareobis albaTobisagan. amitom ufro zusti iqneba Tu vityviT, rom sistema TiTqmis mTel

dros

(praqtikulad

yovelTvis)

maqsimaluri

makromdgomareobaSia.

281

albaTobis

mis

maxlobel

i zo li re b u l

s is te m a Si

p ro ce s e b i

m im di n are ob s

a lb aT ob is

z r dis m i m a rT u le b iT . Termodinamikis

II kanonis statistikuri xasiaTi miuTiTebs,

rom es

kanoni ar aris iseTi universaluri, rogorc energiis mudmivobis kanoni. misi gamoyeneba ar SeiZleba calkeuli nawilakebis an nawilakTa mcire ricxvisagan Sedgenili sistemisaTvis. garda amisa, statistikuri xasiaTi u k ar g av s

k an on s k a te gor iu l ob as , vinaidan is faqti, rom TavisTavad mimdinare process sistema gadahyavs meti Termodinamikuri albaTobis mdgomareobaSi, ar niSnavs, rom procesis sxva mimarTuleba SeuZlebelia. es niSnavs mxolod imas, rom sistemis miaxloeba wonasworobis mdgomareobasTan ufro albaTuria, vidre misgan daSoreba. amitom Termodinamikis II kanoni ase unda gavigoT: Tu sistemas ama Tu im mdgomareobaSi garkveuli entropia aqvs, maSin yvelaze ufro albaTuria is procesi, romelic iwvevs entropiis zrdas, Tumca principulad SesaZlebelia is procesic, romelic gamoiwvevs sistemis entropiis Semcirebas. Termodinamikis

kanonidan

gadaxris

mkafio

magaliTia

sxvadasxva

fluqtuacia (simkvrivis, energiis, temperaturisa da sxva fizikuri sidideebis). magaliTad, simkvrivis fluqtuacia aris nawilakebis Tanabari ganawilebis mdgomareobidan gadaxra, rasac axlavs Termodinamikuri albaTobisa da, maSasadame, entropiis Semcireba. sabolood, xazgasmiT aRvniSnoT Semdegi: marTalia, Termodinamikuri albaToba saSualebas gvaZlevs ganvsazRvroT realuri procesis mimarTuleba, magram am sididis gamoyeneba mouxerxebelia, vinaidan igi ar aris aditiuri sidide. magaliTad, Tu sistema Sedgeba ori nawilisagan, mTliani sistemis mdgomareobis

albaToba

tolia

Semadgeneli

nawilebis

albaTobaTa

namravlisa (da ara jamisa):

  1   2 . bevrad ufro xelsayrelia entropiis gamoyeneba. igi, iseve rogorc albaToba, axasiaTebs

procesis

mimarTulebas

marTlac,

vinaidan

namravlis

da,

amave

logariTmi

dros,

S  k ln   k ln 1   2   k ln 1  k ln  2 , S  S1  S 2 . 282

sididecaa.

ln 1   2   ln 1  ln  2 ,

bolcmanis (83.1) formulis Tanaxmad anu

aditiuri

amitom

ยง 84. hipoTeza `samyaros siTburi sikvdilis~ Sesaxeb me-19 saukunis meore naxevarSi, Termodinamikis meore kanonis aRmoCenasTan dakavSirebiT wamoiWra e.w. `samyaros siTburi sikvdilis~ problema. misi arsi Semdegia. Termodinamikis II kanonis Tanaxmad, izolirebul sistemaSi mimdinare yvela realuri procesi iwvevs sistemis entropiis zrdas. entropiis zrda grZeldeba sistemaSi Termodinamikuri wonasworobis damyarebamde. am momentidan entropia maqsimaluria da sistemaSi yovelgvari makroskopuli procesi Sewyvetilia. Termodinamikis erT-erTma Semqmnelma, germanelma fizikosma klauziusma (1822-1888) samyaro ganixila rogorc Caketili (izolirebuli) sistema da gaavrcela masze Termodinamikis II kanoni. am gavrcelebis Sedegebi man Semdegnairad Camoayaliba: `samyaros energia ucvleli rCeba, samyaros entropia miiswrafvis maqsimumisaken~. klauziusis am daskvnas, romelsac Semdgom mravalma idealistma mecnierma dauWira mxari, Soreul Sedegebamde mivyavarT. marTlac, Tu samyaros entropia ganuwyvetliv izrdeba, maSasadame meqanikuri, eleqtruli da sxva `maRalfasovani~ energia gardaiqmneba yvelaze ufro `naklebfasovan~ energiad _ siTbur energiad. amitom, marTalia, samyaros sruli energia rCeba ucvleli, magram adgili aqvs energiis gaufasurebas. siTburi energia Tbogamtarobisa da sxva procesebis gamo gaibneva samyaroSi da gamoiwvevs samyaros yvela sxeulis temperaturis gaTanabrebas. damyardeba sruli siTburi wonasworoba. SeuZlebeli iqneba siTburi manqanis muSaoba da saerTod, yovelgvari makroskopuli procesi samyaroSi Sewydeba, vinaidan aRar iarsebebs am procesis gamomwvevi mizezebi. samyaros aseT mdgomareobas, romelic faqtiurad mis dasasruls moaswavebs, ewoda `siTburi sikvdilis~ mdgomareoba. `samyaros siTburi sikvdilis~ reaqciuli Teoria mswrafl aitaca samRvdeloebam, raTa `mecnierulad daesabuTebina~ RmerTis arseboba. marTlac, Tu arsebobs samyaros dasasruli, maSasadame arsebobda misi dasawyisic, roca Seiqmna arawonasworuli mdgomareoba, saidanac samyaro TandaTan gadadis wonasworul

mdgomareobaSi.

magram,

vinaidan

arawonasworuli

mdgomareoba

(vTqvaT, temperaturis sxvaobis warmoeqmna) TavisTavad ar SeiZleba Seiqmnas, amitom igi odesRac unda warmoqmniliyo raRac zematerialuri Zalis, e.i. RmerTis mier. 283

`samyaros siTburi sikvdilis~ cru mecnieruli Teoria gamanadgureblad gaakritika f.engelsma. zogadfilosofiuri mosazrebebidan gamomdinare, man aCvena, rom es Teoria ewinaaRmdegeba bunebis ZiriTad kanons _ energiis gardaqmnisa da mudmivobis kanons. engelsi Tavis filosofiur naSromSi `bunebis dialeqtika~ werda: `moZraobis mospobis SeuZleblobis gamoxatulebas warmoadgens energiis gardaqmnisa da mudmivobis kanoni. es kanoni gagebuli unda iqnes ara marto raodenobrivad, aramed agreTve Tvisobrivadac. moZrav materias aqvs ganusazRvreli SesaZlebloba Tvisobriv gardaqmnebSi, romelmac araviTar SemTxvevaSi ar SeiZleba Zala dakargos. amitom bunebaSi energiis fantvasTan erTad unda warmoebdes da faqtiurad xdeba gabneuli siTbos gardaqmna sxva saxis energiaSi~. klauziusisa da misi mimdevrebis Secdoma isaa, rom maT entropiis zrdis principi gaavrceles usasrulod did sistemaze _ samyaroze, maSin roca Termodinamikis II kanoni dadgenil iqna mravali cdis safuZvelze mxolod

makroskopuli

sxeulisaTvis,

romelic

mTeli

samyaros

mcire

nawilia. Termodinamikis II kanonis meqanikuri gavrceleba mTel samyaroze ar SeiZleba, vinaidan es kanoni marTebulia izolirebuli sistemisaTvis, maSin rodesac mtkicdeba (fardobiTobis zogadi Teoriis safuZvelze), rom samyaro hgavs sxeuls, romelic mudmiv zemoqmedebas ganicdis, da, maSasadame, igi ar SeiZleba ganvixiloT rogorc izolirebuli sistema. amrigad, e nt r o p i i s c ne b is g am oy e ne b a m T e l i s am y aro s aT v i s u m arT e b u l oa , amitom umarTebuloa `sayaros siTburi sikvdilis~ Teoriac. marTlac, uaxlesi astrofizikuri monacemebi mowmobs, rom samyaroSi energiis gabnevasTan erTad adgili aqvs mis koncentraciasac. amis eqsperimentuli dadasturebaa galaqtikebis nislovanebebSi gabneuli materiidan axali varskvlavebis warmoqmna. marTalia, amJamad araa cnobili varskvlavTa warmoqmnisas mimdinare procesebis zusti suraTi, magram mecniereba ganuwyvetliv viTardeba da dabejiTebiT SeiZleba iTqvas, rom bunebis es saidumloebac adre Tu gvian amoixsneba. `samyaros

siTburi

sikvdilis~

hipoTezis

sapirispirod

molcmanma

gamoTqva e.w. f lu q tu ac iu r i h i p oT e za . am hipoTezis Tanaxmad bolcmani ar uaryofda Termodinamikis II kanonis marTebulobas mTeli samyaros masStabiTac, magram miuTiTebda, rom Termodinamikis II kanoni aris statistikuri 284

kanoni, romlis Tanaxmadac gadaxra Termodinamikuri wonasworobis mdgomareobidan,

anu

fluqtuacia,

ara Tu

SesaZlebeli,

aramed aucilebelicaa.

bolcmani Tvlida, rom samyaros amJamindeli arawonasworuli mdgomareoba sxva ara aris ra, Tu ara giganturi fluqtuacia. droTa ganmavlobaSi es fluqtuacia unda gaqres, e.i. samyaro unda mivides `siTburi sikvdilis~ mdgomareobamde, magram ramdenime xnis Semdeg kvlav warmoiqmneba giganturi fluqtuacia da samyaro TavisTavad gamova `siTburi sikvdilis~ mdgomareobidan da a.S. magram dro or gigantur fluqtuacias Soris gacilebiT metia TiToeuli maTganis arsebobis droze, amitom am hipoTezis Tanaxmadac samyaro `TiTqmis yovelTvis~ unda imyofebodes `siTburi sikvdilis~ mdgomareobaSi. amrigad,

Cven

vxedavT,

rom

bolcmanis

fluqtuaciuri

hipoTeza,

marTalia, radikalurad gansxvavdeba klauziusis hipoTezisagan, magram masac mivyavarT TiTqmis igive saboloo Sedegamde. aqedan daskvna _ miuxedavad Termodinamikis II kanonis statistikuri bunebisa, misi eqstrapolireba mTeli samyarosaTvis marTebuli ar aris. maSasadame, Termodinamikis II kanonis gamoyeneba SezRudulia ara marto qvemodan (calkeuli nawilakebi, nawilakTa mcire jgufisagan Sedgenili sistema), aramed zemodanac (samyaro mTlianad).

XI Tavi

m y a r i s xe u l i s T v i s e b e b i Ă&#x2030;Â§ 85. kristaluri da amorfuli sxeuli molekulur-kinetikuri Teoriis Tanaxmad yvela sxeuli Sedgeba Zalian didi raodenobis umciresi nawilakebisagan (ionebis, atomebis an molekulebisagan), romelTa Soris moqmedebs urTierTmizidvisa da ganzidvis Zalebi. imis mixedviT, Tu ra sididisaa es urTierTqmedebis Zalebi, nivTierebis agregatuli mdgomareoba SeiZleba iyos samgvari: myari, Txevadi an airadi. myar agregatul mdgomareobaSi nawilakebs Soris manZili Zalian mcirea, urTierTqmedebis Zalebi ki _ Zalian didi. es Zalebi nawilakebs gadataniTi moZraobis

saSualebas

aZlevs, amitom maT garkveuli mdebareoba ukaviaT 285

sivrceSi da asruleben mxolod siTbur rxevebs am (wonasworuli) mdebareobis

mimarT.

amiT

aixsneba,

rom

myar

sxeuls

gaaCnia

rogorc

sakuTari

moculoba, aseve sakuTari forma. arsebobs myari agregatuli mdgomareobis ori forma _ k ri s t alu r i da a m o r fu l i .

kristaluri sxeulebia: qvamarili, kvarci, almasi, yvela

liToni da sxv. amorfuli sxeulebia: mina, qarva, parafini da sxv. myari agregatuli mdgomareobis es ori forma mkveTrad gansxvavdeba erTmaneTisagan TavianTi aRnagobiT. kristalur myar sxeulSi nawilakebi sworad, kanonzomieradaa ganlagebuli da adgens e.w. kristalur mesers. k r is ta lu r i m e s e r i warmoadgens sivrcul bades, romlis kvanZebSic ganlagebulia kristalis Semadgeneli nawilakebi (atomebi, ionebi an molekulebi). aseTi meseri gamosaxulia 85.1 a suraTze. msxvili xazebiT gamoyofilia umciresi paralelepipedi, romlis Tanamimdevruli gadaadgilebiT x, y, z RerZis gaswvriv miiReba mTeli kristali. am paralelepipeds mesris elementaruli ujredi ewodeba. misi calsaxad daxasiaTebisaTvis saWiroa 6 sididis codna: sami wibosi ( a, b, c ) da RerZebs Soris sami kuTxisa (  ,  ,  ). am sidideebs m e s ri s

p ar am e tre b i

ewodeba

(sur. 85.1 b). a m o r fu l

myar

sxe-

ulSi nawilakebi sruliad uwesrigod, qaosuradaa ganlagebuli. amorfuli sxeulis aRnagoba arafriT gansxvavdeba misi warmomqmneli a

siTxis

b suraTi 85.1

aRnagobisagan,

ami-

tom amorful myar sxeuls

zogjer g a d a civ e b u l s iT xe s uwodeben. mkveTri gansxvaveba kristaluri da amorfuli sxeulebis aRnagobas Soris ganapirobebs maTi Tvisebebis mkveTr gansxvavebas: kristaluri myari sxeulis fizikuri Tvisebebi (meqanikuri, siTburi, eleqtruli, optikuri da sxv.) sxvadasxva mimarTulebiT sxvadasxvaa (amboben, rom kristaluri sxeuli a n iz ot r o piu li a ), amorfuli myari sxeulisa ki _ yvela mimarTulebiT 286

erTnairia (amorfuli sxeuli i z ot r o p i u l i a ). es advili misaxvedria, vinaidan kristalur myar sxeulSi sigrZis erTeulze AB da AC mimarTulebiT nawilakebis rao-

denoba

sxvadasxvaa

(sur. 85.2),

amorful

myar sxeulSi ki nawilakebis qaosuri ganlagebis gamo maTi raodenoba yvela mimarsur. 85.2

TulebiT erTnairia.

magaliTad,

ganvixiloT

Tbogamtarobis

movlena.

minis

kvarcis

firfita davfaroT parafinis Txeli feniT da SevexoT maT gaxurebuli nemsis wveriT.

firfitis

zedapiruli

fena

gaTbeba

yvela

mimarTulebiT

gamoiwvevs parafinis dnobas. amasTan, parafinis fenis gamdnari nawili kvarcis

firfitaze

formas

miiRebs

elifsis (sur.85.3a),

minisaze ki _ wrisas (sur. 85.3b). rom

sur. 85.3

cda

kristalur

gviCvenebs, sxeulSi

(kvarcSi) siTbo sxvadasxva

mimarTulebiT sxvadasxvanairad vrceldeba, amorful sxeulSi (minaSi) ki _ erTnairad. mkveTri gansxvaveba kristaluri da amorfuli sxeulis Tvisebebs Soris Tavs iCens dnobis procesSic. d n o b a ewodeba nivTierebis myari mdgomareobidan Txevad mdgomareobaSi gadasvlis process. temperaturas, romelzedac es procesi mimdinareobs, d n o b i s t e m p e r a tura

ewodeba. dnobis Sebrune-

bul process g a m y a r e b a ewode-

ba.

aRmoCnda,

rom

kristaluri

sxeuli dneba da myardeba mocemu-

Tdn

Tgamy

li nivTierebisaTvis erT garkve-

ul temperaturaze (mocemuli wne-

K II

dnobis aseTi garkveuli tempera-

D dro a

dro b

vis dros), amorful sxeuls ki tura ar gaaCnia _ temperaturis gazrdiT igi TandaTanobiT rbildeba da gadadis Txevad mdgomare-

sur. 85.4 85.4 sur.

287

obaSi. kristaluri da amorfuli

sxeulis dnobis mrudebi gamosaxulia 85.4 a suraTze (I _

kristaluri sxeulis, II ki _ amorfulisa). suraTidan Cans, rom kristaluri sxeulis temperatura siTbos raodenobis gadacemis Sedegad izrdeba mxolod dnobis dawyebamde. dnobis dawyebis momentidan ( B wertili) misi temperatura ucvleli rCeba garkveuli drois ganmavlobaSi _ sanam masa mTlianad ar gadneba ( C wertili), B wertils Seesabameba dnobis dawyeba, C wertils ki _ dnobis damTavreba.

BC ubani gamosaxavs orfaza mdgomareobas _ nivTierebis nawili Txevad mdgomareobaSia, nawili ki _ myarSi.

B -dan C wertilamde sxeuls siTbos raodenobas vawvdiT, misi temperatura ki ar izrdeba, radgan gadacemuli siTbos raodenoba xmardeba kristaluri mesris daSlas. es daSla iwyeba B wertilSi da mTavrdeba C wertilSi. siTbos raodenobis Semdgomi gadacema, cxadia, gamoiwvevs mdnaris temperaturis gazrdas. im siTbos raodenobas, romelic saWiroa dnobis temperaturamde miyvanili sxeulis gasadnobad, d n o b i s (faruli) s i T b o ewodeba. amorfuli sxeulis dnobis mrudze nacvlad mudmivi temperaturis Sesabamisi horizontaluri monakveTisa gvaqvs gadaRunvis K wertili. siTbos raodenobis miwodebiT amorfuli sxeulis temperatura ganuwyvetliv izrdeba (dnobis momentSi zrdis intensiurobis SemcirebiT), sxeuli TandaTan rbildeba da gadadis Txevad mdgomareobaSi. amorfuli sxeulis temperaturis aseTi zrda gasagebia, vinaidan ar arsebobs kristaluri meseri da mis dasaSlelad araviTari energia ar ixarjeba. amorfuli sxeulis SemTxvevaSi ar arsebobs temperaturis mkveTrad gansazRvruli zRvari, romlis zeviT nivTiereba iqneba Txevad mdgomareobaSi, qveviT ki _ myarSi. aq gamoiyofa mxolod temperatura, romelic Seesabameba gadaRunvis wertils dnobis mrudze. mas pirobiT d a r b i l e b i s t e m p e r a t u r a e w o d e b a . kristaluri da amorfuli sxeulis gamyarebis procesi mimdinareobs sapirispiro mimarTulebiT (sur. 85.4 b). mdnaridan siTbos raodenobis arinebisas amorfuli sxeulis SemTxvevaSi temperatura TandaTan mcirdeba da sxeuli SeumCnevlad gadadis myar mdgomareobaSi. kristaluri sxeulis SemTxvevaSi ki mruds aqvs mkveTrad gamoyofili safexuri _ BC horizontaluri ubani, romelic Seesabameba gamyarebis dawyebas ( B wertili) da damTavrebas ( C wertili).

temperaturis

mudmivoba

Semdegnairad

aixsneba:

kristalis

Casaxva

zrda

SesaZlebelia, Tu siTxis molekulas wavarTmevT energiis garkveul raodenobas. kristalSi molekula asrulebs mxolod rxeviT moZraobas, maSin, rodesac imave temperaturis mdnarSi molekula asrulebs gadataniTs moZraobasac. amitom cxadia, rom kristalizaciis dros nivTierebam unda gamoyos is siTbos raodenoba, romelic Seesabameba molekulebis gadataniT moZraobas. amrigad, kristaluri mesris Seqmnis procesSi gamoiyofa damatebiTi siTbos raodenoba. cdebi gviCvenebs, rom es siTbos raodenoba dnobis siTbos tolia. energiis STanTqmiT yinulis dnobisas da misi gamoyofiT wylis gayinvisas aixsneba daTbobisa da acivebis dagvianeba didi tbebisa da wyalsacavebis maxloblad.

288

bunebaSi arsebuli sxeulebis didi umravlesoba kristaluri sxeulebia, rac imiT aixsneba rom, rogorc gamoTvlebi gviCvenebs, nawilakebis wesier ganlagebas Seesabameba sistemis minimaluri energia. Cven ki viciT, rom yoveli sistema miiswrafvis minimaluri energiis mdgomareobisaken. Tu, miuxedavad amisa, zogierTi nivTiereba mainc myardeba amorfuli masis saxiT, es gamowveulia am nivTierebis Zalian didi siblantiT. maRali siblante (romelic temperaturis adgilebas, amorful

SemcirebiT

rac

mkveTrad

aucilebelia

sxeulebSi,

izrdeba)

maTi

marTalia

wesieri

Zalian

aZnelebs

nawilakebis

ganlagebisaTvis.

nela,

magram

mainc

gada-

mTel

rig

mimdinareobs

nawilakebis wesieri ganlagebis (`kristalizaciis~) procesi. amiT aixsneba cnobili faqti minis gamokristalebisa, roca minaSi warmoiqmneba mcire zomis kristalebi, romelTa sazRvrebzec xdeba sinaTlis mravaljeradi arekvla da gabneva, ris Sedegadac mina kargavs gamWvirvalobas. naTqvamidan cxadia, rom kristaluri mdgomareoba ufro mdgradia, vidre amorfuli. am ukanasknels xSirad metastabilur mdgomareobas uwodeben. amorfuli da kristaluri myari sxeulis garCevisas yuradReba unda mivaqcioT Semdeg garemoebas: SeiZleba makrosxeuli imyofebodes kristalur mdgomareobaSi

da,

miuxedavad

amisa,

savsebiT

izotropiuli

SemTxveva maSin gveqneba, roca makrosxeuli erTobliobaa kristalisa,

romelTa

ganlageba

savsebiT

qaosuria.

iyos.

aseTi

mravali mcire SemTxvevaSi

mas

p o lik r is t alu ri sxeuli ewodeba. polikristaluri sxeulebia, magaliTad, liTonebi, naxSiri da sxv. polikristaluri sxeulis Semadgeneli yoveli mcire kristali (e.w. kristaluri marcvali) anizotropiulia, mTeli sxeuli ki, am mcire kristalebis qaosuri ganlagebis gamo _ izotropiuli. SeiZleba sxeuli warmoadgendes erT mTlian kristals, anu, rogorc mas am SemTxvevaSi uwodeben, m o nok ri s t al s .

aseTi monokristalis saxiT bunebaSi gvxvdeba

mineralebi: almasi, kvarci, mTis broli da sxv. monokristalebi SeiZleba xelovnuradac miviRoT, risTvisac mravali sxvadasxva meTodi arsebobs. bolo wlebSi monokristalis gaumjobesebam

misaRebi aparaturisa da teqnologiuri procesis

SesaZlebeli

monokristalebis

miReba,

gaxada

ramac

sxvadasxva

xeli

289

Seuwyo

nivTierebis am

(didi

nivTierebis

zomis)

fizikuri

Tvisebebis

ufro

Rrmad

Seswavlas

SesaZlebeli

gaxada

maT

bazarze

mravali, metad saintereso Tanamedrove fizikuri xelsawyos Seqmna. myar sxeulSi nawilakebs Soris manZili daaxloebiT 1A –is tolia*, polikristaluri marcvlis zoma ki 10 3  10 4 A –is rigisaa. maSasadame, polikristalur

sxeulSic

gavrceldeba

ramdenime

nawilakebis

aTasi

mesris

wesieri,

regularuli

parametris

tol

ganlageba

manZilze.

amis

safuZvelze amboben, rom kristaluri sxeulisaTvis damaxasiaTebelia S o r i w e s r i g i nawilakebis wyobaSi. amitom k r is ta li

Se i Z le b a

ga ni m a r to s

r o gor c is e T i s x e u l i, rom e l Si c a d gi li aq v s s am g an zom il e b ia n S o r w e s r ig s m is i Se m a dge ne li na w i l ak e b i s g an la ge b a S i.

§86. kristalis bunebrivi forma da misi simetria. simetriis klasebi rogorc aRvniSneT, kristalis erT-erTi Tviseba, ganpirobebuli misi Semadgeneli nawilakebis kanonzomieri ganlagebiT, aris an iz otr o pi a , e.i. mimarTulebis Sesabamisad Tvisebebis Secvla. garda anizortopiisa, Semadgeneli nawilakebis mowesrigebuli ganlageba ganapirobebs kristaluri nivTierebis Semdeg ZiriTad Tvisebebs: erTgvarovnebas, TviTSemofargvlis unars da simetrias. e r T gv a rov ne b a.

kristali

erTgvarovani

sxeulia.

kristaluri

nivTierebis erTgvarovneba imiT gamoixateba, rom misi erTnairi formisa da orientaciis ori sxvadasxva ubani TvisebebiT erTnairia. T v iT Se m o f a rgv li s u n ar i . kristaluri nivTierebis mesame ZiriTadi Tvisebaa swori zedapirebiT TviTSemofargvlis, anu maravalwaxnagas saxiT warmoqmnis unari. kristali, romelic izrdeba raime xelSemSleli mizezis gareSe, iRebs amozneqili mravalwaxnagas formas. sxvadasxva nivTierebis kristalur formebze dakvirvebisas 1669 wels danielma kanoni,

mecnierma

romelic

stenonma

SemdegSi

(1638-1686)

aRmoaCina

mdgomareobs:

erTi

8

1 A  10 sm. 

290

e.w. da

kuTxeTa i m av e

mudmivobis

n iv T i e r e b i s

k r is t ale b s , im i s m i xe dv iT , T u r a pir ob e b S i m ox d a m a T i g a zr da , S e iZ le b a h q o nde s s u l s xv a d as xv a g a re ga ni s a xe , m a gr a m k u T x e e b i S e s ab am is w a xn a ge b s So ris m u dm iv ia.

sur. 86.2

sur. 86.1

sur. 86.3

es imas niSnavs, rom SeiZleba erTi da imave nivTierebis ori sxvadasxva kristalis orientireba ise, rom saTanado waxnagebi paraleluri iyos. 86.1 suraTze naCvenebia kvarcis sami kristali. Sesabamisi (paraleluri) waxnagebi erTi da imave asoebiTaa aRniSnuli. kristalis meoTxe mniSvnelovani Tvisebaa s im e tri a . gamoCenili rusi kristalografis e. feodorovis (1853-1917) moswrebuli gamoTqmiT `kristali brwyinavs Tavisi simetriiT~. kristalis

simetriis

ganxilva

daviwyoT

simetriuli

sxeulis

gansazRvriT. s xe u li

s im e tr iu l i a,

m is i

r aim e

g a da n acv le b iT

(mobrunebiT, brtyel sarkeSi arekvliT da a.S.) i g i T an xv de b a T av is T av s . davakvirdeT

mTis

brolis kristals

(sur. 86.2).

SevamCnevT,

rom

igi

simetriuli mravalwaxnagaa. misi mobrunebisas OO' RerZis garSemo 60 ď Ż â&#x20AC;&#x201C;iani kuTxiT

kristali

Tanxvdeba

Tavis

Tavs.

sruliad

msgavsia,

agreTve,

sibrtyis zeviT da qveviT arsebuli nawilebi. operacias, romlis saSualebiTac sxeuli Tanxvdeba Tavis Tavs, s im e t r iu l i g ar d a q m na ewodeba. yovel simetriul gardaqmnas Seesabameba raRac geometriuli saxe. am geometriul saxeebs s im e tr iis e le m e n t e b i ewodeba. gavecnoT mokled simetriis ZiriTad elementebs.

291

s im e tri is s ib r ty e . ganvixiloT

wesieri

xuT-

ABCDE

kuTxedi (sur. 86.3). xuTkuTxedis erT-erT

wverosa da mis O

centrze gavavloT nakvTis sibrtyis marTobuli BOB' sibrtye. davuSvaT,

sibrtye

orive

sur. 86.4

sur. 86.5

mxridan amrekli sarkea. advili misaxvedria, rom am sarkeSi arekvlis Sedegad nakvTis marjvena nawili gadava marcxenaSi da piriqiT ( C wertili gadava A – Si, A gadava C –Si da a.S.). cxadia, rom aseTi arekvlis Semdeg xuTkuTxedi ar Seicvleba. aseT SemTxvevaSi amboben, rom xuTkuTxedi simetriulia sibrtyeSi

arekvlis

mimarT.

TviT

BOB'

sibrtyes,

s i m e t ri is

BOB'

s ib r ty e

ewodeba. ganxilul nakvTs aqvs xuTi simetriis sibrtye, romlebic gadis xuTkuTxedis TiTeul wverosa da centrze ( BOB' , AOA' da a.S.). s im e tri is

R e rZ i.

ganvixiloT

igive

xuTkuTxedi.

mis

centrze

gavavloT suraTis sibrtyis marTobuli O RerZi da miviCnioT igi brunvis RerZad. cxadia, am RerZis irgvliv xuTkuTxedis

2 5

kuTxiT Semobrunebisas

nakvTi Tavis Tavs Tanxvdeba _ misi saxe ucvleli darCeba. am SemTxvevaSi amboben, rom nakvTi simetriulia O RerZis irgvliv brunvis mimarT. O RerZs simetriis RerZi ewodeba. amrigad, s i m e tr iis R e rZ i ewodeba RerZs, romlis irgvliv garkveuli kuTxiT mobrunebis Sedegad nakvTi Tavis Tavs Tanxvdeba. nakvTis SeTavsebaTa raodenobas misi 3600-iT Semobrunebis dros R e rZ is r i g i ewodeba. Cvens SemTxvevaSi saqme gvaqvs me-5 rigis simetriis RerZTan, vinaidan xuTkuTxedis 3600-iT Semobrunebisas igi Tavis Tavs xuTjer Tanxvdeba. ganxilul nakvTs, garda me-5 rigis simetriis RerZisa, gaaCnia xuTi meore rigis simetriis RerZi. es RerZebi mdebareobs suraTis sibrtyeSi da gadis xuTkuTxedis wverosa da centrze ( BOB' , AOA' da a.S.). s im e tri is

c e n tr i .

iseT

simetriul

gardaqmnas,

romlis

drosac

romelime wertilSi arekvlis Sedegad nakvTi Tavis TavSi gadadis, i n v e r s i a ewodeba. TviT am wertils, romelSic inversiis Sedegad nakvTi Tavis Tavs Tanxvdeba _ s im e tr iis c e n tri ewodeba. 292

ganvixiloT magaliTad, ABCD nakvTi (sur. 86.4). am nakvTis geometriuli centri _ O wertili simetriis centria, vinaidan O wertilSi arekvlis Sedegad nakvTi Tavis TavSi gadadis ( A wertili gadadis C –Si, B gadadis D –Si, K  K ' -Si da a.S.). kristalSi, romelsac gaaCnia simetriis centri, mopirdapi-

re waxnagebi wyvil-wyvilad erTmaneTis tolia. 86.5 suraTze naCvenebia erTi aseTi

kristali

(TvalsaCinoebisaTvis

erTi

wyvili

inversiulad

toli

waxnagebisa daStrixulia). arsebobs simetriis sxva elementebic (magaliTad, inversiuli RerZi, sarkuli RerZi* da sxv.), magram yvela isini (garda, meoTxe rigis inversiuli RerZisa) daiyvaneba ukve ganxilul simetriis elementebamde, amitom maT calke ar

ganvixilavT.

ufro

metic,

simetriis

RerZic

simetriis

centric

SeiZleba Seicvalos simetriis ramdenime sibrtyis kombinaciiT, magram meti TvalsaCinoebisaTvis iyeneben RerZsac da centrsac. geometriuli nakvTebis, maT Soris kristalebis, simetria gamoisaxeba maTi simetriis elementebis erTobliobiT. m oc e m u li n ak v T is s im e tr ii s e le m e nt e b is s ru l e rT ob l iob as n ak v T is s im e t ri is s ax e an s im e tr ii s k la s i e w o de b a . geometriuli nakvTebis simetriis saxe uamravia, magram ara usasrulod bevri, rac imis Sedegia, rom nakvTis simetriis elementebis kombinacia nebismieri ar SeiZleba iyos (amboben, rom adgili aqvs simetriis elementebis SeuTavsebadobas). magaliTad, SeiZleba damtkicdes, rom SeuZlebelia iseTi nakvTis arseboba, romelsac erTdroulad eqneba meoTxe da meeqvse rigis simetriis RerZebi. kristalebSi simetriis elementebis SesaZlo SeTavsebaTa (kombinaciaTa) ricxvi, an rac igivea, simetriis klasebis raodenoba kidev ufro SezRudulia, vinaidan geometriuli nakvTebisagan gansxvavebiT (romelTaTvisac SesaZloa nebismieri rigis simetriis RerZebis arseboba pirvelidan ∞–mde), kristalebs ar SeiZleba gaaCndeT me-5 rigis da me-6-ze maRali rigis simetriis RerZi. (es

inversiuli RerZi ewodeba iseT RerZs, romlis irgvliv garkveuli kuTxiT mobruneba da

Semdgomi inversia amave RerZze mdebare wertilSi iwvevs sxeulis Tavis TavTan Tanxvedras. sarkul RerZSi wertilSi arekvla Secvlilia sibrtyeSi arekvliT.

293

movlena dakavSirebulia kristalis Sinagan agebulebasTan da Cven mas ar ganvixilavT). garegnulad sruliad sxvadasxva formis kristalebs SeiZleba hqondes simetriis elementebis erTi da igive erToblioba da maSasadame, ekuTvnodnen erTsa da imave simetriis klass. magaliTad, kubi (aseTi formis kristali aqvs sufris marils _ NaCl) da, oqtaedri (aseTi formis kristali aqvs magnetits _ Fe2O3)

miekuTvneba

erTsa

imave

klass,

radgan

maT

aqvT

simetriis

elementebis erTi da igive erToblioba. germanelma mecnierma geselma da rusma mecnierma gadolinma daadgines, rom bunebaSi SeiZleba arsebobdes mxolod 32 simetriis klasi. es klasebi iyofa 7 kristalografiul sistemad (singoniad)*. TiToeul sistemaSi erTiandeba is klasebi, romelTac erTnairi simetriis RerZebi aqvs: 1. t rik li nu r i

s i s te m a . am sistemaSi Semaval klasebs arc erTi

simetriis RerZi ara aqvs. triklinur sistemaSi Sedis ori klasi. am sistemis kristalis elementaruli ujredi gamosaxulia 86.6 suraTze. 2. m on ok l i nuri

s is t e m a .

sistemaSi

Semaval

klasebs mxolod erTi meore rigis simetriis an inversiuli RerZi aqvs (sami klasi).

sistemis

kristalis elementa-

sur. 86.6

sur. 86.7

ruli ujredi gamosaxulia 86.7 suraTze. 3. r om b u li s i s te m a . am sistemaSi Semaval klasebs mxolod meore rigis simetriis an inversiuli RerZi aqvs, amasTan, monoklinuri sistemisagan

termini `singonia~ pirvelad Semoitana frangma mecnierma s o r e m 1893 wels da sityvasityviT

niSnavs `msgavskuTxas~. amiT xazgasmulia, rom mocemul singoniaSi erTiandeba is kristalebi, romelTac erTnairi maxasiaTebeli kuTxeebi aqvT.

294

gansxvavebiT, maTi ricxvi erTze metia (sami klasi). elementaruli ujredi gamosaxulia 86.8 suraTze. 4. t ri g onu ri

(romboedruli)

sistema.

sistemaSi

Semavali

klasebisaTvis damaxasiaTebelia erTi mesame rigis simetriis RerZis arseboba (xuTi klasi). am sistemis kristalis elementaruli ujredi gamosaxulia 86.9 suraTze.

sur. 86.8

5. t e tr a go nu ri

sur. 86.9

s i s te m a .

sistemaSi

sur. 86.10

erTiandeba

klasebi,

romelTac erTi meoTxe rigis simetriis an inversiuli RerZi aqvs (Svidi klasi). am sistemis kristalis elementaruli ujredi gamosaxulia 86.10 suraTze. 6. h e q s ag onu ri s i s te m a . am sistemaSi erTiandeba is klasebi, romelTac erTi meeqvse rigis simetriis an inversiuli RerZi aqvs (Svidi klasi). am sistemis kristalebis elementaruli ujredi gamosaxulia 86.11 suraTze*. 7. k u b u ri s i s te m a . am sistemaSi Sedis is klasebi, romelTac sami (erTmaneTis marTobi) meoTxe rigis simetriis RerZi aqvs (xuTi klasi). am sistemis kristalis elementaruli ujredi gamosaxulia 86.12 suraTze. zemoT CamoTvlili Svidive elementaruli ujredi Seicavs rva-rva atoms, romlebic paralelepipedis rva wveroSia moTavsebuli. vinaidan TiToeuli wvero-kvanZi erTdroulad ekuTvnis rva elementarul ujreds, amitom TiToeul ujredze modis erTi atomi. aseT elementarul ujreds (romlis SigniT kvanZebi ar aris) primitiuli an martivi ujredi *

heqsagonuri sistemis elementaruli ujredic paralelepipedia (romelic 35.11 suraTze sqeli

wiriTaa gamoyofili). magram mesris simetrias ufro srulad da TvalsaCinod gamosaxavs eqsvwaxnaga prizma, romelic sami aseTi paralelepipedis SeerTebiT miiReba.

295

ewodeba. Tu elementaruli ujredi wveroebs garda sxva adgilebSic Seicavs kvanZebs, mas rTuli ewodeba. rTuli ujredi SeiZleba iyos moculobadacentrili, waxnagdacentrili, da fuZedacentrili.

moculobadacentrili

elementaruli

ujredi

kvanZebs

wveroebs

garda

Seicavs agreTve paralelepipedis moculobis centrSi. cxadia, rom aseT ujredze modis ori atomi.

waxnagdacentril

waxnagebis

centrebSi,

elementarul

xolo

ujredSi

fuZedacentrilSi

kvanZebi _

wveroebs

fuZeebis

(zeda

garda da

moTavsebulia

qveda

waxnagebis)

centrebSi.

sur. 86.11

ramdeni

rTuli

sur. 86.12

elementaruli

ujredi

SeiZleba

arsebobdes

bunebaSi?

cxadia,

elementaruli ujredi ar SeiZleba iyos nebismieri. igi unda iyos iseTi,

rTuli

rom Tavisi

simetriis elementebis erTobliobiT aucileblad Tavsdebodes erT-erT kristalografiul sistemaSi. magaliTad, ganvixiloT kuburi sistemis elementaruli ujredi. igi warmoadgens martiv kubs. movaTavsoT mis centrSi nawilaki. miviRebT rTul, moculobadacentril ujreds, romelsac iseTive simetria aqvs, rogoric martiv ujreds. Tu nawilakebs movaTavsebT kubis waxnagebis centrebSi, miviRebT rTul, waxnagdacentril elementarul ujreds, romelsac agreTve gaaCnia am sistemisaTvis damaxasiaTebeli simetria. amrigad, kubur sistemaSi sami elementaruli ujredi gvaqvs: erTi martivi da ori rTuli. kubis sxva adgilebSi nawilakebis damateba ar SeiZleba, vinaidan es mkveTrad Secvlis mesris (ujredis) simetrias. analogiuri msjelobis Catareba sxva sistemebis elementaruli ujredebis mimarT gvaZlevs, rom bunebaSi 7 martivi elementaruli ujredis garda arsebobs amdenive rTuli elementaruli ujredi. e.i. bunebaSi SeiZleba arsebobdes mxolod 14 sxvadasxva saxis elementaruli ujredi (meseri). es Sedegi pirvelad 1855 wels miiRo frangma kristalografma b r a v e m (1811-1863) da amitom maT b r a v e s m e s r e b i ewodeba. 86.13 suraTze gamosaxulia braves 14-ve meseri. ar

unda

vifiqroT,

rom

braves

erTi

meseri

yovelTvis

amowuravs

mocemuli

kristalis yvela atoms (ions, molekulas). rTuli struqturis kristalis meseri braves ramdenime, erTmaneTSi Cadgmuli mesris erTobliobaa.

296

sur. 86.13

297

§87. kristaluri mesris fizkiuri tipebi. wina paragrafSi movaxdineT kristalebis klasifikacia simetriis elementebis mixedviT. aseTi klasifikacia, marTalia, warmodgenas gvaZlevs kristalis geometriul maxasiaTeblebze, magram ar ixilavs sakiTxs im Zalebis bunebis

Sesaxeb,

romliTac

erTmaneTTan

dakavSirebulia

myari

sxeulis

nawilakebi. kristalebis klasifikacia SeiZleba movaxdinoT sxva principis mixedviTac, saxeldobr, nawilakTa m ak av S ire b e li Z ale b is (bmis Zalebis) bunebis mixedviT. am TvalsazrisiT bunebaSi arsebuli kristalebi (kristaluri mesrebi) iyofa oTx ZiriTad tipad (klasad). esenia ionuri, atomuri, liTonuri da molekuluri kristalebi. i on u r i , anu h e te ro po l aru l i kristalis kvanZebSi moTavsebulia dadebiTi da uaryofiTi ionebi. aseTia, magaliTad sufris marilis (NaCl) kristali (sur. 87.1). kavSiri dadebiT da uaryofiT ionebs Soris xorcieldeba kulonis eleqtrostatikuri

mizidulobis

ZaliT.

aseT sur. 87.1

kavSirs i onu ri k av S ir i ewodeba. kose lis

(1888-1956) Teoriis Tanaxmad, yoveli atomi qimiur naerTSi miiswrafvis

hqondes uaxloesi inertuli elementis eleqtronuli garsis msgavsi garsi. Na–is atoms gaaCnia 11 eleqtroni, qlorisas ki _ 17. natriumis atomis gare Sreze aris erTi savalento eleqtroni, qlorisaze ki _ Svidi. amrigad, imisaTvis, rom am atomebis gare eleqtrinuli Sre gaxdes inertuli airis atomis gare eleqtronuli Sris msgavsi (e.i. iyos mTlianad Sevsebuli da, maSasadame, mdgradi), saWiroa, natriumma gasces, xolo qlorma miierTos erTi eleqtroni. Na–isa da Cl–is atomebis urTierTdaaxloebisas asec xdeba: natriumi gadaiqceva dadebiT ionad,

qlori

uaryofiTad

maT

Soris

warmoiqmneba

ionuri

kavSiri.

advili

misaxvedria, rom ionuri kavSiri SeiZleba ganxorcieldes mxolod sxvadasxva nivTierebis atomebs Soris, amitom mas xSirad h e t e r o p o l a r u l kavSirs uwodeben (berZnuli sityvidan `hetero~ _ sxvadasxva).

a t om u r i,

anu h om e o p ol aru li

kristalis kvanZebSi moTavsebulia

neitraluri atomebi. aseTia almasis, germaniumis, siliciumis da sxva nivTierebis kristalebi. atomur kristalSi yoveli ori mezobeli atomi erTmaneT298

Tan dakavSirebulia e.w. k ov ale ntu r i

k av Si riT , romelic xorcieldeba

ganzogadebuli (saerTo) eleqtronebis wyviliT. am kavSiris warmoqmnisaTvis yoveli urTierTqmedi atomi gascems TiTo eleqtrons eleqtronuli wyvilis Sesaqmnelad, romelic erTdroulad ekuTvnis orive atoms. kovalenturi ganzogadebuli,

kavSiri

ewodeba

sawinaaRmdego spinebis

iseT mqone

kavSirs,

romelic

eleqtronebis

ganxorcielebulia

wyviliT. kvanturi

meqanikis

TvalsazrisiT antiparaleluri spinebis mqone ori eleqtronis yofna ori atombirTvis velSi energetikulad xelsayrelia. amitom, aseTi kavSiris warmoqmnisas adgili aqvs energiis mniSvnelovan gamoyofas, rac ganapirobebs kavSiris maRal simtkices. rogorc vxedavT, kovalentur kavSirs safuZvlad udevs gacvliTi efeqti (atomebis mier eleqtronebis gacvla), romelsac wminda kvanturi buneba aqvs. kovalenturi kavSiris warmoqmnac (iseve rogorc ionuris) Sedegia imisa, rom atomi miiswrafvis hqondes 8 an 18eleqtroniani

(inertuli

elementis

msgavsi)

mdgradi

garsi.

ionuri

kavSiris

dros

xorcieldeba Warbi (8-ze an 18-ze meti) savalento eleqtronebis gacemiT, an gareTa garsze arasakmarisi (rvamde) eleqtronebis miRebiT. kovalenturi kavSiris dros ki _ eleqtronebis erTi (an ramdenime) wyviliT erTdrouli `sargeblobis~ Sedegad. mdgradi eleqtronuli garsebis miRwevis es saSualeba damaxasiaTebelia, magaliTad, halogenebis molekulebisaTvis, an gogirdis, selenis da sxva molekulebisaTvis (sur.87.2 a, b). rogorc suraTidan Cans, qloris molekulaSi kavSiri atomebs Soris xorcieldeba erTi eleqtronuli gogirdis

wyviliT,

molekulaSi

ki _ oriT. ganvixiloT

almasis

eleqtroni

da srul oqtetamde aklia kidev oTxi

eleq-

troni. maSasadame, naxSirbadis atoms SeuZlia oTxi kovalenturi kavSiris warmoqmna, e.i. oTx

sur. 87.2

sxva atomTan dakavSireba. rogorc suraTidan Cans, almasSi TiToeuli atomi dakavSirebulia oTx mezobel atomTan, romlebic ganlagebulia wesieri tetraedris wveroebSi. TiToeuli kavSiri

ganxorcielebulia

ori

ganzogadebuli

kavSirSi myofi erTi atomi, meores ki _ meore.

299

eleqtroniT,

romelTagan

erTs

iZleva

advili misaxvedria, rom kovalenturi kavSiris ganxorcieleba damaxasiaTebelia erTnairi (erTi da imave) atomebisaTvis, amitom am kavSirs xSirad h o m e o p o l a r u l kavSirs uwodeben (`homeo~ _ msgavsi).

l iT onu ri kristalis kvanZebSi moTavsebulia dadebiTad damuxtuli ionebi, romelTa Soris Tavisuflad moZraobs airis kanonebs daqvemdebarebuli eleqtronebi _ `eleqtronuli airi~. dadebiTi ionebi da `Tavisufali~ eleqtronebi warmoiqmneba liTonis orTqlis (neitraluri atomebis) myar sxeulad erTmaneTs,

kondensirebis rom

procesSi.

urTierTqmedebis

dros

Sedegad

atomebi

xdeba

maTi

imdenad

uaxlovdeba

ionizacia

isini

kargaven savalento eleqtronebs. amis gamo atomebi gadaiqcevian dadebiT ionebad da Tavsdebian mesris kvanZebSi, `ganTavisuflebuli~ eleqtronebi ki moZraoben kvanZebs Soris sivrceSi. `Tavisufali~ eleqtroni kargavs individualurobas, igi konkretul atomTan dakavSirebuli ar aris, `koleqtivizebulia~. Tavisufali eleqtronebis erToblioba, anu `eleqtronuli airi~ erTmaneTTan akavSirebs dadebiT ionebs, `aduRabebs~ maT, romlebic winaaRmdeg SemTxvevaSi erTmaneTisagan ganizidebodnen. kavSirs, romelic ganxorcielebulia eleqtronuli airis saSualebiT, liT o nu r i k av S ir i ewodeba. zemoaRniSnulidan gamomdinareobs, rom Tavisufali eleqtronebis ricxvi liTonSi Zalian didia da es ricxvi ar aris damokidebuli gareSe faqtorebze (ganaTeba, temperatura, wneva da sxv.), rac ganapirobebs liTonis maRal eleqtro- da Tbogamtarobas. rogorc vxedavT, liTonuri kavSiri waagavs kovalentur kavSirs, vinaidan orives safuZvlad udevs savalento eleqtronebis ganzogadeba, `koleqtivizeba~. magram kovalenturi kavSiris dros eleqtronebis ganzogadebaSi monawileobas iRebs atomebis uaxloesi wyvilebi da

ganzogaddeba

mxolod

eleqtronebis

wyvilebi.

eleqtronebi

yovelTvis

imyofeba

mocemul atomebs Soris. liTonuri kavSiris dros ki savalento eleqtronebis ganzogadebaSi monawileobs kristalis yvela atomi da ganzogadebuli eleqtronebi TavianT atomebTan ki araa lokalizebuli, aramed Tavisuflad moZraoben kristalis mTel moculobaSi.

m ole k u lu r i kristalis kvanZebSi moTavsebulia neitraluri molekulebi, romlebic inarCuneben individualobas ara marto airad, aramed Txevad da myar mdgomareobebSic (H2, N2, Cl2, I2, CH4, CO2, H2O). molekulis Semadgeneli atomebi erTmaneTTan dakavSirebulia kovalenturi bmiT, molekulebs Soris 300

kavSiri ki ganxorcielebulia n a rC e n i , anu v an - de r -v a a ls i s ZalebiT. am Zalebis buneba daiyvaneba molekuluri dipolebis urTierTqmedebamde. pirvelad es Zalebi SemoRebul iqna idealuri airis mdgomareobis gantolebidan realuri airis gadaxvevis asaxsnelad. aqedan warmodgeba maTi dasaxeleba. vander-vaalsis gantolebaSi

   p  a V  b   RT  V2   es Zalebi gansazRvravs

a Sesworebas. b Sesworeba gansazRvravs ganzidvis V2

Zalebs. inertuli airis SemTxvevaSi van-der-vaalsis Zala erTaderTi Zalaa, romlis

moqmedebac

SesaZlebels

xdis

nivTierebis

myari

Txevadi

mdgomareobebis arsebobas. van-der-vaalsis ZalebiT dakavSirebul molekulebs Soris asxvaveben urTierTqmedebis sam saxes: orie ntaci uls, da

sur. 87.3

in duqciurs

di sp ersi uls.

orientaciul

urTierTqmedebas adgili aqvs arasi-

metriul (e.w. polarul) molekulebs Soris, romelTac amis gamo gaaCniaT dipoluri momenti. induqciur

urTierTqmedebas

adgili

aqvs

simetriul

(arapolarul)

molekulebs

Soris,

romelTac Cveulebriv pirobebSi dipoluri momenti ar gaaCniaT, magram daaxloebisa da urTierTqmedebis Sedegad iZenen mas. dispersiuli urTierTqmedebac aRiZvreba simetriul molekulebs Soris, romlebic daaxloebis Sedegad ar polarizdebian, magram maT Soris mizidvis Zala aRiZvreba eleqtronebis `SeTanxmebuli~ moZraobis Sedegad (sur. 87.3). molekuluri mier

kavSiri

yvelaze

ganxilul

kavSirebs

sustia

Cven

Soris.

gansakuTrebiT

sustia

kavSiri

arapolarul molekulebs Soris (aseTebia inertuli airis erTatomiani molekulebi, agreTve

wyalbadis,

Jangbadis,

azotis,

qloris da sxv. oratomiani molekulebi, parafinebisa da mTeli rigi sxva organuli nivTierebebis mravalatomiani molekulebi), amitom aseTi molekulebisagan Sedgenili

sur. 87.4

301

kristalebi, Cveulebriv, mcire simtkicisaa da dneba dabal temperaturaze. SedarebiT didia urTierTqmedebis Zala polarul molekulebs Soris, amitom, aseTi molekulebisagan Sedgenili kristali ufro mtkicea. aseT kristalebs miekuTvneba, magaliTad, yinulis (H2O) kristali, romelsac aqvs aJuruli (faSari) struqtura. molekulaSi Semavali

Jangbadis

atomi

(uaryofiTi

ioni)

wyalbadis

atomebis

(dadebiTi

ionebis)

saSualebiT uerTdeba Jangbadis oTx sxva atoms (sur. 87.4 a). amrigad, yinulis kristalur meserSi wylis yoveli molekula kavSirSia oTx mezobel molekulasTan. yinulis kristalis aJuruli struqturis modeli naCvenebia 87.4 b suraTze, sadac simartivisaTvis Jangbadis (TerTi burTulebi) da wyalbadis (Savi burTulebi) molekulebi erTnairi zomisaa. suraTidan Cans, rom mTel kristals gahyveba sakmaod farTe arxebi _ carieli adgilebi. yinulis aJuruli struqturiT aixsneba misi naklebi simkvrive wyalTan SedarebiT da misi dnoba maRali wnevis pirobebSi, roca es struqtura irRveva. Cven movaxdineT kristalebis klasifikacia makavSirebeli Zalebis xasiaTis mixedviT. aseTi klasifikacia pirobiTia, vinaidan xSirad nivTierebis nawilakebs Soris kavSiris xasiaTi gamokveTili ar aris, an kavSiri Sualeduria, kombinirebuli. magaliTad, grafitis struqturaSi (sur. 87.5) erTdroulad sami tipis bma moqmedebs. rogorc suraTidan Cans, grafits aqvs fenovani struqtura*. naxSirbadis atomebs Soris TiToeul fenaSi moqmedebs ori

tipis

bma:

sami

eleqtroni

qmnis

mocemuli

fenis

sibrtyeSi sam simetriulad ganlagebul kovalentur bmas, meoTxe eleqtroni ki ganzogadebulia fenis farglebSi, e.i. qmnis

liTonur

kavSirs.

mezobeli

fenebi

erTmaneTTan

dakavSirebulia van-der-vaalsis ZalebiT. fenebs Soris susti molekuluri sriali

kavSiris

erTmaneTis

Sedegia

mimarT.

maTi

grafitis

metad gaadvilebuli meserSi

sxvadasxva

tipis bmebis arseboba ganapirobebs am nivTierebis saocari Tvisebebis Serwymas: mas aqvs Zalian dabali sisale (fenebs Soris

gaadvilebuli

srialis

gamo)

Zalian

maRali

ď Ż

sur. 87.5

(~ 4000 C ) dnobis temperatura.

mesris tipi ganisazRvreba ara marto nawilakebs Soris urTierTqmedebis xasiaTiT, aramed siTburi moZraobis intensivobiTac. es ukanaskneli sxvadasxva gareSe pirobebis (temperatura, wneva) dros sxvadasxvaa, amitom erTi da igive nivTiereba sxvadasxva pirobebSi sxvadasxva struqturad kristaldeba. am movlenas po lim or f izm i ewodeba. e.i. p ol im o r fi zm i e w o de b a m oc e m u l i niv T ie re b i s s xv a d as xv a k r is t a lu r f or m a S i y o fn is u n ar s . amis *

fenovani, anu Sreebrivi struqtura ewodeba iseT struqturas, sadac mocemul fenaSi

atomebs Soris manZili gacilebiT naklebia fenebs Soris manZilze.

302

TvalsaCino

magaliTia

naxSirbadis

ori

kristaluri

forma:

almasi

grafiti. am formebidan TiToeuls po li m or fu li m od i fik a ci a ewodeba. calkeuli modifikaciebi xSirad Tavisi atomuri aRnagobiT da fizikuri TvisebebiT Zlier gansxvavdeba erTmaneTisagan. magaliTad, almasi yvelaze sali

mineralia,

mas

iyeneben

minis

saWrelad,

saxexi

qargolebis

gasasworeblad da sxv. grafiti ki erT-erTi yvelaze ufro rbili masalaa. misgan, magaliTad, mzaddeba fanqris guli. almasi gamWvirvalea, grafiti ki Savi ferisaa, gaumWviria. almasis simkvrivea 3,5 g/sm3, grafitisa ki _ 2,1 g/sm3. almasi kargi dieleqtrikia, grafiti ki dens atarebs. almasis kristali msxvrevadia, grafitisa ki Runvadi. polimorfizmis movlena Zlier gavrcelebulia. garkveul pirobebSi SeiZleba miRebul iqnes TiTqmis yvela nivTierebis polimorfuli modifikacia.

§88. myar sxeulSi nawilakebis siTburi moZraobis xasiaTi nawilakebis

kanonzomieri

ganlageba

ganapirobebs

kristaluri myari

sxeulis sxvadasxvagvar fizkiur Tvisebebs _ meqanikurs, siTburs, eleqtruls, magniturs da optikurs. kursis mocemul nawilSi mokled ganvixilavT myari sxeulis siTbur Tvisebebs. myari

sxeulis

nawilakebis

siTburi

moZraobis

xasiaTi

arsebiTad

gansxvavdeba airis molekulebis siTburi moZraobisagan. myari sxeulis nawilakebs gadataniTi moZraobis Sesruleba ar SeuZlia, isini mxolod irxevian wonasworobis mdebareobis (kristaluri mesris kvanZebis) maxloblad. radgan nivTiereba myar mdgomareobaSi SeiZleba iyos sakmaod dabal temperaturaze, roca siTburi rxevebis energia, e.i.

kT , gacilebiT naklebia nawilakebis

urTierTqmedebis potenciur energiaze, amitom am rxevebis amplituda metad mcirea da, Cveulebriv, ar aRemateba 0,1A –s, rac atomebs Soris wonasworuli manZilis 5-7%-ia. nawilakebis rxeviTi moZraobis energia warmoadgens swored myari

sxeulis

Sinagan

energias,

xolo

gansazRvravs sxeulis temperaturas.

303

misi

saSualo

mniSvneloba

temperaturis SemcirebiT nawilakebis rxevis intensiuoba klebulobs da sxeulis

Sinagani

energia

mcirdeba.

magram

rxeviTi

moZraobis

kvanturi

Teoriis Tanaxmad, roca T ď&#x201A;Ž 0 , kristaluri myari sxeulis energia miiswrafvis ara nulisaken, aramed energiis garkveuli U 0 mniSvnelobisaken, romelsac n u l ov a ni e ne r gi a ewodeba. am energiis warTmeva nawilakisaTvis araviTar pirobebSi ar SeiZleba. misi sidide miT metia rac mcirea nawilakis masa. nulovani energiis arseboba ar ewinaaRmdegeba imas, rom roca T ď&#x201A;Ž 0 , sistemis entropiac miiswrafvis nulisaken, vinaidan es energia ganpirobebulia e.w. n u l ov a ni

r xe v e b iT

aris

dakavSirebuli

siTbur

moZraobasTan.

entropia ki ganisazRvreba sistemis nawilakebis uwesrigo, siTburi moZraobiT. amitom

absolutur

nulze,

rodesac

siTbur

moZraobas

adgili

aqvs,

idealuri (e.i. uminarevo da udefeqto) kristali savsebiT mowesrigebuli sistemaa da misi entropia nulis tolia. Cveulebriv, nulovani energia gacilebiT naklebia nawilakebis urTierTqmedebis energiaze. ase rom, igi ar uSlis xels nivTierebis gamyarebas. mxolod erTaderTi nivTierebis _ heliumis SemTxvevaSi aRwevs nulovani energia mniSvnelovan sidides. amitomaa heliumis gamyareba Zalian Zneli da amisaTvis

Zalian

maRali

wnevaa

saWiro.

yvela

sxva

nivTiereba

myardeba

gacilebiT ufro adre, vidre nulovani energia maT Tvisebebze gavlenis moxdenas daiwyebdes. siTburi rxevebis xasiaTi metad rTulia, vinaidan nawilakebi erTmaneTTan dakavSirebulia drekadi

ZalebiT.

Zalebis

warmosadgenad

xSirad sargebloben gamartivebuli modeliT

nawilakebs ganixilaven rogorc birTvebs, romlebic SeerTebulia daWimuli rezinis zonriT da SekumSuli

drekadi

zambariT.

zonari

birTvebs

sur. 87.1

miizidavs, zambara ki _ ganizidavs. aseTi gamartivebuli kavSiris organzomilebiani modeli gamosaxulia 87.1 suraTze. sinamdvileSi, suraTi gacilebiT rTulia, vinaidan Zalebis bunebaa ufro rTuli. garda amisa, kristalis mesers sami ganzomileba aqvs. manZili nawilakebs Soris sxvadasxva mimarTulebiT sxvadasxvaa, rac iwvevs kavSiris Zalebis anizotropiulobas da sxva. magram am gamartivebuli modeliTac naTlad Cans, rom nawilakebis rxevebi ar SeiZleba erTmaneTisagan damoukidebeli iyos, rom es rxevebi urTierTkavSirSia. erTi romelime nawilakis rxeva TandaTan gadaecema sxva nawi-

304

lakebsac. amasTan cxadia, rom Tu temperaturis gazrdis Sedegad am nawilakis rxevis amplituda gaizarda, danarCeni nawilakebis rxevis amplitudebic gaizrdeba. aseTnairad xsnis klasikuri

Teoria

eleqtronuli

aragamtari

Tbogamtaroba,

myari

sxeulis

romelic

Tbogamtarobas,

ganpirobebulia

(liTonSi

siTbos

gadataniT

mas

emateba

Tavisufali

eleqtronebis mier). rxevebis gadacema kristalSi warmoqmnis drekad talRas. am talRis meqanizmi kristalSi analogiuria bgeriTi talRis meqanizmisa da amitom mas akustikuri talRa ewodeba. Tbogamtarobis klasikurma Teoriam Semdgomi ganviTareba hpova ingliseli mecnieris p a i e r l s i s (1907) SromebSi, romelmac dauSva, rom drekadi siTburi talRebis (`normaluri rxevebis~) energia, eleqtromagnituri talRis energiis msgavsad, ikvanteba: talRis energia nebismieri ki ar aris, aramed energiis garkveuli porciebis _ siTburi kvantebis mTeli ricxvis tolia. siTbur kvantebs f o n o n e b i ewodeba (amitom Tbogamtarobas, ganpirobebuls siTbos gadataniT kristaluri mesris nawilakebis siTburi

rxevebis

xarjze,

fononuri

Tbogamtaroba ewodeba). fononis energia ganisazRvreba tolobiT

  h , sadac h plankis mudmivaa, xolo



_ akustikuri rxevis sixSire.

amrigad imis msgavsad, rogorc sinaTle warmoadgens fotonebis nakads da sinaTlis talRis gavrceleba daiyvaneba fotonebis gavrcelebamde, siTburi drekadi talRa myar sxeulSi warmoadgens fononebis nakads da am talRis gavrceleba (e.i. Tbogadacema) daiyvaneba fononebis gavrcelebamde. mTeli myari sxeuli avsebulia fononebiT, romelTa erToblioba fononur airs Seadgens. gavrcelebisas xdeba fononebis gabneva erTmaneTze da kristalis struqturul defeqtebze. fononebi

miekuTvneba

kv azin aw ilakebis

kategorias.

ZiriTadi

gansxvaveba

kvazinawilaksa da Cveulebriv nawilaks (eleqtroni, protoni, neitroni, fotoni da sxv.) Soris is aris, rom kvazinawilaki ar SeiZleba arsebobdes vakuumSi. misi warmoqmnisa da gavrcelebisaTvis aucilebelia nivTieri garemo. dadgenilia, rom siTburi talRebis kvanturi xasiaTi yovelTvis ar vlindeba. arsebobs garkveuli temperaturuli zRvari, e.w. debais temperatura, romlis zeviTac siTburi rxevebis dakvanta ar iCens Tavs*. debais temperatura ganisazRvreba tolobiT:

Θ sadac



maqs

cxadia,

hνmaqs k

mocemul sxeulSi nawilakebis rxevis maqsimaluri sixSirea.

rom

kategoriulad

SeiZleba

gamijnavs

arsebobdes

klasikur

mkacrad

kvantur

gansazRvruli

areebs.

debais

temperatura,

temperatura

romelic

miaxloebiT

gamosaxavs sazRvars maRali temperaturis aresa, romelzedac rxevebi aiwereba klasikuri kanonebiT da dabali temperaturis ares Soris, romelzedac mniSvnelovan rols asrulebs kvanturi efeqtebi.

305

§89. myari sxeulis siTburi gafarToeba airis Tvisebebis ganxilvisas Cven gavecaniT airis siTbur gafarToebas, romelic gei-lukasis kanoniT aiwereba. gaTbobisas farTovdeba ara marto airi, aramed myari sxeulic, amasTan airisagan (da siTxisagan) gansxvavebiT, romelTac sakuTari forma ar gaaCnia, myari sxeulis SemTxvevaSi ganixilaven rogorc moculobiT, ise wiriT gafarToebas. davuSvaT, sxeulis sigrZe T0 temperaturaze iyo l 0 , xolo T temperaturamde gaTbobis Sedegad gaxda lT . sidide lT  l0  l absolutur,

xolo

lT  l 0 l0

farTobiT

gansazRvravs sxeulis

wagrZelebas

T  T0  TK -Ti

misi

gaTbobis Sedegad. cda gviCvenebs, rom arc Tu ise did temperaturul SualedSi ( T  100K ) s xe u l i s

f a rd ob i T i

w a grZ e le b a

misi

te m pe r a tu r is

cv li le b i s

p r op or c iu li a :

proporciulobis

lT  l 0 l    T . l0 l0 koeficients ewodeba

(89.1) w ir iT i

g a f arT oe b i s

k o e f ic ie nt i . Tu (89.1) formulaSi davuSvebT, rom T  1 , maSin   wiriTi

ga f ar T oe b is

k o e f ic ie nt i

r ic xob riv a d

to li a

l , e.i. l0

s xe u li s

f a r dob iT i w ag rZ e le b is a m is i e r T i g ra du s iT g aT b ob is as . (89.1) formulidan sxeulis sigrZe T temperaturaze Tu

sxeuli

lT  l0 1   T  . polikristaluria, misi moculobis

(89.2) damokidebuleba

temperaturaze analogiuri formuliT gamoiTvleba: sadac 

VT  V0 1   T  *, (89.3) warmoadgens moculobiTi gafarToebis koeficients da gamosaxavs

sxeulis moculobis fardobiT namats misi erTi gradusiT gaTbobisas. davamyaroT kavSiri  –sa da  -s Soris. davuSvaT, sxeuli warmoadgens kubs, romlis wiboa l . maSin V  l 3 da (89.2) da (89.3) tolobebidan miviRebT: *



xSirad sawyis temperaturad miiCneven 0 C da (2) da (3) formulebi iRebs Semdeg saxes: lt  l0 1   t  da Vt  V0 1   t  .

306

V0 1   T   l03 1   T  , 3

tolobis V0  l 03 –ze Sekveca da frCxilebis gaxsna gvaZlevs:

  3  3 2 T   3 T 2 .

(89.4)

Cveulebriv, wiriTi gafarToebis koeficicienti Zalian

mcirea _ (10 -5

_10-6) grad-1-is rigisaa. amitom (89.4) formulaSi  2 –isa da  3 –is Semcveli wevrebi SeiZleba ugulebelvyoT da davweroT:

  3 , e.i.

izotropiuli

sxeulis

(89.5)

moculobiTi

gafarToebis

koeficienti

gasamkecebuli wiriTi gafarToebis koeficientis tolia. Tu sxeuli monokristaluria, misi anizotropiulobis gamo, wiriTi gafarToebis koeficienti sxvadasxva mimarTulebiT sxvadasxvaa. amitom Tu monokristals mivcemT sferos formas da gavaTbobT (an gavacivebT), zogad SemTxvevaSi igi miiRebs samRerZiani elifsoidis formas (romlis RerZebi dakavSirebulia kristalis kristalografiul RerZebTan). sxeulis wiriTi gafarToebis 1 ,  2 ,  3 koeficientebs am elifsoidis RerZebis mimarT ewodeba kristalis siTburi gafarToebis mTavari koeficientebi. am SemTxvevaSi

  1   2   3 .

(89.6)

kuburi simetriis monokristalisaTvis, iseve rogorc polikristaluri (izotopiuli) sxeulisaTvis, 1   2   3

da (89.6) formulidan miiReba (89.5)

formula. zogierTi kristalis siTburi gafarToebis mTavari koeficientebi didad gansxvavdeba erTmaneTisagan. magaliTad, TabaSirisaTvis (monoklinuri sistema)

1  1,6  10 6 grad-1,  2  42  10 6 grad-1 da  3  29  10 6 grad-1 T  313K  . rogorc aRvniSneT, siTburi gafarToebis koeficienti Zalian mcirea. magaliTad, rkinisaTvis   12  10 6 grad-1, TiTbrisaTvis   19  10 6 grad-1, xolo aluminisaTvis siTburi

  26  10 6 grad-1

gafarToebis

kvarcisaTvis

magram

koeficienti

arsebobs

nivTierebebi,

gansakuTrebiT

mcirea.

romelTa magaliTad,

  0,5  10 6 grad-1, xolo invarisaTvis (rkinisa da nikelis

307

Senadnobi Fe+36% Ni,)   1  10 6 grad-1. aseTi nivTierebebi farTod gamoiyeneba zusti xelsawyoebis dasamzadeblad. avxsnaT gaTbobis Sedegad myari sxeulis gafarToebis movlena. myari sxeulis

molekulebs

Soris

(iseve

rogorc

realuri

airis

siTxis

molekulebs Soris) moqmedebs mizidvisa da gazidvis Zalebi, amasTan, manZilis gazrdiT ganzidvis Zalebi ufro swrafad mcirdeba, vidre mizidvisa. amitom, Zalian

mcire

manZilze

sWarbobs

ganzidvis

Zala,

xolo

SedarebiT

did

manZilze _ mizidvis Zala. Ceulebriv, Tu sxeuli deformirebuli ar aris, nawilakebi iseT

r  r0

manZilze

arian erTmaneTisagan,

rom mizidvisa da

ganzidvis Zalebi erTmaneTs awonasworeben. 89.1 a suraTze gamosaxulia ori nawilakis urTierTqmedebis jamuri Zalis

damokidebuleba

manZilze.

suraTidan

Cans,

rom

roca

didia,

nawilakebi saerTod ar urTierTqmedeben. manZilis Semcirebisas nawilakebs Soris

aRiZvreba

m iz idv is

Zala

(Zalis

uaryofiTi

mniSvneloba

imis

maCveneblia, rom igi mizidvis xasiaTisaa). am Zalis absoluturi mniSvneloba manZilis SemcirebiT izrdeba da aRwevs maqsimalur mniSvnelobas roca r  r1 .

r -is Semdgomi Semcirebisas mizidvis Zala mcirdeba da roca r  r0 nulis toli xdeba. manZilis Semdgomi Semcirebisas nawilakebs Soris aRiZvreba g a nz i dv i s Zala, romelic swrafad izrdeba r –is SemcirebiT da miiswrafvis usasrulobisaken, roca r  0 . ori nawilakis urTierTqmedebis gamosaxva ufro mosaxerxebelia urTierTqmedebis potenciuri mrudiT, romelic mocemulia 89.1 b suraTze*. igi advilad miiReba

F r 

damokidebulebidan,

vinaidan,

ro-

gorc cnobilia, F  

dU . dr

sur. 89.1

rogorc U r  grafikidan Cans, r  r0 manZilze, roca urTierTqmedebis Zala nulis tolia, potenciuri energia minimaluria (aseT mdgomareobaSi

89.1 a, b suraTze gamosaxuli mrudebi analogiuria 68.2 a, b suraTze gamosaxuli mrudebisa.

308

myofi atomis Sesaxeb amboben, rom igi imyofeba `potenciuri ormos~ fskerze). dU am dros nawilaki wonasworobaSia. roca r  r0 , maSin  0 da F  0 , anu dr adgili aqvs nawilakebs Soris mizidvas, xolo roca r  r0 , maSin F  0 , anu adgili aqvs nawilakebs Soris ganzidvas. amrigad nawilakebi rom uZravi yofiliyo, maSin isini iqnebodnen

erTmaneTisagan

daSorebuli.

magram

myari

manZiliT

sxeulis

nawilakebi

uZravi araa, isini irxevian wonasworobis mdebareobis maxloblad. simartivisaTvis davuSvaT, rom erTi nawilaki uZravadaa damagrebuli koordinatTa saTaveSi, meore ki irxeva mis mimarT Ox RerZis gaswvriv. r0 –is marjvniv gadaxrisas warmoiqmneba mizidvis Zala, romelic cdilobs daabrunos nawilaki wonasworobis mdgomareobaSi, xolo r0 –is marcxniv gadaxrisas _ ganzidvis Zala. am SemTxvevaSi r0 gansazRvravs nawilakebs Soris saSualo manZils. kristalis

temperaturis

gazrdisas

izrdeba

rxevis

energia,

rasac

Seesabameba rxevis amplitudis zrda. magram, sazogadod, amplitudis zrda ar aris sxeulis gafarToebis erTaderTi mizezi. nawilakebis

rxeva

rom

mkacrad

harmoniuli

iyos,

maSin

gadaxra

wonasworbis mdebareobidan marjvniv da marcxniv iqneboda zustad erTnairi. am SemTxvevaSi, cxadia, amplitudis gazrdiT nawilakebs Soris saSualo manZili ar Seicvleboda da, maSasadame, gaTbobisas sxeuli ar gafarTovdeboda. magram, sinamdvileSi, nawilakebis rxeva myar sxeulSi an h a rm o niu li a (e.i. araharmoniuli), ris gamoc nawilakis gadaxra marjvniv

da marcxniv

erTnairi ar aris. es ganpirobebulia nawilakebs Soris urTierTqmedebis Zalis manZilze damokidebulebis xasiaTiT, kerZod imiT, rom SedarebiT did manZilze

nawilakebs

Soris

moqmed

urTierTqmedebis

jamur

Zalas

aqvs

mizidvis xasiaTi, mcire manZilze ki _ ganzidvisa. amasTan (da es mTavaria), manZilis SemcirebiT ganzidvis Zala izrdeba gacilebiT ufro swrafad, vidre manZilis gadidebiT _ mizidvis Zala. amis gamo sxeuli gaTbobisas nawilakis rxevis amplituda marjvniv da marcxniv erTnairad ar izrdeba _ igi marjvniv ufro metad gadaixreba,

vidre marcxniv. es, cxadia, iwvevs nawilakebs Soris

saSualo manZilis gazrdas, e.i. sxeulis gafarToebas. es naTlad Cans potenciuri energiis grafikidan (sur. 89.1 b), romelic nawilakebs Soris moqmedi urTierTqmedebis Zalis xasiaTis gamo asimetrulia 309

r0 mdebareobis mimarT. suraTze horizontaluri wrfeebiT gamosaxulia dasaS-

vebi rxeviTi energetikuli doneebi. nulovan dones Seesabameba nulovani energia, romelic nawilakebis nulovani rxevebiTaa gamowveuli (§88). temperaturis zrdasTan erTad nawilakis rxeviTi energia izrdeba da igi gadadis ufro maRal energetikul doneze. suraTidan Cans, rom donis nomris zrdasTan erTad nawilakebs Soris saSualo manZili (jvrebiTaa aRniSnuli) izrdeba, rac iwvevs sxeulis gafarTovebas. amrigad,

sxeulis

siTburi

gafarToebis

mizezia

nawilakebis

rxevis

anharmoniuloba, rac ganpirobebulia urTierTqmedebis Zalis xasiaTiT.

§90. myari sxeulis siTbotevadoba myari sxeulis siTbotevadoba mWidrodaa dakavSirebuli kristaluri mesris dinamikasTan. amitom siTbotevadobis kerZo sakiTxis sworma Teoriulma aRweram didi roli Seasrula myari sxeulis fizikis ganviTarebaSi. airis

siTbotevadobis

ganxilvisas

siTbotevadoba mudmivi moculobisa

CV 

(§64)

ganvasxvaveT

da mudmivi wnevis

erTmaneTisagan

C  p

dros. myari

sxeulis SemTxvevaSi, maTi umniSvnelo siTburi gafarToebis gamo (§89), gansxvaveba C p –sa da CV –s Soris metad mcirea*. amitom myari sxeulis siTbotevadobas ( romlis gazomva, Cveulebriv, xdeba mudmivi wnevis dros) aRvniSnoT C asoTi indeqsis gareSe. ganmartebis Tanaxmad, siTbotevadoba Q ; T siTburi

gafarToeba

Zalian

daxarjuli

siTbos

raodenoba

C

magram,

vinaidan

gafarToebis

myari

dros

sxeulis

muSaobaze

(90.1) mcirea,

SeiZleba

ugulebelvyoT da CavTvaloT, rom gadacemuli siTbos raodenoba mTlianad xmardeba sxeulis Sinagani energiis gazrdas, e.i. Q  U . amitom C *

U . T

(90.2)

temperaturis gazrdiT es gansxvaveba izrdeba. mag. oqrosaTvis oTaxis temperaturaze

C p  5,8 kal/K.mol. da CV  5,7 kal/K.mol. xolo t  500  C –ze es cifrebi Sesabamisad 6,7 da 6,2-is tolia.

310

cda gviCvenebs, rom sazogadod, siTbotevadoba damokidebulia temperaturaze, amitom (90.1) da (90.2) formulebi gvaZlevs siTbotevadobis saSualo mniSvnelobas T temperaturul SualedSi. siTbotevadobis ufro zusti mniSvnelobis misaRebad sxeuls unda gadavceT usasrulod mcire siTbos raodenoba, rac gamoiwvevs misi temperaturis gazrdas usasrulod mcire sididiT. maSin (90.1) da (90.2) formulebi miiRebs Semdeg saxes:

Q*

dU (90.2’). dT dT siTbotevadobis klasikur TeoriaSi erTgvarovani myari sxeuli ganixiC

leba

rogorc

(90.1’)

erToblioba

C

erTmaneTisagan

nawilakebisa, romlebic asruleben h a rm o niu l sixSiriT.

TiToeul

aseT

nawilaks sami

sruliad

damoukidebeli

rxevebs e rT i

Tavisuflebis

xarisxi

im av e

aqvs

(mas

SeuZlia rxeva sami urTierTmarTobi mimarTulebiT). vinaidan nawilaki harmoniulad irxeva, energiis Tanabrad ganawilebis kanonis Tanaxmad, rogorc kinetikuri, ise potenciuri energiis saSualo mniSvneloba, romelic modis Tavisuflebis erT xarisxze, tolia

1 kT –si. e.i. rxevis erT Tavisuflebis xarisxs 2

eTanadeba kT energia, amitom erTi nawilakis saSualo energia iqneba 3kT 3 ( kT kinetikuri da amdenive potenciuri). 2 nawilaks, romelic asrulebs harmoniul rxeviTs moZraobas, h a r m o n i u l i o s c i l a t o r i ewodeba. vinaidan myari sxulis yoveli nawilakis rxeviTi moZraoba SeiZleba daiSalos sam damoukidebel urTierTmarTob rxevad, amitom TiToeul nawilaks eTanadeba sami damoukidebeli wrfivi oscilatori, xolo N nawilakisagan Sedgenil myar sxeuls _ 3N damoukidebeli wrfivi oscilatori. amasTan, erTi wrfivi oscilatoris sruli energia aris kT .

qimiurad martivi myari sxeulis (e.i. iseTi myari sxeulis, romlis kristaluri mesris kvanZebSi moTavsebulia erTnairi atomebi) erTi moli Seicavs nawilakebis (atomebis) N A raodenobas, sadac N A avogadros ricxvia, amitom aseTi sxeulis erTi molis Sinagani energia U   3kT  N A  3RT , xolo am energiis cvlileba sxeulis temperaturis

T sididiT cvlilebisas

U   3RT

gavixsenoT, rom siTbos raodenoba ar aris mdgomareobis funqcia, amitom

warmoadegns

Q –s warmoebuls T - s mimarT 311

Q dT

U  -s miRebuli mniSvnelobis Casma (90.2) formulaSi qimiurad martivi myari sxeulis moluri (atomuri)* siTbotevadobisaTvis gvaZlevs Cμ 

3RΔT  3R , ΔT

(90.3)

magram R  8,31

j kal , 2 K  mol. K  mol.

amitom j kal . 6 K  mol. K  mol. m ar tiv i m y a r i k ri s t alu ri C   24,9

e.i. y v e l a

q im iu r ad

s xu l is

( a t om u r i ) s iT b ot e v a d ob a e rT i d a i giv e a d a t ol ia 6

m o lu r i

kal –is. es K  mol.

Sedegi, romelic Cven miviReT molekulur-kinetikuri Teoriis safuZvelze, am Teoriis

Camoyalibebamde

gacilebiT

adre,

1819

wels,

eqsperimentulad

daadgines frangma fizikosebma d iu l on gm a (1785-1838) da p t i m (1791-1820) da igi maT saxels atarebs. davuSvaT myari sxeuli qimiuri naerTia, mag.

NaCl. misi kristaluri meseri Sedgeba

sxvadasxva tipis atomebisagan. magram energiis Tanabrad ganawilebis kanonisaTvis ara aqvs mniSvneloba erTnairia atomebi Tu sxvadasxva. TiToeul atoms gaaCnia sami tavisuflebis xarisxi da masze modis saSualod 3kT energia. NaCl–is kristalis erTi moli Seicavs NaCl– is

molekulebis

raodenobas,

amitom

raodenobas, misi

Sinagani

anu

atomebis

energia

(natriumisa

iqneba

2 N A 3kT ,

qloris

xolo

ionebis)

2 NA

siTbotevadoba

2 N A  3k  2  3R . zogadad, erTi moli myari qimiuri naerTis Sinagani energia U μ  nN A  3kT , xolo misi siTbotevadoba C   n3R , sadac n aris atomebis ricxvi molekulaSi. maSasadame, m y a r i Sem av ali

e leme ntebis

qimiuri atom ur

naerTis

moluri

siT botev adoba

siT botev adobaT a

tolia

m asSi

j a m i s a . Cven mier Teoriulad

miRebuli es Sedegi agreTve adre (1844 w.) iqna eqsperimentulad dadgenili ingliseli mecnieris joulis (1818-1889), xolo Semdeg germaneli mecnieris kopis (1817-1892) mier da mas joulisa da kopis kanoni ewodeba.

qimiurad martivi myari kristaluri sxeulis erTi molis siTbotevadobas xSirad atomur

siTbotevadobas uwodeben.

312

Semdgomi gazomvebiT gamoirkva, rom diulongisa da ptis kanoni ar aris zusti.

cxrilSi

mocemulia

qimiurad

martivi

myari

sxeulebis

atomuri

siTbotevadobebi oTaxis temperaturaze. cxrili gviCvenebs, rom marTalia, bevri elementisaTvis es kanoni sruldeba sakmaod zustad, magram aris elementebi (almasi, bori), romelTa C  SesamCnevad naklebia 6

kal –ze. garda K  mol.

amisa, gamoirkva rom myari sxeulis siTbotevadoba damokidebulia temperaturaze. saxeldobr, temperaturis zrdis dros, sakmaod maRal temperaturaze siTbotevadoba izrdeba da 6

kal –ze meti xdeba, temperaturis SemcirebiK  mol.

sas ki swrafad klebulobs da absolutur nulTan miaxloebisas nulisaken miiswrafvis. 90.1 suraTze mocemulia sxvadasxva nivTierebis moluri sitbotevadobis temperaturaze damokidebulebis grafiki. wyvetili wiriT mocemulia klasikuri Teoriis Sedegi. cxrili

nivTiereba

Cu Al

sur. 90.1

kal K  mol.

alumini

6,14

rkina

6,39

oqro

6,36

TuTia

6,10

kal

6,63

kadmiumi

6,11

platina

6,29

vercxli

6,13

spilenZi

5,90

siliciumi

4,64

bori

2,51

almasi

1,35

unda aRiniSnos, rom temperaturis zrdiT gamowveuli siTbotevadobis zrda aixsneba klasikuri TeoriiT, Tu mxedvelobaSi miviRebT im garemoebas, rom metad maRal temperaturaze nawilakebis rxevis amplituda imdenad izrdeba, rom rxeva anharmoniuli xdeba. rac Seexeba siTbotevadobis Semcirebas temperaturis Semcirebisas, es cdiseuli faqti klasikurma Teoriam ver axsna. 313

siTbotevadobis klasikuri Teoriis es nakli ganpirobebulia ori mizeziT. jer

erTi,

TeoriaSi daSvebulia,

rom

myari sxeulis

atomebi irxeva

erTmaneTisagan damoukideblad da erTnairi sixSiriT. sinamdvileSi ki myari sxeulis atomebi erTmaneTTan dakavSirebulia imdenad mtkiced, rom adgenen erTian

sistemas,

romelsac

gaaCnia

sixSireTa

farTo

speqtri

(rxevis

sakuTari sixSireebi). meorec, merxevi atomi unda ganvixiloT ara rogorc klasikuri, aramed rogorc kvanturi oscilatori, romelsac gaaCnia diskretuli energetikuli speqtri. aseTi oscilatoris saSualo energia unda gamoiTvalos plankis formuliT da ara energiis Tanabrad ganawilebis kanonis gamoyenebiT, romelic bolcmanis ganawilebis klasikuri kanonis Sedegia. myari sxeulis siTbotevadoba Teoriis Semdgomi ganviTareba warimarTa am garemoebaTa gaTvaliswinebiT. siTbotevadobis pirveli kvanturi Teoria 1905 wels Seqmna ainStainma (1879-1955). am TeoriaSi,

iseve

rogorc klasikurSi,

myari sxeuli ganixileba rogorc erTmaneTisagan

damoukidebeli atomebis erToblioba, romlebic irxevian erTi da imave



sixSiriT. magram

erT Tavisuflebis xarisxze mosuli saSualo energia kT –s tolad ki ar aris miRebuli, aramed

gamoiTvleba

wrfivi

oscilatoris

saSualo

energiis

gamomsaxveli

plankis

formuliT:*

ε 

hν hν e kT

(90.4)

1

qimiurad martivi nivTierebis erT mols gaaCnia 3N A Tavisuflebis xarisxi, amitom misi energia

U μ  3N A ε  3N A sidides



h k

aqvs

temperaturis

hν hν kT e

 3N A k

1

ganzomileba,

hν k hν kT e

mas

t e m p e r a t u r a uwoda. rogorc misi gamosaxulebidan Cans,

(90.5)

1 ainStainma

max asi aTebe li

 uSualodaa dakavSirebuli

myari sxeulis atomebis rxevis sixSiresTan da amrigad, warmoadgens mis damaxasiaTebel mudmivas (konstantas). *

es formula miRebulia daSvebiT, rom merxevi nawilakis (oscilatoris) energia ikvanteba:

 n  h n  1 2 ,

sadac, n  0,1,2,....

 0  1 2 h -s

ewodeba nulovani energia. igi temperaturis

funqcias ar warmoadgens, amitom siTbotevadobis ganxilvisas SeiZleba ugulebelvyoT.

314

maxasiaTebeli temperaturis gamoyenebiT (90.5) formula miiRebs Semdeg saxes:

U μ  3N A k

Θ Θ eT

 3R

1

Θ Θ eT

(90.6)

1

xolo moluri siTbotevadoba

Cμ 

aseTia

myari

formulebi.

sxeulis

davinaxavT,

dU μ dT

Θ   3RΘe T 



siTbotevadoba rom

Θ Θ2 T   e 2  T   3R T 2 . 2 2  Θ   Θ   e T  1  e T  1        

ainStainis

ukanasknels

aqvs

mixedviT.

upiratesoba,

(90.7)

SevadaroT rac

imiT

(90.3)

gamoixateba,

(90.7) rom

siTbotevadoba mudmivi sidide ki ar aris, aramed damokidebulia temperaturaze. dabal

temperaturaze,

 T

T   ,

roca

e >>1,

mniSvnelSi

erTiani

SeiZleba

ugulebelvyoT da (90.7) formula miiRebs Semdeg saxes: 

  C   3 R  e T . T  2

(90.8)

    T roca T  0 ,   , xolo e  0 . vinaidan e T klebulobs ufro swrafad, vidre T

 izrdeba   , amitom T  2

C  0 . maRal temperaturaze, roca T   , e



 T

 1

 da (90.7) formula iRebs (90.3) formulis T

saxes. amrigad, ainStainis Teoriis Sedegebi emTxveva eqsperiments, magram Tvisobrivad da ara raodenobrivad. (90.7) formuliT gamoTvlili Sedegebi dabal temperaturaze gansxvavdeba eqsperimentuli Sedegebisagan. kerZod, (90.7) formulis Tanaxmad siTbotevadoba miiswrafvis nulisaken

eqsponencialuri

miiswrafvis nulisaken C ~ T

kanoniT 3

~ e

 T

,

eqsperimentis

Tanaxmad

siTbotevadoba

kanoniT. es gansxvaveba ganpirobebulia uxeSi daSvebiT, rom

myari sxeulis atomebi irxeva erTmaneTisagan damoukideblad erTi da imave sixSiriT. ainStainis Teoriis es nakli gamosworebulia debais (1884-1966) TeoriaSi, romelSic myari sxeuli ganxilulia, rogorc erTmaneTTan dakavSirebuli N atomisagan Sedgenili erTiani sistema, romelsac gaaCnia 3N Tavisuflebis xarisxi. aseT sistemaSi, Cveulebriv, aRiZvreba sxvadasxva sixSiris 3N rxeva. am rxevebs si stemi s sak uT ar i rx evebi ewodeba. xolo sixSireebs _ sakuTari six Sireebi . debais Teoriis detalur ganxilvas ar SevudgebiT. aRvniSnavT mxolod, rom am TeoriiT miRebuli Sedegebi zustad emTxveva eqsperimentul Sedegebs.

315

am TeoriaSic SemoRebulia maxasiaTebeli temperatura p e r a t u r a ewodeba. aq

 maqs

h maqs k

, romelsac d e b a i s t e m -

mocemul nivTierebaSi atomTa siTburi rxevis maqsimaluri sixSi-

rea. Teoriidan gamomdinareobs, rom dabal temperaturaze, roca T   , siTbotevadoba ic-

vleba absoluturi temperaturis kubis proporciulad C ~ T ; maRal temperaturaze T   siTbotevadoba temperaturaze damokidebuli ar aris da sruldeba diulongisa da ptis kanoni. amrigad, debais temperatura, rogorc es aRniSnuli iyo (§88), SeiZleba CaiTvalos 3

garkveul sazRvrad dabal da maRal temperaturas Soris. siTbotevadobis kvanturi Teoriis TvalsazrisiT is faqti, rom zogierTi nivTiereba (almasi, bori) ar emorCileba diulongisa da ptis kanons ukve oTaxis temperaturaze, imiT aixsneba,

rom

nivTierebisaTvis

debais

temperatura

Zalian

maRalia.

amitom

oTaxis

temperatura maTTvis dabal temperaturad iTvleba. magaliTad, Tu tyviisaTvis   90 C , 

vercxlisaTvis   210 C , aluminisaTvis   400 C da a.S., almasisaTvis   2000 C . 



§91. kristalis defeqtebi aqamde vixilavdiT iseT kristalebs, romlebSic nawilakebis ganlagebis kanonzomiereba ar iyo darRveuli. aseTi kristalebi bunebaSi ar arsebobs. maT i de a lu r i kristalebi ewodeba. realur kristalebSi es kanonzomiereba irRveva, an rogorc amboben, realuri kristalebis wyobaSi adgili aqvs defeqtebs. realuri kristalebis defeqtebi, anu struqturuli arasrulyofiloba arsebiT gavlenas axdens kristalebis mTel rig meqanikur, eleqtrul da optikur Tvisebebze. imis mixedviT, Tu damaxinjebis ra ares warmoqmnis nawilakebis ganlagebis periodulobis darRveva kristalSi, defeqtebi iyofa oTx jgufad: nulganzomilebiani (wertilovani), erTganzomilebiani (wrfivi), organzomilebiani (sibrtyiTi, zedapiruli) da samganzomilebiani (moculobiTi) defeqtebi. wertilovani defeqtebi Tavis mxriv iyofa energetikul, eleqtrul da atomur defeqtebad. yvelaze ufro gavrcelebuli energetikuli defeqtebia f on one b i

siTburi moZraobiT gamowveuli kristalis mesris regularobis droebiTi damaxinjeba. energetikul defeqtebs miekuTvneba agreTve kristalze sxvadasxva radiaciis (rentgenis sxivebi, nawilakebi da sxv.) moqmedebiT gamowveuli kristaluri mesris droebiTi damaxinjeba (agzneba). 316

eleqtrul defeqtebs miekuTvneba: Warbi eleqtronebi, xvrelebi (Seuvsebeli valenturi kavSirebi) da e q s i to ne b i . eqsitoni wyvili defeqtebia, romelic Sedgeba kulonis ZalebiT dakavSirebuli eleqtronisa da xvrelisagan. atomur defeqtebs miekuTvneba v ak ans ie b i

_ kristaluri mesris

Seuvsebi kvanZebi (sur. 91.1 a), romelic idealur SemTxvevaSi atomebiT unda

yofiliyo

dakavebuli

kvanZTaSorisi atomebi (sur. 91.1b). maTi warmoqmnis meqanizmi Sem-

sur. 91.1

degia:

energiis

ganawileba

myari

sxeulis nawilakebs Soris, airis molekulebis msgavsad, araTanabaria. myar sxeulSi mocemul temperaturaze yovelTvis mravladaa atomebi, romelTa energia saSualoze metia. isini sZleven mezobeli atomebis SeWidulobis Zalebs

(potenciur

jebirs)

gadadian

kvanZebs

Soris

sivrceSi.

aseTi

gadasvlis Sedegad erTdroulad warmoiqmneba wyvili defeqti _ vakansia da kvnZTaSorisi atomi. cxadia, rogorc vakansia, ise kvanZTaSorisi atomi gadaadgildeba kristalSi, maT Soris manZili

ganuwyvetliv

icvleba.

aseT wyvil defeqts f re nk e l is d e fe q t i ewodeba. defeqtebis dros

SeiZleba

kristalSi,

gadaadgilebis vakansia

darCes

kvanZTaSorisi

atomi

ki `aorTqldes~ _ amovides kristalis fal~

sur. 91.2

zedapirze. vakansias,

aseT

`Tavisu-

romelsac

kris-

talSi aRar Seesabameba kvnZTaSorisi atomi, S otk is de fe q ti ewodeba (sur. 91.2). Sotkis defeqti SeiZleba sxva gziTac warmoiqmnas: kristalis zedapirul fenaSi atomis aorTqlebis Sedegad gaCndeba vakansia, romelic Semdeg ufro Rrmad mdebare atomebiT Senacvlebis Sedegad d i fu n d ir d e b a kristalis moculobaSi. 317

atomur defeqts warmoqmnis agreTve meserSi arsebuli atomebis sxva atomebiT Senacvleba. zemoT CamoTvlil defeqtebs atomuri defeqtebi ewodeba, vinaidan maTi warmoqmnis mizezi calkeuli atomebia. wrfiv

defeqtebs

miekuTvneba

sxvadasxva

saxis

dislokaciebi

(`dislokacia~ _ wanacvleba). zedapirul

defeqtebs

miekuTvneba,

pirvel

rigSi,

TviT

kristalis

zedapiri. garda amisa, aseT defeqtebs warmoqmnis kristaluri marcvlebis sazRvrebi (polikristalSi) da blokebs Soris sazRvrebi (monokristalSi). saqme isaa, rom rogorc rentgenostruqturulma da eleqtronomikroskopulma gamokvlevebma aCvena, monokristali ar warmoadgens erTian mesers, mas aqvs e.w. m oz aik u r i mimarT

s t ru q tu r a . igi Sedgeba mcire zomis (10-6-10-4 sm) erTmaneTis

odnav

dezorientirebuli

calkeuli

blokebisagan (sur. 91.3). kuTxe blokebs Soris metad

mcirea

igi

icvleba

ramdenime

sekundidan ramdenime minutamde. urTierTSemxeb blokebSi

meseri

TandaTan

gadadis

erTi

blokis Sesabamisi orientaciidan meoreSi. dabolos, kristalSi zogjer gvxvdeba mo-

sur. 91.1

culobiTi defeqtebi forebisa da bzarebis saxiT ganvixiloT mokled dislokaciis ori saxe _ kiduri (sasazRvro) dislokacia da xraxnuli dislokacia. k i d u r i d i s l o k a c i a . ganvixiloT kristali, romelSic gare Zalebis moqmedebis Sedegad moxda misi erTi nawilis gasrialeba meoris mimarT garkveuli

sur. 91.4

ABCD sibrtyis gaswvriv,

ď ˛ b veqtoris mimarTulebiT. sxvanairad rom vTqvaT, kristalma ganicada Zvris deformacia. ď ˛ ABCD sibrtyes ewodeba s r i a l i s s i b r t y e , xolo b veqtors w a n a c v l e b i s v e q t o r i .

Tu aseTi deformaciis Sedegad srialis sibrtyis mezobeli zeda atomuri sibrtyis atomebi gadaadgildeba qveda sibrtyis

mimarT

zustad atomTaSoris manZilze,

maSin kristalSi

araviTari defeqti ar warmoiqmneba. deformacia tolia srialis erTi erTeulisa, kristalis moculoba ki ucvleli rCeba (sur. 91.4).

318

magaram analogiur atomebis

cdebi

gviCvenebs,

moqmedebas

rom

ganicdis

gadaadgileba

rodesac

realuri

araTanabaria,

Zalis

kristali,

e.i.

srialis

erTeulze wanacvlebas adgili aqvs atomuri sibrtyis erT nawilze, danarCen nawilze ki _ mTlianad gamoricxulia; 91.5 suraTze swored aseTi SemTxvevaa gamosaxuli: ABCD

ď ˛

sur. 91.5

sibrtyeSi b â&#x20AC;&#x201C;is mimarTulebiT gasrialeba moxda mxolod

AHED ubanze, HECB ubanze ki ara. HE wrfes, romelic sazRvaria kristalis am or nawils Soris, k i d u r i (sasazRvro) d i s l o k a c i a ewodeba. 91.6 suraTze naCvenebia kiduri dislokaciis irgvliv atomebis ganlagebis perspeqtiuli (sivrculi) gamosaxuleba (a) da atomebis ganlageba dislokaciis ( HE wrfis) marTobul sibrtyeSi (b). suraTze Cans, rom zemoaRniSnuli Zvris Sedegad kristaluri mesris zeda nawilSi aRmoCnda erTi atomuri sibrtyiT meti vidre qvedaSi ( OM sibrtye), ris gamoc srialis sibrtyis zemoT moTavsebuli atomuri mwkrivi (1) Seicavs erTi atomiT mets, vidre qvemoT moTavsebuli mwkrivi (2). *

O wertilis d i s l o k a c i i s c e n t r i ewodeba. O wertilis maxloblobaSi manZili atomebs Soris (1) mwkrivSi normalurze naklebia (meseri SekumSulia), (2) mwkrivSi ki _ normalurze

metia

(meseri

gaWimulia).

magram

damaxinjeba

dislokaciis

centridan

daSorebisas Zalian swrafad mcirdeba da misgan ukve ramdenime atomuri diametris tol manZilze imdenad mcirea, rom kristals TiTqmis srulyofili struqtura aqvs. saerTod, yvela

saxis

Taviseburebaa

dislokaciis is,

moicavs

metad

wrfis

uSualo

dislok aciis

rom

mcire

mniSvnelovani

Zlieri ares,

damaxinjeba

dislokaciuri

siaxloveSi. birTvi

ewodeba.

ares

(suraTze

Semofarglulia wyvetili wiriT). xraxn uli

sur. 91.6

Zalebis

moqmedebiT

nawilebi

SeiZleba

di slok aci a . kristalis ise

daiZras

gare

calkeuli erTmaneTis

mimarT, rom kristali mTlianad warmogvidges xraxnuli kibis msgavsad dagrexili atomuri sibrtyeebis erTobliobad (sur. 91.7 a). ukeTrom vTqvaT, yvela atomi erT zedapirzea. es zedapiri xraxnulia, igi iwyebakristalis erTi bolodan da mTavrdeba meoreze. swored aqedan warmodgeba dislokaciis saxelwodeba _ xraxnuli. xraxnis mimarTuleba SeiZleba iyos rogorc marjvena, ise marcxena, xolo xraxnis biji SeiZleba iyos erTi an ramdenime atomTaSorisi manZilis toli.

miiReba

iseTi

suraTi,

TiTqos

kristalis

moculobaSi

Setanilia

zedmeti

atomuri

naxevarsibrtye. am sibrtyis kide qmnis kristalSi struqturul damaxinjebas. swored aqedan warmodgeba dislokaciis dasaxeleba _ kiduri.

319

91.7 b romelic

suraTze

iwvevs

naCvenebia

xraxnuli

Zvra,

dislokaciis

warmoqmnas. Zvra moxda ABCD sibrtyeze.

AB sazRvari warmoadgens xraxnuli dislokaciis RerZs. rogorc suraTidan Cans, kiduri dislokaciisagan gansxvavebiT, aq Zvris veqtori dislokaciis RerZis para-

sur. 91.7

leluria.

bolo wlebSi intensiurad ganviTarda kristalis zrdis meqanizmisa da plastikuri deformaciis dislokaciuri Teoriebi. am ukanasknelma mniSvnelovnad Secvala winandeli warmodgenebi Tanaxmad,

plastikuri

deformaciis

meqanizmis

Sesaxeb.

adre

arsebuli

Sexedulebis

deformacia

xdeba kristalis mezobeli atomuri fenebis urTierTgasrialebis Sedegad, Tanac ise, rom erTdroulad gadainacvlebs

fenaSi

arsebuli

yvela atomi. Teoriuli gamoTvlebi

sur. 91.8

gviCvenebs, rom aseTi gadanacvlebi-

saTvis saWiroa sakmaod didi (Zvris modulis meaTedi rigis) mxebi Zabva. sinamdvileSi ki, rogorc amas cdebi gviCvenebs, plastikuri deformacia iwyeba gacilebiT adre, Teoriul mniSvnelobaze 100-jer da 1000-jer naklebi Zabvis dros. magaliTad, Teoriuli gamoTvlebiT srialis

Zabva

sufTa

TuTiaSi



F  350 kgZ/mm2, sinamdvileSi ki sriali iwyeba ukve S

τ  1 kgZ/mm2 Zabvis dros. swored am Seusabamobam Teoriasa da eqsperiments Soris gamoiwvia plastikuri deformaciis dislokaciuri Teoriis warmoqmna. am Teorias safuZvlad daedo daSveba, rom sriali iwyeba atomebis mxolod mcire jgufis gadanacvlebiT, rac warmoqmnis dislokacias da mTeli fenis gasrialeba xdeba am dislokaciis TandaTanobiT gadaadgilebiT kristalSi (sur. 91.8). sabolood miiReba iseTive deformacia, rogorsac adgili eqneboda atomuri sibrtyis (rogorc erTi mTlianis) gasrialebisas. magram cxadia, rom dislokaciis gadaadgilebiT fenis gasrialebas dasWirdeba gacilebiT naklebi Zabva, vidre mTeli fenis erTdroulad gasrialebisaTvis iyo saWiro.

320

XII Tavi

s i T xi s T v i s e b e b i ยง 92. Txevadi mdgomareobis zogadi daxasiaTeba IX da XI TavebSi ganvixileT airisa da myari sxeulis Tvisebebi. fizikuri Tvisebebis mxriv siTxes Sualeduri adgili ukavia airsa da myar sxeuls Soris. siTxis umniSvnelovanesi Tviseba _ d e na d ob a _ aaxloebs mas airTan. denadobis gamo siTxe, airis msgavsad, iRebs im WurWlis formas, romelSic igi asxia, gadascems masze warmoebul wnevas yvela mimarTulebiT Tanabrad (paskalis kanoni), moqmedebs masSi CaZirul sxeulze amomgdebi ZaliT (arqimedes kanoni), airis msgavsad siTxec izotropiulia. myar sxeulTan siTxes aaxloebs sakuTari moculobis arseboba da mcire kumSvadoba. myari sxeulis msgavsad siTxec ganicdis gaWimvis deformacias. im

faqtma,

rom

van-der-vaalsis

gantolebis

gamoyeneba

nawilobriv

SesaZlebelia Txevadi mdgomareobisaTvisac, warmoSva azri, rom siTxesa da airs Soris araviTari principuli gansxvaveba ar unda arsebobdes arc molekuluri aRnagobis, arc molekulebis moZraobis xasiaTis mixedviT _ rogorc erTi, ise meore, unda iyos sruliad qaosuri. gansxvaveba unda iyos mxolod urTierTqmedebis Zalebis sidideSi, romelic siTxeSi, didi simkvrivis gamo, gacilebiT metia. am azrs ganamtkicebda is faqti, rom siTxe, airis msgavsad, izotropiulia. magram aRmoCnda, rom aseTi Sexeduleba siTxis struqturisa da misi molekulebis moZraobis Sesaxeb marTebuli ar aris. piriqiT, rogorc cdebma aCvena, nivTierebis Txevadi mdgomareoba molekuluri aRnagobiTac da TvisebebiTac ufro axlosaa myar mdgomareobasTan, vidre airadTan. aseT cdiseul faqts pirvel rigSi miekuTvneba nivTierebis moculobis mcire cvlileba (daaxloebiT 10%-iT) dnobisa da gamyarebis procesSi, rac udavod miuTiTebs imas, rom siTxis struqtura ar SeiZleba Zalian gansxvavdebodes

myari

sxeulis

struqturisagan.

marTlac,

rentgeno-struqturulma

gamokvlevebma aCvena, rom siTxeSi SeimCneva nawilobrivi wesrigi nawilakebis ganlagebSi, rac imiT gamoixateba, rom nebismieri molekulis maxloblad mezobeli molekulebi mowesrigebuladaa ganlagebuli. magram kristalisagan gansxvavebiT, sadac es wesrigi vrceldeba mTel moculobaSi (adgili aqvs 321

Sor wesrigs nawilakebis wyobaSi, § 85), siTxeSi,

mocemuli nawilakidan daSo-

rebisas, es mowesrigebuloba TandaTan mcirdeba da sakmaod swrafad sruliad qreba. maSasadame, siTxeSi SeiniSneba wesrigi mxolod uaxloesi molekulebis ganlagebaSi an, rogorc amboben, adgili aqvs a x l o w e s ri gs nawilakebis wyobaSi. axlo wesrigis arsebobis gamo siTxis struqturas uwodeben k v azi k r is t a lu r s (kristaluris magvars). dnobis procesSi irRveva Sori wesrigi _ kristaluri meseri iSleba, magram ZalaSi rCeba axlo wesrigi. garda amisa,  dnobis kuTri siTbo gacilebiT naklebia L orTqladqcevis kuTr

siTboze

 j j 6  magaliTad, natriumisaTvis   1,1  10 5 da L  4 , 6  10  K  kg K  kg 

 ,  

maSasadame dnobis procesSi SeWidulobis Zalebi nawilakebs Soris gacilebiT ufro mcired icvleba (mcirdeba), vidre orTqladqcevis procesSi. da bolos, mTeli rigi Tvisebebi, magaliTad, kuTri siTbotevadoba, myarsa da Txevad mdgomareobebSi umniSvnelod gansxvavdeba erTmaneTisagan  j j  natriumisaTvis Sesabamisad aris 1,2  10 3 da 1,4  10 3  K  kg K  kg 

  , maSin rodesac  

Txevadsa da airad mdgomareobaSi es gansxvaveba didia. siTxis Sualeduri mdgomareoba ganapirobebs misi Tvisebebis sirTules. amitom nivTierebis Txevadi mdgomareobis Teoria gacilebiT ufro naklebadaa damuSavebuli, vidre kristaluri an airadi mdgomareobisa. srulyofili da sayovelTaod aRiarebuli Teoria dRemde ar arsebobs. siTxis Teoriis mTeli rigi

problemebis

damuSavebaSi,

Teoriis

safuZvlebis

SeqmnaSi,

didi

damsaxureba miuZRvis sabWoTa fizikoss i.frenkels (1894-1952). frenkelis Tanaxmad, siTxis yoveli molekula, myari sxeulis nawilakebis msgavsad, irxeva wonasworobis mdebareobis maxloblad. magram siTxeSi molekulebis wonasworobis mdgomareobebi fiqsirebuli (uZravi) ki ar aris, aramed swrafad da uwesrigod icvleba. amrigad, siTxis molekulis moZraoba SeiZleba warmovidginoT rogorc rxeviTi da gadataniTi moZraobebis jami. molekulis erT adgilas rxevis dro an rogorc mas frenkelma uwoda, `binadari cxovrebis~ dro (  ), magaliTad wylis molekulisaTvis oTaxis temperaturaze saSualod 10-11 wm-is tolia da sagrZnoblad aRemateba erTi rxevisaTvis saWiro dros (10-12_10-13 wm). am drois gasvlis Semdeg molekula 322

akeTebs `naxtoms~, romlis sigrZe molekulis diametris rigisaa da gadadis wonasworobis axal mdgomareobaSi (`icvlis binas~) da a.S. amrigad, siTxis molekula, Cerdeba ra garkveul dros wonasworobis TiToeul mdgomareobaSi, nela gadaadgildeba siTxeSi. frenkelis xatovani gamoTqmiT, siTxis molekula `xetialobs~ mTel moculobaSi, eweva `momTabare cxovrebas~, romlis drosac xanmokle gadasvlebi icvleba SedarebiT xangrZlivi `binadari cxovrebiT~. `binadari cxovrebis~ dro damokidebulia siTxis gvarobasa da temperaturaze. temperaturis gazrdiT  mcirdeba. Teoria iZleva Semdeg formulas: U

    e KT ,

(92.1)

sadac U aris aqtivaciis energia, anu im potenciuri jebiris simaRle, romelic unda gadalaxos molekulam wonasworobis erTi mdebareobidan meoreSi gadasasvlelad. (92.1) formuliT aixsneba siTxis qcevis sxvadasxva xasiaTi gare Zalis moqmedebis mimarT. marTlac,  drois ganmavlobaSi molekula asrulebs rxeviTs moZraobas da, maSasadame, ganicdis drekadi Zalis moqmedebas. am drois gasvlis Semdeg molekula akeTebs `naxtoms~ (siTburi moZraobis xarjze) da gadadis wonasworobis axal mdgomareobaSi. rodesac molekulaze gare Zala ar moqmedebs, es gadasvlebi yvela mimarTulebiT Tanabaria. davuSvaT, siTxeze moqmedebs gare Zala (Cveulebriv, esaa dedamiwis mizidvis Zala an moZravi myari sxeuliT gamowveuli wnevis Zala). Tu am Zalis moqmedebis dro t   , maSin imisaTvis, rom moxdes molekulebis urTierTwanacvleba, saWiro iqneba molekulebis makavSirebeli drekadi Zalebis daZleva (molekulebis daaxloebisas aRiZvreba ganzidvis Zala, xolo daSorebisas _ mizidvisa). amitom siTxe, gare Zalis xanmokle moqmedebisas, amJRavnebs drekad Tvisebebs (iqceva ise, rogorc myari drekadi sxeuli). Tu t   , maSin molekulebi aswreben erTi wonasworobis mdebareobidan meoreSi gadasvlas, magram es gadasvlebi axla ukve upiratesad Zalis mimarTulebiT xdeba, rac ganapirobebs siTxis denadobas. unda aRiniSnos, rom wamSi Sesrulebuli gadasvlebis raodenoba gare Zalis moqmedebiT SesamCnevad ar icvleba _ igi ganisazRvreba siTburi moZraobis intensivobiT (formula 92.1). gare Zala mxolod upirates mimarTulebas aZlevs am gadasvlebs.

323

§ 93. molekuluri wneva viciT, rom molekulebs Soris urTierTqmedebis

Zalebi eleqtruli

bunebisaa da manZilis gazrdiT metad swrafad mcirdebian nulamde. amitom, praqtikulad siTxis yoveli molekula urTierTqmedebs mxolod uaxloes molekulebTan. aRvniSnoT r –iT is udidesi manZili, romelzec mocemuli A molekula kidev amJRavnebs Tavis moqmedebas danarCen molekulebze. Tu

molekulis irgvliv SemovavlebT r –radiusian sferos, maSin sferos SigniT moxvedril yvela molekulas metnaklebad miizidavs molekula. sferos gareT moTavsebul molekulebze ki A molekula ar imoqmedebs. aseT sferos ewodeba m o le k u lu ri

q m e d e b is

s fe ro ,

xolo mis radiuss _ molekuluri

qmedebis radiusi. gamoTvlebi gviCvenebs, rom molekuluri qmedebis r radiusi 10-9 m-is rigisaa. ganvixiloT a molekula, romlis molekuluri qmedebis sfero mTlianad siTxis SigniTaa (sur. 93.1). am sferos SigniT myofi molekulebi a molekulaze yoveli mxridan erTnairad moqmedeben, amitom masze moqmedi Zalebis tolqmedi nulis tolia. sxva

mdgomareobaSia

molekulebi,

romlebic moTavsebulia siTxis zedapirze an masTan

axlos.

molekula,

magaliTad,

romelic

ganvixiloT

moTavsebulia

sur. 93.1

siTxis

zedapirze.

misi

molekuluri

qmedebis sferos naxevari moTavsebulia siTxeSi, meore naxevari ki _ haerSi (siTxis orTqlSi). vinaidan orTqlis simkvrive gacilebiT naklebia siTxis simkvriveze, amitom cxadia, rom Zala romliTac siTxis molekulebi moqmedebs b molekulaze, gacilebiT metia im Zalaze, romliTac airis molekulebi moq-

medebs masze. am Zalebis F tolqmedi mimarTulia siTxeSi, siTxis zedapiris marTobulad.

advili

misaxvedria,

rom

met-naklebi

sididis

aseTi

Zala

moqmedebs siTxis yovel molekulaze, romlis daSoreba siTxis zedapiridan molekuluri qmedebis r radiusze naklebia.

324

am Zalis moqmedebis Sedegad zedapiruli fenis molekulebis ganlageba gansxvavdeba siTxeSi moTavsebuli molekulebis ganlagebisagan. zedapiruli fenis molekulebi siTxis siRrmisaken

mimarTulia

mxriT,

romelzec

yvelaze

metad

moqmedebs

mizidvis

Zala.

magaliTad, molekulebi, romelTac aqvT wagrZelebuli, jaWviseburi forma, ganlagdebian erTmaneTis paralelurad zedapiris marTobulad.

amrigad, siTxis zedapiruli fena, romlis sisqe molekuluri qmedebis sferos

radiusis

rigisaa,

danarCen

siTxeze

aviTarebs

wnevas,

romelsac

m ole k u lu r i , anu Siga wneva ewodeba. molekuluri wneva iwvevs siTxis SekumSvas, ris gamoc molekulebi erTmaneTs uaxlovdebian. es daaxloeba grZeldeba manam, sanam warmoqmnili ganzidvis Zalebi ar gaawonasworeben molekuluri wnevis Zalebs. molekuluri wneva siTxeSi iseTive bunebisaa, rogorc Sinagani wneva airSi. amitom mis gamosaTvlelad SeiZleba visargebloT formuliT Pi 

a . V2

vinaidan siTxis kuTri moculoba gacilebiT naklebia airis kuTr moculobaze, amitom Pi Siga wneva siTxeSi ramdenime milionjer metia, vidre airSi. magaliTad,

a  5,47

wylisaTvis

atm.l 2 mol.

, 0  C -ze

erTi

molis

moculoba

V0  18 sm 2 mol.  0,018l mol. , amitom

Pi 

a 5,47  atm  17000atm. 2 V 0 0,0182

aseTive didia molekuluri wneva sxva siTxisaTvisac. amitomaa, rom siTxe praqtikulad ukumSvelia. misi moculobis odnav Sesamcirebladac ki saWiroa Zalian didi gare wneva. aRsaniSnavia, rom molekuluri wnevis uSualod gazomva SeuZlebelia, vinaidan igi yovelTvis mimarTulia siTxis SigniT, zedapiris marTobulad da WurWlis kedlebze an siTxeSi moTavsebul sxeulze ar moqmedebs. misi gamoTvla, garda zemoT moyvanili meTodisa. SeiZleba im muSaobiTac, romelsac asrulebs molekulebi siTxidan gareT gamosasvlelad. molekulebis siTxidan gareT gamosvla xdeba orTqladqcevis procesSi, amitom gamosvlis muSaoba

gansazRvravs

orTqladqcevis

siTbos.

gamoiTvalos molekuluri wneva.

325

ukanasknelis

gazomviT

SeiZleba

§ 94. zedapiruli daWimuloba ganvixiloT ufro dawvrilebiT zedapiruli fenis Tvisebebi. imisaTvis, rom

molekula

avides

zedapirul fenaSi

siTxis

siRrmidan

zedapirze,

man

unda

moqmedi Zalebis winaRoba da Seasrulos

dasZlios

muSaoba.

muSaobas molekula asrulebs Tavisi kinetikuri energiis xarjze, ris gamoc misi potenciuri energia izrdeba. maSasadame, molekuluri wnevis arseboba ganapirobebs imas, rom siTxis zedapiruli fenis molekulebs meti potenciuri energia aqvT, vidre siTxis siRrmeSi myof (siTxis danarCen) molekulebs. am zedmet energias, romelic Seadgens siTxis Sinagani energiis nawils da damaxasiaTebelia mxolod zedapiruli fenisaTvis, ze d a pi ru l i ( ES )

ewodeba.

cxadia,

rom

zedapiruli

energia

zedapiris

e ne r g i a farTobis

piradpirproporciulia: E S  ε 0 S , sadac ε0 kuTri zedapiruli energiaa. molekulis zedapiridan siTxeSi gadasvlisas xdeba piriqiT _ izrdeba misi kinetikuri energia potenciuri energiis klebis xarjze. zemoTqmulidan gamomdinareobs, rom siTxis zedapiris adiabaturi gazrdisas siTxe unda gacivdes da piriqiT _ zedapiris adiabaturi Semcirebisas igi unda gaTbes. cda gviCvenebs, rom zedapiris adiabaturi cvlilebisas temperaturis cvlilebas marTlac aqvs adgili, magram igi umniSvneloa _ Seadgens gradusis memilioned nawils. Tu siTxis zedapiri icvleba Zalian nela, maSin garemosTan Tbocvlis gamo siTxis temperaturis cvlileba ar xdeba _ procesi izoTermulia.

amrigad, siTxis zedapiris gazrda iwvevs potenciuri energiis gadidebas da piriqiT. vinaidan wonasworobis mdgomareobas, romliskenac miiswrafvis yoveli sistema, Seesabameba potenciuri energiis minimumi, amitom, T u s iT xe z e g ar e Z ale b i ar m oq m e d e b s , Si g a Z ale b is m oq m e de b iT m a n is e T i f o r m a u nd a m i iR os , rom li s ze da p ir i u m c ire s i iq ne b a. mocemuli

moculobis

sxvadasxva

formis

sxeulebs

Soris

umciresi

zedapiri aqvs sferos, e.i. Tu siTxeze gare Zalebi ar moqmedebs, man sferuli forma unda miiRos. Cveulebriv, siTxeze Siga Zalebis garda yovelTvis moqmedebs dedamiwis mizidulobis Zala, amitom siTxe iRebs iseT formas, romelsac Seesabameba jamuri energiis (mizidulobis potenciuri energiisa da zedapiruli energiis 326

jamis) minimumi. sxeulis zomis gazrdisas moculoba izrdeba radiusis kubis proporciulad. amitom sxeulis zomis cvlilebisas misi energia simZimis Zalis velSi (romelic moculobis proporciulia) ufro swrafad icvleba vidre zedapiruli energia (romelic zedapiris proporciulia). mcire zomis wveTebisaTvis mTavar rols asrulebs zedapiruli energia, ris gamoc siTxis mcire wveTi iRebs sferos formas. didi zomis wveTebisaTvis mTavar rols asrulebs mizidulobis potenciuri energia, romelic miT naklebia, rac ufro dablaa sxeulis masaTa centri. amitom aseTi wveTi simZimis Zalis moqmedebiT iRebs SeWyletil formas (an ganiRvreba sxeulis zedapirze) miuxedavad imisa, rom zedapiruli energia am dros izrdeba. Tu didi masis siTxis simZimis Zalas gavawonasworebT sxva ZaliT, magaliTad, arqimedes amomgdebi ZaliT, maSin isic sferos formas

miiRebs.

mosazreba

dasturdeba

belgieli mecnieris platos (1801-1883) mier Catarebuli cdiT. plato amzadebda spirtisa da wylis iseT narevs, rom misi simkvrive ze-

sur. 94.1

Tis simkvrivis toli yofiliyo. aseT narevSi CaSvebuli zeTis mozrdili wveTi sferos formas iRebda (sur. 94.1). igive Sedegi SeiZleba miviRoT sxva mosazrebiTac: siTxeSi molekula saSualod garSemortymulia 12 molekuliT, zedapirze ki (Tu ar CavTvliT orTqlis molekulebs) _ mxolod eqvsiT. Tu siTxis or molekulas Soris urTierTqmedebis potenciur energias aRvniSnavT U 0 –iT, maSin siTxeSi myofi molekulis potenciuri energia iqneba U1  12U 0 , xolo zedapirze myofi molekulisa _ U 2  6U 0 . vinaidan U 0  0 , amitom U 2  U 1 . maSasadame, molekulis

siTxidan

zedapirze

amosayvanad

unda

SevasruloT

muSaoba

A  U 2  U1  6U 0  12U 0  6U 0  6 U 0 . siTxis zedapiridan siRrmeSi gadasvlisas molekulis potenciuri energia mcirdeba imave  6U 0

sididiT. simZimis velSi moTavsebuli sxeulis msgavsad, romelic miiswrafvis

daikavos umciresi potenciuri energiis Sesabamisi mdgomareoba, siTxis zedapirze myofi molekulebic miiswrafvian gadavidnen siTxis siRrmeSi. vinaidan molekulebis raodenoba zedapirze siTxis zedapiris proporciulia, amitom zedapiruli molekulebis miswrafeba Cavidnen siTxis siRrmeSi niSnavs siTxis miswrafebas Seamciros Tavisi zedapiri.

327

siTxis miswrafeba _ Seamciros Tavisi zedapiri, imis damadasturebelia, rom siTxis zedapiruli fena imyofeba daWimul mdgomareobaSi, riTac igi waagavs daWimul drekad afsks, magaliTad, daWimul rezinis afsks*. amitom siTxis zedapirze, ise rogorc daWimuli drekadi afskis SigniT, unda moqmedebdes daWimulobis Zalebi, romlebic cdiloben Seamciron siTxis zedapiri. aseTi Zalebis arsebobaSi advilad

davrwmundebiT

Semdegi

cdiT: mavTulis marTkuTxa CarCo, romelzedac Tavisuflad SeuZlia moZraoba msubuq AB Reros, amovava

loT sapnian wyalSi. mas gadaekvre-

b sur. 94.2

ba Txeli afski (sur. 94.2 a). Tu AB Reros qveviT afsks gavwyvetT, Rero zeviT aiwevs (sur. 94.2 b). imisaTvis, rom AB Rero wonasworobaSi gaCerdes, masze unda vimoqmedoT perpendikularuli

f ZaliT**, romelic siTxis zedapiris mxebia (sur. 94.2 g). wonasworobis mdgomareobaSi TviT afski moqmedebs AB Reroze f Zalis toli da sawinaaRmdego ZaliT, romelsac ze d a p iru l i d aW im u l o b is Z al a e w o de b a . amrigad, zedapiruli daWimulobis Zala moqmedebs siTxis zedapiruli afskis napirze, igi zedapiris mxebia da afskis napirisadmi perpendikularuli. magram unda gvaxsovdes, rom am Zalis warmoqmnis pirvelwyaros warmoadgens zedapiruli fenis molekulebze moqmedi, zedapiris marTobulad siTxeSi mimarTuli Zalebi. zedapiris daWimulobis Zalebis arsis damatebiT garkvevaSi dagvexmareba Semdegi martivi cdebi.

msgavsebasTan erTad, daWimul drekad afsksa da siTxis zedapirul fenas Soris aris

arsebiTi gansxvavebac: afskis farTobis gazrda dakavSirebuli nawilakebs Soris manZilis gazrdasTan,

amitom

masSi

moqmedi

daWimulobis

Zala

miT

metia,

rac

ufor

metadaa

afsikiigaWimuli. gauWimav afskSi drekadi Zalebi saerTod ar aRiZvreba. siTxis zedapiris gazrda ki dakavSirebulia ara molekulebs Soris manZilis gazrdasTan, aramed siTxis zedapirze axali molekulebis amosvlasTan, amitom siTxis zedapiris gazrda ar iwvevs misi daWimulobis gazrdas, romelic cxadia, arasdros xdeba nulis toli. **

Tu CarCo vertikaluradaa dayenebuli, maSin Reroze SeiZleba CamovkidoT Sesabamisi

tvirTi. am SemTxvevaSi daWimulobis Zala tolia TvirTisa da Reros simZimis ZalaTa jamisa.

328

mavTulis

rgoli,

romlis

wertilSi

damagrebulia

dauWimavi

abreSumis Zafi, amovavloT sapnian wyalSi. rgols gadaekvreba Txeli afski, xolo Zafi miiRebs SemTxveviT formas (sur. 94.3 a). Tu Zafis erT mxares afsks SevexebiT gaxurebuli nemsiT, afski gawydeba da Zafi daiWimeba (sur. 94.3 b). pirvelad, afskis gawyvetamde, Zafis TiToeul elementze, mis marTobulad, orive mxridan moqmedebda toli da sapirispirod mimarTuli zedapiruli daWimulobis Zalebi, amitom Zafi iyo wonasworobaSi. afskis gawyvetis Semdeg Zafze zedapiruli Zalebi moqmedebs cali mxridan, rac iwvevs Zafis daWimvas. igi axla ukve iRebs iseT formas, romlis drosac afskis zedapiris farTobi minimaluria. igive cda SeiZleba CavataroT sxva saxiTac. mavTulis skis

rgolze

gadakvris

af-

Semdeg

masze frTxilad davdoT sur. 94.3

sur. 94.4

Zafis maryuJi (sur.94.4a)

Tu maryuJis SigniT afsks gavwyvetT, afskis darCenili nawili zedapiruli Zalebis moqmedebiT Semcirdeba, Zafs daWimavs da maryuJi miiRebs wrewiris formas (sur. 94.4 b). cda gviCvenebs, rom zedapiruli daWimulobis Zala miT metia, rac metia afskis napiris l sigrZe (es gasagebicaa, vinaidan sruli Zala, rogorc es 94.3 a, b suraTebidan Cans, aris sigrZis calkeul elementebze moqmedi Zalebis jami da igi miT meti iqneba, rac metia elementebis raodenoba, e.i. rac metia afskis napiris sigrZe):

koeficients ze d ap iru li

f  σ l . d aW im u lob is

k o e f ic ie nt i

(94.1) ewodeba. igi

ricxobrivad tolia siTxis zedapiris SemomsazRvreli konturis erTeul sigrZeze moqmedi Zalisa. erTeulTa SI sistemaSi misi erTeulia n/m, xolo CGS sistemaSi _ dn/sm. zedapiruli daWimulobis koeficienti damokidebulia siTxis gvarobasa da temperaturaze. temperaturis gazrdiT  TandaTan mcirdeba da kritikul temperaturaze, roca ispoba gansxvaveba siTxesa da mis orTqls Soris, xdeba nulis toli. zedapiruli daWimulobis koeficienti damokidebulia agreTve 329

siTxis SemomsazRvrel garemoze. magaliTad, wylis zedapiruli daWimulobis koeficienti, rodesac mas esazRvreba misive orTqli (an haeri) aris 73  10 3 n/m, xolo wylisa da benzolis sazRvarze   34 103 n/m.  –s sidideze mniSvnelovan gavlenas axdens minarevebic. zedapiruli

daWimulobis

koeficienti

SeiZleba

meorenairadac

ganvsazRvroT.

S

gamovTvaloT muSaoba, romelic unda Sesruldes afskis zedapiris farTobis raRac

sididiT gasazrdelad. amisaTvis odnav gavadidoT AB Reroze moqmedi Zala. Rero Zalian nela daiwevs qveviT h sididiT (sur. 94.5 ), afski gaiWimeba da misi zedapiris farTobi gaizrdeba 2 ABh  S –iT. orze gamravleba imitomaa saWiro, rom afsks (qaRaldis furclis msgavsad) ori zedapiri aqvs _ wina da ukana. gare Zalebis mier Sesrulebuli muSaoba

A  fh   2 ABh   S , aqedan



(94.2)

A , S

Δh

(94.3)

e.i siTxis zedapiruli daWimulobis koeficienti ricxobrivad tolia im muSaobisa, romelic unda Seasrulos gare Zalam siTxis zedapiris farTobis erTi erTeuliT gasadideblad. axali ganmartebiT



–s erTeuli

sistemaSi aris j/m2.

vinaidan 1 j=1 n.m, amitom (94.1) da (94.3) formulebidan



sur. 94.5

–saTvis erTi

da igive ganzomileba miiReba. siTxis

zedapiris

farTobis

gasazrdelad

Sesrulebuli

A

muSaoba

xmardeba

zedapiruli fenis energiis gazrdas. zedapiris Semcirebisas muSaoba sruldeba am damatebiTi energiis xarjze. siTxis zedapiris Semcireba, Cveulebriv, izoTermuli procesia. sistemis Sinagani energiis im nawils, romlis xarjzec sruldeba muSaoba izoTermuli procesis dros,

Tav isuf ali

nawilisagan,

romelic

energia muSaobad

(F ) ar

ewodeba (amiT SeiZleba

mas

asxvaveben

gardaiqmnas

Sinagani energiis

romelsac

bmuli

energia

ewodeba). amrigad, siTxis zedapiris gazrdisas izrdeba Tavisufali energia. Tu zedapiris farTobis S –iT gazrdisas Tavisufali energia gaizarda F sididiT, maSin A  F da (94.3) formulidan miviRebT:



F . S

(94.4)

(94.4) formulidan viRebT zedapiruli daWimulobis koeficientis kidev erT ganmartebas: igi ricxobrivad tolia zedapiris Tavisufali energiis nazrdisa zedapiris farTobis erTi erTeuliT gazrdisas.

330

§ 95. wneva siTxis mrude zedapirze. laplasis formula wina paragrafSi gavarkvieT, rom siTxis zedapirul fenaSi, gaWimuli drekadi afskis msgavsad, moqmedebs zedapiruli daWimulobis Zalebi, romlebic cdiloben siTxis zedapiris Semcirebas. Tu zedapiri mrudea, Semcirebis es miswrafeba vlindeba

imiT,

rom

igi

miiswrafvis miiRos brtyeli forma. amis gamo amoz-

sur. 95.1

neqili

zedapiri

awveba

siTxis

qveda

fenebs

(sur.95.1a), xolo Cazneqi-

sur. 95.2

zedapiri Wimavs maT

(sur. 95.1 b). amis Sedegia is, rom siTxis yoveli mrude zedapiri awarmoebs damatebiT p Ywnevas im

p 0 wnevasTan SedarebiT, romelic moqmedebs siTxis

brtyel zedapirze*. amrigad, sruli wneva siTxis mrude zedapirze p  p0  p ,

(95.1) sadac niSani `+~ Seesabameba amozneqil zedapirs, xolo niSani ` -~ Cazneqils. gamovTvaloT damatebiTi p wneva im SemTxvevisaTvis, roca zedapirs sferos forma aqvs. sferos radiusia R . gamovyoT mcire sferuli segmenti, romelic eyrdnoba r –radiusian wrewirs. am wrewiris yovel dl elementze siTxis danarCeni nawili moqmedebs df

ZaliT, romelic siTxis zedapiris mxebi

da elementisadmi perpendikularulia (sur. 95.2). (95.1) formulis Tanaxmad

df   dl . am Zalis mdgeneli, romelic mimarTulia siTxis SigniT

(95.2)

df n  df sin    dl sin  ,

magram sin  

r , amitom R

damatebiT

wnevas

zogjer

kapilarul

wnevasac

uwodeben,

radgan

misi

moqmedeba

gansakuTrebiT SesamCnevia viwro (erT milimetrze naklebi diametris) milSi, romelsac k a p i l a r u l i mili ewodeba. (termini `kapilari~ warmodgeba laTinuri sityvisagan capillus _ Tma).

331

r dl . R konturis

df n   aseTi

Zala

moqmedebs

segmentis

(95.3) TiToeul

elementze.

mTel

konturze moqmedi, siTxis SigniT vertikalurad mimarTuli Zala miiReba (95.3) formulis integrebiT: 2 πr

2π r 2  dl  σ R . 0

r f n   df n  σ R damatebiTi

wnevis

misaRebad

unda

gavyoT

wris

farTobze,

romelsac eyrdnoba sferuli segmenti:

fn 2 r 2 2 . (95.4) p    R  r2 R r 2 (95.4) formula gviCvenebs, rom damatebiTi wneva zedapiris simrudis radiusis ukuproporciulia. zedapiris simrudiT gamowveuli damatebiTi wnevis arseboba ganapirobebs, magaliTad, imas, rom haeris wneva sapnis buStSi ufro metia, vidre buSts gareT da es gansxvaveba miT metia, rac naklebia buStis radiusi. suraTi sawinaaRmdegoa imisa, rac miiReba rezinis burTis gabervis dros. burTis zomis gazrdiT izrdeba masSi moTavsebuli airis wneva. es gansxvaveba gamowveulia imiT, rom sapnis buStis afskis zedapiruli daWimuloba ar aris damokidebuli mis zomebze, rezinis burTis daWimva ki gabervasTan erTad izrdeba. damatebiTi wnevis damokidebuleba simrudis radiusze SeiZleba SevamowmoT martivi cdiT: minis milis boloebze gamovberoT

sxvadasxva

diametris

buSti

(sur. 95.3).

95.4

formulis Tanaxmad patara buStSi haeris wneva meti iqneba vidre didSi, amitom, Tu onkans gavxsniT, haeri gadava patara buStidan didSi, ris gamoc patara buSti mcirdeba, didi ki izrdeba. buStis

haeris

gadasvla

darCenili

Sewydeba

nawilis

simrude

maSin,

roca

gautoldeba

patara

sur. 95.3

didi

buStis simrudes.

zogad

SemTxvevaSi,

rodesac

siTxis

zedapiri

sferuli ar aris, aseve miiReba, rom damatebiTi wneva

sadac R1 da R2

 1 1   , (95.5) p      R1 R2  siTxis zedapiris ori urTierTmarTosur. 95.5

332

bi sibrtyiT miRebuli kveTis (`normaluri kveTis~) simrudis radiusebia (sur. 95.4).

1 1 gansazRvravs zedapiris saSualo simrudes 0 wertilSi.  R1 R2

(95.5) formulas ewodeba la pl as is formula. Tu zedapiri sferulia

R1  R2  R

(95.5)

formulidan

miiReba

(95.4)

formula.

zedapiri

cilindrulia, misi RerZis perpendikularuli normaluri kveTi aris wrewiri, romlis radiusi R1  R , RerZze gamavali normaluri kveTi ki _ wrfe, romlis radiusi R2   da laplasis formula iRebs aseT saxes: p 

 R

§ 96. damsvelebeli da aradamsvelebuli siTxe. kapilaroba cnobilia, rom siTxe, myar sxeulTan Sexebisas asvelebs an ar asvelebs mas. amasTan, dasveleba an ar dasveleba fardobiTi cnebebia: siTxe, romelic erT myar sxeuls asvelebs, meore SeiZleba ar daasvelos. magaliTad, wyali minas asvelebs, parafins ki ara, vercxliswyali rkinas asvelebs, minas ki ara da a.S.

4 3

a1 2 θ

f1 a

b suraTi 96.1.

gavarkvioT am movlenis molekuluri meqanizmi. amisaTvis ganvixiloT siTxis zedapiris a molekula, romelic kedels exeba. misi qmedebis sfero gavyoT oTx nawilad (sur. 96.1 a). pirvel nawilSi imyofeba mxolod airis molekulebi, meoreSi _ mxolod siTxis molekulebi, xolo mesame da meoTxeSi mxolod myari sxeulis molekulebi. amrigad, molekulaze erTdroulad moqmedebs, rogorc airis,

ise siTxisa da myari sxeulis molkulebi. airis mole-

kulebis moqmedeba a molekulaze, airis mcire simkvrivis gamo, SeiZleba ar 333

gaviTvaliswinoT. siTxis molekulebis moqmedeba a molekulaze gamovsaxoT

f 1 ZaliT, xolo myari sxeulis molekulebisa _ f 2 ZaliT. f1 Zala yovelTvis mimarTulia

siTxeSi,

xolo

Zala

siTxis

gareT

WurWlis

kedlis

marTobulad. Tu Zalebis f

f 2  f1 , maSin siTxe asvelebs myar sxeuls. am SemTxvevaSi

f1 da

tolqmedi mimarTulia myari sxeulisaken. magram siTxis wonaswo-

robis dros zedapirze myof molekulaze moqmedi f Zala yovelTvis mimarTuli unda iyos am wertilSi zedapiris marTobulad. winaaRmdeg SemTxvevaSi f Zalas eqneboda zedapiris paraleluri mdgeneli, romelic nawilaks gadaadgilebs zedapiris gaswvriv _ siTxe ar iqneba wonasworobaSi. maSasadame, kedlis maxloblad siTxis zedapiri ise unda gamruddes, rom gaxdes

tolqmedi

Zalis marTobuli, e.i. Cazneqili (sur. 96.1 b). Tu

f 2  f1 , siTxe myar sxeuls ar asvelebs. tolqmedi Zala axla ukve

siTxis SigniTaa mimarTuli (sur. 96.2 a), ris gamoc siTxis zedapiri kedlis maxloblad amozneqilia (sur. 96.2 b).

α1 2

4 3

θ f

b suraTi 96.2 amrigad, dasvelebis an ar dasvelebis movlena, romelic iwvevs siTxis zedapiris gamrudebas myari sxeulis zedapirTan ganpirobebulia siTxisa da myari sxeulis molekulebis urTierTqmedebis xasiaTiT. Tu mizidvis Zala siTxis molekulebs Soris metia, vidre siTxisa da myari sxeulis molekulebs Soris, siTxe miiswrafvis Seamciros Sexebis sazRvari (farTobi) myar sxeulTan da ar asvelebs mas. aseTi siTxis wveTi horizontalur zedapirze iRebs *

Sebrtyelebuli*

sferos

formas

(sur. 96.3 a).

Sebrtyeleba gamowveulia simZimis Zalis moqmedebiT.

334

Tuki

mizidvis

Zala

siTxisa da myari sxeulis molekulebs Soris metia, vidre siTxis molekulebs

Soris,

maSin

siTxe

miiswrafvis

gazardos

Sexebis

farTobi

myar

sxeulTan da iRebs 96.3 b suraTze naCveneb formas.

b suraTi 96.3.

 kuTxes,

romelsac

adgens

siTxis

zedapiris

mxebi

myari

sxeulis

zedapirTan, k i du ri kuTxe ewodeba. 96.1_96.3 suraTebidan Cans, rom damsvelebeli siTxisaTvis kiduri kuTxe maxvilia (siTxe ekvris myar sxeuls), aradamsvelebeli siTxisaTvis ki blagvi (siTxe Sordeba myar sxeuls). siTxes, romlisTvisac

  0 , ewodeba srulad an absoluturad damsvelebeli. aseTi

siTxe srulad gaerTxmis myari sxeulis zedapirs. Tu    , maSin siTxe absoluturad ar asvelebs myar sxeuls. magaliTad, navTi srulad asvelebs minis an rkinis zedapirs, wyali minis zedapirs da sxv. sruli dasvelebis efeqti iwvevs imas, rom Tavdaxurul minis WurWelSi moTavsebuli wyali uwonobis pirobebSi nawildeba WurWlis mTel zedapirze, haeri ki grovdeba WurWlis SuaSi. es faqti eqsperimentulad pirvelad iqna dadgenili 1962 wels sabWoTa kosmonavtis a. nokolaevisa da p. popoviCis kosmosuri frenis dros. dasvelebisa da aradasvelebis movlena farTod gamoiyeneba yoveldRiur cxovrebasa da teqnikaSi. magaliTad, am movlenazea damyarebuli madnis gamdidrebis f l o t a c i u r i * meTodi. meTodis arsi Semdegia: fxvnilis saxiT aRebul qans, saidanac unda gamoiyos saWiro madani, ureven iseT siTxeSi, romelic asvelebs cariel qans da ar asvelebs madans. paralelurad siTxeSi uberaven haers (xdeba siTxis haeriT gaqreva). carieli qani, romelic madans ar Seicavs, sveldeba da iZireba fskerze. madnis nawilakebs ki garSemoertymeba haeris buStulebi da arqimedes ZaliT amotivtivdeba zedapirze,

sadac xdeba maTi Segroveba. haeris buStu-

lebiT garSemortyma xdeba imitom, rom siTxe romelic ar asvelebs madans, miiswrafvis Seamciros Sexebis farTobi madnis nawilakTan da `miaWers~ mas haeris buStulebs.

warmodgeba inglisuri sityvisagan flottation _ amotivtiveba.

335

siTxis

zedapiri

mruddeba

mxolod

uSualod

WurWlis

kedlebTan.

amitom, Tu WurWeli ganieria, siTxis zedapiris udidesi nawili brtyelia. WurWlis daviwroebasTn erTad siTxis zedapiris brtyeli nawili TandaTan mcirdeba metad viwro milis _ k a p il ari s

SemTxvevaSi mTlianad qreba*.

maSasadame, kapilarSi siTxis zedapiri mTlianad mrudia. aseT mrud zedapirs m e ni s k i * *

ewodeba.

damsvelebeli

siTxis

SemTxvevaSi

meniski

Cazneqilia,

aradamsveleblis SemTxvevaSi ki _ amozneqili. imis

gamo,

rom

meniskis

simrude

Zalian

didia,

warmoiqmneba

mniSvnelovani damatebiTi p wneva, rac iwvevs kapilarSi siTxis donis awevas (Tu siTxe asvelebs kapilaris kedlebs) an dawevas (Tu siTxe ar asvelebs kapilaris

kedlebs) (sur. 96.4 a,b).

movlenas kapilaruli movlena an

kapilaroba ewodeba.

b suraTi 96.4 a, b.

gamovTvaloT siTxis donis cvlileba. ganier WurWelSi CavasxaT siTxe da

CavuSvaT

masSi

r –radiusiani

kapilari,

romlisTvisac

siTxe

damsvelebelia. cxadia, meniski kapilarSi Cazneqili iqneba, maSin rodesac mis gareT brtyelia (sur. 96.5). viciT (§95), rom wneva brtyel zedapirze aris p 0 , xolo

Cazneqil

zedapirze

p  p0  p .

e.i.

kapilarSi

wneva

p

sididiT

naklebia, vidre mis gareT. wnevaTa es sxvaoba gamoiwvevs kapilarSi siTxis donis awevas, rac gagrZeldeba manam, sanam warmoqmnili h simaRlis siTxis svetis wneva (  gh ), ar gaawonasworebs kapilarSi wnevis danakliss. amrigad, siTxis wonasworobisas gveqneba:

es maSin xdeba, rodesac milis kedlebs Soris manZili siTxis zedapiris simrudis radiusis rigisaa. ** warmodgeba berZnuli sityvisagan meniskos _ naxevarmTvare.

336

 gh  p ,

(96.1)

Tu kapilaris kveTi wriulia, maSin siTxis gamrudebuli zedapiri SeiZleba CavTvaloT R –radiusiani sferos nawilad da p 

 gh 

2 , amitom R

2 , R

saidanac

2 . (96.2)  gR simrudis radiusi, Cveuleb-

h meniskis o

riv, cnobili ar aris. amitom (96.2) formuliT sargebloba gaadvildeba, Tu masSi

R –s Sev-

cvliT milis r radiusiT. suraTidan Cans, rom

R

r , sadac  kiduri kuTxea. R –is am mniScos 

vnelobebis Setana (96.2) formulaSi mogvcems:

2 (96.3) cos  .  gr (96.3) formula pirvelad miiRo 1718 wels inh

sur. 96.5

gliselma mecnierma J u re nm a da mis saxels atarebs. igi gansazRvravs kapilarul milSi damsvelebeli siTxis awevis simaRles. formula gamosadegia aradamsvelebeli siTxisaTvisac. am dros  blagvia, cos   0 da h  0 , e.i. (96.3) formula mogvcems siTxis dawevis siRrmes. Jurenis formula SeiZleba gamoviyenoT siTxis zedapiruli daWimulobis koeficientis gansazRvrisaTvis. amisaTvis unda SevarCioT iseTi mili, romelic gamosakvlevi siTxiT srulad sveldeba. maSin   0 da (96.3) formulidan  

 grh 2

. milis radiusi winaswar cnobilia da siTxis awevisas h simaR-

lis gazomviT ganvsazRvravT  –s. (96.3) formula gviCvenebs, rom kapilarSi damsvelebeli siTxis awevis h simaRle milis radiusis ukuproporciulia. amitom Zalian wvril kapilarSi awevis simaRle metad didia. magaliTad, 1 mkm diametris minis kapilarSi

kg   wyali    10 3 3 ,   0,073n/m ,   0  aiwevs m   337

h

2 cos  2  0,073  1  3  30 m  gr 10  9,8  5  10 7

simaRleze. kapilarobis movlena mniSvnelovan rols asrulebs bunebaSi, teqnikasa da yofa-cxovrebaSi. sakmarisia iTqvas, rom iseTi procesi, rogoricaa mcenaris mier niadagidan tenis miReba, am movleniTaa ganpirobebuli (kapilars qmnis mcenaris ujredis kedlebi). niadagis kapilarebis saSualebiT wyali amodis siRrmis fenebidan zedapirul fenebSi, saidanac igi orTqldeba da iwvevs niadagis gaSrobas. niadagis datkepniT kapilaris diametri mcirdeba da gaSrobis procesi Cqardeba. niadagis moxvnisas, piriqiT, xdeba kapilaris wyveta, wylis aorTqleba ferxdeba da niadagis gaSrobis procesi neldeba. Cveulebrivi aguri, agreTve betonis fila, forovani sxeulebia da tens kargad iwovs. amitom, Senobis sinestisagan dasacavad axdenen misi qveda nawilis hidroizolacias. pirsaxociT xelis SemSraleba, saSrobiT melnis amoSroba, navTis Sewova patruqiT, futkris mier yvavilidan neqtris amoReba da mravali sxva kapilarobis movlenazea dafuZnebuli.

§ 97 aorTqleba da kondensacia molekulur-kinetikuri Teoria ara marto arkvevs Tu ratom SeuZlia nivTierebas iyos myar, Txevad an airad mdgomareobaSi, aramed xsnis nivTierebis erTi mdgomareobidan meoreSi gadasvlis processac. realuri airis eqsperimentuli izoTermis ganxilvisas vnaxeT (§69), rom siTxes SeuZlia imyofebodes wonaworobaSi Tavis orTqlTan. ganvixiloT dawvrilebiT orTqladqcevisa da wonasworobis meqanizmi molekulur-kinetikuri Teoriis safuZvelze. viciT, rom siTxis zedapiruli Txeli fena imyofeba gansakuTrebul mdgomareobaSi

_ masze

moqmedebs

siTxeSi

mimarTuli

molekuluri

Zala,

romelic siTxis danarCeni molekulebis mizidviTaa gamowveuli. molekula rom siTxidan amovides, man unda Seasrulos muSaoba am Zalebis winaaRmdegobis dasaZlevad. Cveulebriv, molekulis saSualo kinetikuri energia araa sakmarisi am muSaobis Sesasruleblad, magram maqsvelis ganawilebis kanonis Ta338

naxmad siTxeSi yovel temperaturaze mravladaa iseTi molekulebi, romelTa siCqare (da, maSasadame, kinetikuri energiac) saSualoze metia. aseTi swrafi molekulebi, siTxis zedapirul fenaSi moxvedrisas, sZleven molekuluri mizidulobis Zalebs da amodian siTxidan. siTxidan amosuli molekulebis erTobliobas am siTxis orTqli ewodeba, xolo nivTierebis Txevadi mdgomareobidan airad mdgomareobaSi gadasvlis process orTqladqceva (orTqlwarmoqmna) ewodeba. o r T q l a dq c e v a s , r om e l ic w arm oe b s ne b is m ie r t e m pe r atu r aze s iT xi s T av i s u f al i z e d a pi ri d an a orT q le b a e w o de b a. orTqldeba ara marto siTxe, aramed myari sxeulic, magaliTad, nafTalini, qafuri, yinuli. yinulis aorTqlebiT aixsneba cnobili faqti sveli TeTreulis gaSrobisa yinvaSi. myari sxeulis aorTqlebis process s u b l im a c i a ewodeba. vinaidan aorTqlebisas siTxes tovebs didi energiis mqone molekulebi, siTxeSi darCenili molekulebis saSualo kinetikuri energia mcirdeba. amitom, Tu siTxe garedan ar iRebs siTbos raodenobas, igi civdeba. siTxis gaciveba aorTqlebisas farTod gamoiyeneba praqtikaSi, kerZod macivar-danadgarebSi,

sadac

xdeba

Txevadi

amiakis

naxSirorJangis

aorTqleba. maSasadame, aorTqlebisas siTxe rom ar gacivdes, e.i. procesi rom izoTermuli iyos, siTxes garedan unda mivawodoT siTbos raodenoba. s iT b os r a o d e n ob a s , r om e l ic s aW i ro a e r T e u l i m as is s iT xi s a s ao r T q le b l a d m u dm iv te m pe r atu ra ze , o rT q la dq ce v i s k u T ri s iT b o e w o d e b a * . Tu m masis siTxis asaorTqleblad saWiroa Q siTbos raodenoba, maSin, ganmartebis sistemaSi

Tanaxmad,

misi

orTqladqcevis

erTeulia

sistemgareSe erTeuli

j , kg

CGS

kuTri sistemaSi

siTbo _

Lď&#x20AC;˝

ergi ; g

Q ; m

erTeulTa

xSirad

ixmareba

kal kkal . ď&#x20AC;˝ g kg

orTqladqcevis siTbos (iseve, rogorc dnobis siTbos) xSirad orTqladqcevis farul siTbosac uwodeben. `faruls~ uwodeben imitom, rom misi miwodebisas ar xdeba sxeulis temperaturis gazrda.

339

temperaturis gazrdiT aorTqlebis intensivoba izrdeba, xolo orTqladqcevis

kuTri

siTbo

0  C temperaturaze L  2,5 106

mcirdeba.

magaliTad,

wylisaTvis

j j , xolo 100  C temperaturaze L  2,25  10 6 . krikg kg

tikul temperaturaze, roca ispoba gansxvaveba siTxesa da mis orTqls Soris, L  0.

cxadia, rom orTqladqcevis siTbo warmoadgens siTxis molekulebs Soris moqmedi SeWidulobis Zalebis raodenobriv maxasiaTebels: rac metia es Zalebi, miT metia orTqladqcevis siTbo da piriqiT. Termodinamikis pirveli kanonis Tanaxmad sistemaze gadacemuli siTbos raodenoba

Q  U  A (97.1). Cven SemTxvevaSi Q  L; A  pV , sadac p aris wneva, romelzedac V

mimdinareobs orTqladqceva, xolo

_ nivTierebis kuTri moculobis cvlileba misi

Txevadi mdgomareobidan airadSi gadasvlisas. U aris nivTierebis erTeuli masis Sinagani energiis

cvlileba

misi

Txevadi

mdgomareobidan

airadSi

gadasvlisas.

SeWidulobis Zalebis dasaZlevad Sesrulebuli muSaobiT, e.i. zedapiruli fenis sisqe,

igi

izomeba

U  n' fr , sadac r aris

_ am fenaSi moqmedi saSualo Zala, xolo

n' molekulaTa

raodenoba erTeul masaSi. (97.1) formulaSi CasmiT miviRebT:

L  n' fr  pV . amrigad, ixarjeba

orTqladqcevis

molekuluri

kuTri

SeWidulobis

siTbo

ori

nawilakisagan

(mizidulobis)

Zalebis

Sedgeba.

dasaZlevad

erTi

(97.2) nawili

Sesrulebul

muSaobaze, meore nawili _ gafarToebisas gare p wnevis dasaZlevad Sesrulebul muSaobaze. temperaturis gazrdiT mcirdeba rogorc f , ise V , amitom mcirdeba L –ic.

U  n' fr -is gamoTvla bevr daSvebasTanaa dakavSirebuli da amitom miaxloebiTia. L – is

cdiT

gansazRvris

gavrcelebulia

ramdenime

saimedo

kalorimetruli

meTodi).

meTodi

arsebobs

A  pV –s

(romelTagan

gamoTvlac

did

yvelaze

ufro

siZneles

warmoadgens (vinaidan Zneli ar aris kuTri moculobis cvlilebis gamoTvla nivTierebis Txevadi mdgomareobidan airadSi gadasvlisas). amitom (97.1) toloba, rogorc wesi, gamoiyeneba molekuluri SeWidulobis Zalebis dasaZlevad Sesrulebuli muSaobis gamosaTvlelad (rac saSualebas

gvaZlevs

vimsjeloT

temperaturaze, L  2260  10

Zalebis

sidideze).

magaliTad,

wylisaTvis

100  C

j 3 j 3 j da A  167,145  10 , amitom nfr  2093  10 . rogorc kg kg kg

vxedavT, orTqladqcevis siTbos udidesi nawili ixarjeba molekuluri SeWidulobis Zalebis dasaZlevad Sesrulebul muSaobaze.

340

Tu aorTqleba mimdinareobs Ria WurWlidan, orTqlis molekulebi, difuziisa da haeris nakadis moZraobis gamo, ifantebian sivrceSi. amis gamo, aorTqleba Ria WurWlidan warmoebs manam, sanam siTxe mTlianad ar aorTqldeba. davuSvaT,

aorTqleba

mimdinareobs

daxurul

WurWelSi

mudmiv

temperaturaze. maSin orTqlis molekulebs gafantvis saSualeba aRar eqnebaT. siTburi moZraobis gamo orTqlis molekulebis nawili miuaxlovdeba siTxis zedapirs da dabrundeba siTxeSi. am movlenas, e.i. nivTierebis airadi mdgomareobidan Txevad mdgomareobaSi gadasvlis process, k o nd e ns a ci a ewodeba*. orTqlis molekulas meti energia aqvs, vidre siTxisas, amitom orTqlis kondensaciis dros xdeba energiis gamoyofa siTbos saxiT. siTbos es raodenoba, romelic gamoiyofa orTqlis kondensaciis dros, orTqladqcevis siTbos tolia. cdiT Cveulebriv sazRvraven kondensaciis kuTr siTbos. Tavdapirvelad, molekulebis raodenobas.

orTqlis

raodenoba es

iwvevs

mcire

sWarbobs orTqlis

simkvrivis siTxeSi

simkvrivis

gamo,

siTxidan

dabrunebuli zrdas.

amosuli

molekulebis

orTqlis

simkvrivis

zrdasTan erTad izrdeba siTxeSi dabrunebuli molekulebis raodenobac. bolos

dadgeba

momenti,

rodesac

drois

erTeulSi

siTxidan

amosuli

molekulebis raodenoba gautoldeba siTxeSi dabrunebuli molekulebis raodenobas. am momentidan dawyebuli orTqlis simkvrive mudmivi iqneba.damyardeba e.w.

dinamikuri

(moZravi)

wonasworoba

siTxesa

orTqls

Soris.

aseT

orTqls Cven najeri orTqli vuwodeT. najeri orTqlis wneva an, rogorc xSirad uwodeben xolme, n aje ri o rT q lis d re k a dob a , damokidebulia temperaturaze _ temperaturis gazrdiT igi swrafad izrdeba, vinaidan, erTi mxriv, izrdeba siTxidan amosuli molekulebis raodenoba da, maSasadame, orTqlis simkvrive, meore mxriv izrdeba orTqlis molekulebis saSualo kinetikuri

energia

nivTierebis

2    p  n0W  . 3  

gvarobazec.

najeri

magaliTad,

orTqlis

oTaxis

wneva

temperaturaze

damokidebulia wylis

najeri

orTqlis wnevaa 17,5 tori, xolo eTerisa _ 48,7 tori. *

sublimaciis

Sebrunebul

processac

(e.i.

mdgomareobaSi gadasvlas) kondensacia ewodeba.

341

nivTierebis

airadi

mdgomareobidan

myar

najeri orTqlis wneva, rogorc es aRniSnuli iyo 67-e paragrafSi, araa damokidebuli

moculobaze.

moculobis

gadidebisas

orTqlis

simkvrive

pirvel momentSi Semcirdeba, magram amasTan maSinve dairRveva dinamikuri wonasworoba siTxesa da orTqls Soris. siTxidan amosuli molekulebis raodenoba meti aRmoCndeba kondensirebuli molekulebis raodenobaze. e.i. moxdeba siTxis damatebiTi raodenobis aorTqleba da es gagrZeldeba manam, sanam orTqlis simkvrive da, maSasadame, drekadoba pirvandel sidides ar miaRwevs. moculobis Semcirebisas piriqiT _ Tavdapirvelad simkvrive gaizrdeba, magram amas mohyveba Warbi orTqlis kondensacia, romelis gagrZeldeba manam, sanam orTqlis simkvrive pirvandel sidides ar miaRwevs. 97.1

suraTze

mocemulia

cdiT

miRebuli

najeri orTqlis wnevis temperaturaze damokidebulebis grafiki. mas orTqladqcevis wiri ewodeba. grafikis TiToeuli wertili gamosaxavs najeri orTqlisa da masTan wonasworobaSi myofi siTxis wnevasa da temperaturas. vinaidan mocemul temperaturaze najeri orTqlis wneva metia ujeri orTqlis (airis) wnevaze da naklebia siTxis

sur. 97.1

wnevaze, amitom grafikis qveviT mdebare nebismieri wertili gamosaxavs ujeri orTqlis mdgomareobas, xolo zeviT mdebare nebismieri wertili _ siTxis mdgomareobas. grafiki mTavrdeba kritikul K wertilSi, vinaidan kritikuli wertilis zeviT siTxisa da orTqlis mdgomareobebi ukve aRar gansxvavdeba erTmaneTisagan.

ยง 98. duRili orTqladqcevis metad saintereso saxea duRili. d u R i li e w o de b a in t e n s iu r o rT q la dq c e v a s , rom lic m im d in a re ob s e rT dro u l a d r o g o r c T av is u f al i z e d a pi ri d an , i s e s iT x is m T e l i m o cu l o b id a n. temperaturas, romelzedac duRs siTxe, duRilis temperatura an duRilis wertili ewodeba. duRilis dros siTxis temperatura aRar izrdeba, rac imas

342

miuTiTebs, rom am dros gadacemuli siTbos raodenoba mTlianad ixarjeba orTqladqcevaze. siTxis duRilisaTvis aucilebelia, rom masSi arsebobdes `Tavisufali areebi~,

romelSic

moxdeba

siTxis

molekulebis

gadasvla

(aorTqleba),

orTqlis dagroveba da Semdeg zedapirze amotana. aseTi Tavisufali areebis rols asrulebs haeris (an sxva airis) buStulebi. haeris buStulebi, rogorc wesi warmoiqmneba e.w. aqtiur centrebze _ WurWlis fsersa da kedlebze arsebul mikrouswormaworobebze, bzarebsa da RrmulebSi. haeris buStulebSi xdeba siTxis aorTqleba, amitom buStula, haeris garda, Seicavs wylis najer orTqlsac. duRilis

dawyebamde

sistema

`siTxe-buStulebi~

imyofeba

meqanikur

wonasworobaSi, romelic ganisazRvreba ori pirobiT: 1) buStula ar unda amotivtivdes (wonaworoba simaRlis mixedviT), 2) buStula ar unda gaiWylitos (wonasworoba moculobis mixedviT). buStula ar amotivtivdeba, Tu masze moqmedi arqimedes amomgdebi Zala

 gV (sadac  siTxis simkvrivea, xolo V

_ buStulas moculoba) naklebi

iqneba buStulasa da WurWlis kedels Soris arsebul SeWidulobis Zalaze:

 gV  f , an V  V',

sadac V ' 

g

(98.1)

aris is moculoba, romlis drosac buStula wydeba kedels da

iwyebs amotivtivebas. buStula ar gaiWyliteba, Tu masze warmoebuli p1 wneva gawonaswordeba buStulas SigniT arsebuli p 2 wneviT.

p1 wneva Sedgeba p gare (Cveulebriv, atmosferuli),  gh hidrostatikuri da

2 kapilaruli wnevisagan: r

2 . r p 2 wneva ki aris buStulaSi arsebuli haeris p 0 wnevisa da najeri orTqlis p1  p   gh 

pH wnevis jami: p 2  p0  p H . amrigad, moculobis mixedviT wonasworoba Semdegnairad Caiwereba:

343

p H  p0  p   gh 

2 . r

(98.2)

temperaturis gazrdiT najeri orTqlis wneva buStulaSi izrdeba da buStula farTovdeba, rac iwvevs masSi arsebuli haeris parcialuri

wnevis Semcirebas, buStulas radiusis garkveuli zomisas p H  p0 da ukanasknelis

ugulebelyofa

SeiZleba.

buStulas

zomis

zrdiT

mcirdeba

2 r

kapilaruli wneva. rodesac buStulas moculoba V ' –is toli gaxdeba, igi mowydeba kedels daiwyebs zeviT amosvlas. zeviT amosuli buStula rom gaskdes da gamoiyos masSi arsebuli orTqli, e.i. duRils rom hqondes adgili, saWiroa, rom buStulas SigniT arsebuli wneva toli iyos (ufro zustad, odnav meti iyos) buStulaze moqmedi gare wnevisa. zedapirze

h  0 , amitom hidrostatikuri

wneva nulis tolia. zedapirTan buStulas radiusic iseT zomas aRwevs, rom 2 r

kapilaruli

wneva

mcirea

( p -sTan

SedarebiT)

SeiZleba

misi

ugulebelyofa. amrigad, siTxis duRilis dros buStulaSi arsebuli najeri orTqlis wneva, gare wnevis tolia da (98.2) toloba icvleba tolobiT:

pH  p . amrigad, d r os ac

d u R i li

b u S tu l a S i

m im d in a re ob s ars e b u l i

(98.3) im

s iT xi s

te m pe r at u r aze , na je ri

rom li s

orT q li s

w ne v a

g a u t ol de b a g are w ne v a s s iT xi s T av i s u f al ze d ap ir ze . am ganmartebidan gamomdinareobs, rom duRilis temperatura damokidebuli unda iyos gare wnevaze _ gare wnevis SemcirebiT igi unda Semcirdes da piriqiT. cda adasturebs am daskvnas. marTlac, cnobilia, rom mTaSi wyali ufor dabal temperaturaze duRs, vidre barSi. metad efeqturia haertumbos zarxufis qveS moTavsebul WurWelSi civi wylis aduReba zarxufidan haeris amotumbvisas. kedlidan

buStis

mowyvetis

Semdeg

kedelze

rCeba

patara

buSti,

romelSic kvlav warmoebs siTxis aorTqleba, ris gamoc igi izrdeba da wydeba kedels da a.S. amitom TiToeuli aqtiuri centridan buStebi jaWvebad amodis siTxis zedapirze, rac kargad Cans wylis aduRebisas minis WurWelSi. 344

duRilis

Sedegad

haeris

buStebis

raodenoba

siTxeSi

mcirdeba.

buStebis raodenobis Semcireba SeiZleba siTxis xangrZlivi SenjRreviT (am dros buStebi wydeba WurWlis kedlebs da amodis zeviT) an misi maRali wneviT

(~400

atm)

damuSavebiT

(am

dros

buStebi

`iWyliteba~

masSi

arsebuli haeri ixsneba siTxeSi). aseTnairad damuSavebuli siTxe SeiZleba sagrZnoblad g a d a v axu roT

ise, rom atmosferuli wnevis dros igi ar

aduRdes 170oC temperaturamde. duRilis temperatura damokidebulia siTxis gvarobaze. kerZod, rac ufro maRalia siTxis najeri orTqlis wneva, miT ufro dabal temperaturaze utoldeba igi gare wnevas da miT ufro dabalia siTxis

duRilis

wertili.

magaliTad,

wylis

najeri

orTqlis

wneva

normaluri atmosferuli wnevis (101 325 pa) tolia 100 oC-ze, vercxliswylis najeri orTqlisa ki _ mxolod 357 oC-ze. amitom normaluri atmosferuli wnevis dros wyali duRs 100 oC-ze, vercxliswyali ki 357 oC-ze. dabolos SevniSnoT, rom gaiSviaTebuli areebis (orTqlis buStulebis) warmoqmna siTxeSi principulad SesaZlebelia simkvrivis fluqtuaciis xarjzec. magram vinaidan orTqlis simkvrive gacilebiT naklebia siTxis simkvriveze, aseTi fluqtuaciis albaToba metad mcirea. miT ufro mcirea ramdenadme didi zomis orTqlis buStulis warmoqmnis albaToba. mcire zomis orTqlis buStula ki, didi kapilaruli wnevis gamo, mdgradi ar aris _ myisve iWyliteba, amitomaa, rom siTxis duRilisaTvis aucilebelia masSi airis buStulebis arseboba. ยง 99. fazisa da fazuri gardaqmnis cneba TermodinamikaSi f a z a e w o de b a a r ae rT g v a rov an i s is te m is f iz i k u r a d e r T gv a rov an naw il s * , r om e li c Se iZ le b a m e q an ik u r i s a Su a l e b iT Ca m ov a So roT s xv a e rT g v ar ov an n aw i le b s .

asxvaveben sistemis fizikur da qimiur erTgvarovnebas (an araerTgvarovnebas). sistema fizikurad erTgvarovania, Tu mis yvela nawilSi simkvrive, drekadoba, damagnitebuloba da sxva Tvisebebi erTnairia. Tu sistema Sedgeba ramdenime nivTierebis narevisagan, maSin igi qimiurad araerTgvarovania, Tumca fizkurad SeiZleba erTgvarovani iyos. qimiurad araerTgvarovani sistemaa Senadnobi, xsnari, meqanikuri narevi da sxv. qimiuri naerTi, ra rTulic ar unda iyos, qimiurad erTgvarovania. wyali erTgvarovania fizikuradac da qimiuradac. marilis an mJavas wyalxsnari qimiurad araerTgvarovania fizikurad ki SeiZleba erTgvarovani iyos.

345

erTgvarovan

nivTierebas

h omog enur s

uwodeben,

xolo

araerTgvarovan

sistemas,

romelic Sedgeba ramodenime homogenuri nawilisagan _ h eterog enul s. amitom fazas xSirad ganmartaven rogorc heterogenuli sistemis homogenur nawils.

magaliTad,

daxSul

WurWelSi

moTavsebuli

wyali

mis

zemoT

arsebuli wylis orTqli warmoadgens orfazian sistemas, vinaidan araerTgvarovan sistemaSi (siTxe-airi) gvaqvs ori erTgvarovani nawili (wyali da orTqli), romlebic SeiZleba meqanikuri saSualebiT davaSoroT erTmaneTs (magaliTad, dguSis meSveobiT orTqli gadavdenoT sxva wurWelSi). wylis orTqls rom nebismieri airi, magaliTad, haeri davumatoT, fazaTa raodenoba ar Seicvleba, vinaidan wylis orTqlis da haeris narevi qmnis fizikurad erTgvarovan airs. analogiurad, wyals rom spirti davumatoT, fazaTa raodenoba ar Seicvleba, vinaidan spirti Seereva wyals da mogvcems fizikurad erTgvarovan siTxes. magram Tu wyals davumatebT vercxliswyals, imis gamo, rom vercxliswyali ar Seereva wyals, sistema gaxdeba samfaziani, romelTagan erTi

airadia

(orTqlisa

haeri

narevi),

ori

Txevadi

(wyali

vercxliswyali). Tu wyalSi yinulis natexs CavagdebT, sistema gaxdeba oTxfaziani da Sedgenili iqneba erTi myari (yinuli) ori Txevadi (wyali da verxliswyali) da erTi airadi (wylis orTqlisa da haeris narevi) fazebisagan. fazebis ricxvis dadgenisas ara aqvs mniSvenloba, esa Tu is faza erTian sxeuls warmoadgens Tu ramdenime nawilisagan Sedgeba. magaliTad, wyalSi yinulis erTi natexis nacvlad ramdenime rom CavagdoT, amiT myari fazis ricxvi ar Seicvleba. magram wyalSi Tu qvamarilis natexs davumatebT, sistemaSi ukve ori myari faza gveqneba da sistema gaxdeba xuTfaziani. amrigad, sistemaSi SeiZleba iyos ramdenime myari da Txevadi faza, magram igi ar SeiZleba Seicavdes erTze met airad fazas, vinaidan yvela airi erTmaneTSi ireva. am magaliTebis mixedviT SeiZleboda gvefiqra, rom nivTierebis faza igivea, rac nivTierebis agregatuli mdgomareoba, magram sinamdvileSi es ase ar aris. agregatuli mdgomareobis cneba ufro farToa, vidre fazisa. erTi da igive agregatul mdgomareobaSi nivTiereba SeiZleba imyofebodes sxvadasxva fazaSi, romelTa aRnagoba da Tvisebebi arsebiTad gansxvavdebian erTmaneTisagan.

magaliTad,

airi

SeiZleba

imyofebodes

atomur-molekulur

plazmur fazebSi. mocemuli nivTierebis myari agregatuli mdgomareoba SeiZ346

leba imyofebodes sxvadasxva kristalur modifikaciaSi (polimorfizmi), romlebic am nivTierebis fazebs warmoadgens. almasi da grafiti naxSirbadis sxvadasxva fazaa,  - rkina da  – rkina rkinis sxvadasxva fazaa da sxv. cnobilia yinulis 5 sxvadasxva faza. nivTierebis erTi fazidan meoreSi gadasvlis process fazuri gadasvla an fazuri gardaqmna ewodeba. im temperaturas, romelzedac xdeba fazuri gardaqmna, fazuri gardaqmnis temperatura an fazuri gardaqmnis wertili ewodeba. arsebobs orgvari fazuri gardaqmna _ pirveli gvaris da meore gvaris. pirveli gvaris fazuri gardaqmna ewodeba iseT gardaqmnas, romlis drosac STainTqmeba an gamoiyofa garkveuli siTbos raodenoba (romelsac fazuri gardaqmnis siTbos raodenoba ewodeba). am gardaqmnis dros naxtomiseburad icvleba sxeulis simkvrive, Sinagani energia, entropia. pirveli gvaris fazur gardaqmnas miekuTvneba nivTierebis agregatuli mdgomareobis Secvla (dnoba da gamyareba, sublimacia, orTqladqceva da kondensacia), agreTve myari sxeulis erTi kristaluri modifikaciidan meoreSi gadasvla. am ukanasknels xSirad polimorful gardaqmnasac uwodeben. meore

gvaris

fazuri

gardaqmna

ewodeba

iseT

gardaqmnas,

romelic

mimdinareobs siTburi efeqtis gareSe. am dros ar xdeba siTbos raodenobis STanTqma an gamoyofa. meore gvaris fazuri gardaqmnis dros naxtomiseburad icvleba ara simkvrive, Sinagani energia an entropia, aramed nivTierebis Tvisebis damaxasiaTebeli iseTi sidideebi, rogoricaa siTbotevadoba, siblante, moculobiTi gafarToebisa da kumSvis koeficientebi da sxv. meore gvaris fazuri gardaqmnis magaliTebia dabal temperaturaze zogierTi nivTierebis gadasvla zegamtar mdgomareobaSi, Txevadi heliumis gadasvla zedenad mdgomareobaSi, kiuris temperaturaze feromagnetikis gardaqmna paramagnetikad da sxv. pirveli

gvaris

fazuri

gardaqmnisaTvis

damaxasiaTebelia

is,

rom

fazuri gardaqmnis wertilSi axali fazis warmoqmna mTel moculobaSi uceb ki ar xdeba, aramed jer warmoiqmneba am fazis Canasaxi (mag. siTxis gamyarebis dros kristalizaciis centrebi, orTqlis kondensaciis dros _ kondensaciis centrebi), romelic TandaTan izrdeba `Zveli~ fazis xarjze da ikavebs mTel moculobas. erTi fazis meoreSi TandaTanobiT gadasvlis procesSi adgili aqvs `Zveli~ da `axali~ fazebis wonasworobas. 347

meore gvaris fazuri gardaqmnis wertilSi (romelsac xSirad kiuris wertils an  wertilsac uwodeben) nivTierebis gadasvla erTi fazidan meoreSi mTel moculobaSi xdeba erTbaSad, naxtomiseburad. amitom, cxadia, rom am gardaqmnis dros Zveli da axali fazebis wonasworobaze laparaki ar SeiZleba. rogoria pirveli gvaris fazuri gardaqmnis dros fazaTa wonasworobis pirobebi? ori fazis _ siTxisa da misi najeri orTqlis wonasworobas ukve gavecaniT realuri airis eqsperimentuli izoTermis ganxilvisas. am kerZo magaliTis safuZvelze SeiZleba davaskvnaT, rom urTierTSemxebi ori fazis wonasworobisaTvis

aucilebelia

fazaTa

temperaturebis,

wnevebisa

masebis toloba T1  T2 ; p1  p2 ; m1  m2  *. ori

fazis

intervalSi,

wonasworoba

amasTan,

SesaZlebelia

temperaturis

garkveuli mniSvneloba,

romlis

yovel

drosac

temperaturis

mniSvnelobas SesaZlebelia

garkveul

eTanadeba

wnevis

wonasworoba.

rac

Seexeba sam fazas, erTi da imave nivTierebis sami faza (myari, Txevadi da airadi) wonawororbaSi SeiZleba iyos temperaturisa da wnevis erTaderTi mniSvnelobis dros, romelsac

 p, T 

sibrtyeze Seesabameba wertili, romelsac

s am m a gi w e r ti li ewodeba. TermodinamikaSi mtkicdeba, rom erTi da igive nivTierebis sam fazaze meti wonasworobaSi ar SeiZleba iyos. fazuri gardaqmna TvalsaCino gamovlenaa dialeqtikuri materializmis zogadi kanonisa _ raodenobrivi cvlilebis TvisobrivSi gadasvlis kanonisa. magaliTad,

ganvixiloT

dnobis

procesi.

gaTbobisas

myari

sxeulis

temperaturis TandaTanobiT zrda iwvevs masSi `raodenobis dagrovebas~ _ Semadgeneli nawilakebis kinetikuri energiis TandaTanobiT zrdas ise, rom sxeulis

Tvisobrioba

icvleba

sxeuli

rCeba

myar

mdgomareobaSi.

gaTbobis garkveul stadiaze xdeba raodenobrivi cvlilebebis TvisobrivSi n a xt om is e b u ri gadasvla _ sxeuli dneba da gadadis Tvisobrivad axal (Txevad) mdgomareobaSi.

m1  m2 gamosaxavs im faqts, rom wonasworobis dros erTi fazidan meoreSi (vTqvaT, siTxidan mis najer orTqlSi) gadasuli n12 molekulebis ricxvi tolia imave droSi meore fazidan pirvelSi (orTqlidan siTxeSi) gadasuli n 21 molekulebis ricxvisa: n12 = n 21 348 *

toloba

§ 100. klapeiron-klauziusis gantoleba cnobilia, ratura Rilis

rom

pirveli

damokidebulia temperatura

gvaris

wnevaze.

izrdeba,

nebismieri

magaliTad, xolo

fazuri

wnevis

yinulis

gardaqmnis tempe-

gadidebiT siTxis

dnobis

du-

temperatura mcir-

deba. raodenobrivad es damokidebuleba gamoisaxeba klapeiron-klauziusis gantolebiT. gamoviyvanoT es gantoleba orTqladqcevis da kondensaciis kerZo SemTxvevisaTvis. amisaTvis ganvixiloT karnos elementaruli cikli, romelSic izoTermul procesebad aRebulia siTxis orTqladqcevisa da orTqlis kondensaciis procesebi. davuSvaT, dguSian WurWelSi moTavsebulia duRilis temperaturamde miyvanili erTeulovani masis siTxe. aRvniSnoT misi moculoba V1 –iT, wneva p –Ti da temperatura T –Ti. sawyis momentSi aira-

di faza ara gvaqvs, e.i. WurWlis moculoba mTlianad siTxes ukavia. realuri airis izoTermaze siTxis es mdgomareoba gamosaxulia A wertiliT (sur. 100.1), amasTan abscisaTa RerZze gadazomilia

sur. 100.1

kuTri moculoba. movaTavsoT siTxiani WurWeli saxurebelze, romlis temparaturaa T , siTxe saxureblisagan miiRebs garkveul Q1 siTbos raodenobas, aorTqldeba da gamoiwvevs dguSis gadaadgilebas _ mimdinareobs orTqlis izoTermuli gafarToeba. orTqlis gafarToebis procesi amave dros izobarulic aris, vinaidan, rogorc viciT, mocemul temperaturaze najeri orTlis wneva moculobaze ar aris damokidebuli. procesi gavagrZeloT manam, sanam siTxe mTlianad ar aorTqldeba, e.i. mdgomareobis gamomsaxveli wertili ar aRmoCndeba B

wertilSi ( V2

gafarToebuli orTqli kuTri moculobaa). amisaTvis man

saxureblisagan unda miiRos L

orTqladqcevis kuTri siTbos toli siTbos

raodenoba, e.i. Q1  L . amis Semdeg movaSoroT WurWeli saxurebels da orTqli gavacioT ise, rom misi temperatura Semcirdes dT

sididiT, xolo wneva dp –Ti. orTqlis

349

axal mdgomareobas gamosaxavs

wertili

T  dT

temperaturis Sesabamis

izoTermaze. Semdeg movaTavsoT WurWeli macivarze, romlis temperaturaa T  dT da SevkumSoT najeri orTqli izoTermul-izobarulad manam, sanam igi mTlianad ar kondensirdeba. am dros sistema gadascems macivars Q2 siTbos raodenobas, mdgomareobis gamomsaxveli wertili gahyveba CD

wrfes da aRmoCndeba

wertilSi. cxadia, Q2  Q1 , vinaidan kondensacia mimdinareobs ufro dabal, T  dT temperaturaze.

dabolos, miRebuli siTxe gavaTboT ise, rom is kvlav dabrundes sawyis mdgomareobaSi ( A wertili). vinaidan dp da dT usasrulod mcirea, SeiZleba ugulebelvyoT BC da DA ubnebze gacemuli da siTbos raodenobebi da CavTvaloT es procesebi

TiTqmis adiabaturad. maSin ABCD cikli Sedgenili iqneba ori izoTermisa da ori adiabatisagan, e.i. igi karnos cikls warmoadgens, amitom misi margi qmedebis koeficienti

Q1  Q2 T1  T2 , (100.1)  Q1 T1 magram Q1  Q2  A , sadac A ciklis dros Sesrulebuli muSaobaa da



ricxobrivad

cikliT

SemosazRvruli

ABCD

farTobis

tolia.

ciklis

elementalurobis gamo ABCD oTxkuTxedi SeiZleba miviCnioT marTkuTxedad, amitom

A  dpV2  V1 ; Q1  L; T1  T ,

xolo

T2  T1  dT ,

amitom

T1  T2  dT .

mniSvnelobis Casma (100.1) formulaSi mogvcems dpV2  V1  dT  , L T

saidanac

dp L  dT T V2  V1  an

(100.2)

dT T V2  V1   (100.2’) dp L (100.2) gantolebas, an rac igivea (100.2’) gantolebas, ewodeba klapeiron-

klauziusis gantoleba. es gantoleba Cven gamoviyvaneT orTqladqcevisa da kondensaciis kerZo SemTxvevisaTvis, magram vinaidan karnos ciklis mq koeficienti ar aris damokidebuli muSa sxeulis gvarobaze, igi marTebulia nebis350

mieri pirveli gvaris fazuri gadasvlisaTvis (dnoba, gamyareba, sublimacia, kondensacia).

amrigad,

klapeiron-klauziusis

gantoleba

zogadia,

igi

gamosaxavs pirveli gvaris fazuri gadasvlis temperaturis damokidebulebas wnevaze. (100.2’) gantolebis marjvena nawilSi temperatura da fazuri gadasvlis kuTri siTbo arsebiTad dadebiTi sidideebia. maSasadame,

dT -s niSani dadp

mokidebulia V1 –sa V2 –is Tanafardobaze. orTqladqcevisa da sublimaciis dros yovelTvis V2  V1 , amitom

dT  0 . es imas niSnavs, rom duRilisa da dp

sublimaciis temperatura wnevis gadidebiT izrdeba da piriqiT. rac Seexeba dnobas, nivTierebaTa umravlesobisaTvis V2  V da maTTvis wnevis gadidebiT dnobis temperatura izrdeba. magram zogierTi nivTierebisaTvis, romelTac anomaluri ewodeba, V  0 . aseT nivTierebebs miekuTvneba yinuli, bismuti, germaniumi, agreTve zogierTi xarisxis Tuji. maTTvis

dT  0. dp

e.i. wnevis gadidebiT am nivTierebaTa dnobis temperatura mcirdeba da piriqiT.

§ 101. mdgomareobis diagrama. sammagi wertili 97.1 suraTze moyvanili DK wiris TiToeuli wertili gamosaxavs wnevisa da temperaturis im mniSvnelobas, romlis drosac ori faza_siTxe da misi najeri orTqli imyofeba wonasworobaSi. amitom

wirs xSirad fazuri

wonasworobis wirsac uwodeben. wiris TiToeuli wertilis ageba xdeba eqsperimentulad an klapeiron-kaluziusis gantolebis meSveobiT. cxadia, fazuri wonasworobis analogiuri wirebi SeiZleba aigos dnobisa da sublimaciis p r oc e s is T v is a c (agreTve polimorfuli gardaqmnisaTvis). Tu dnobis, orTqladqcevisa da sublimaciis wirebs avagebT erT naxazze, aRmoCndeba, rom isini ikveTebian erT D wertilSi (sur. 101.1). suraTidan Cans, rom dnobis wiri ( DL ) ufro cicaboa, vidre orTqladqcevisa ( DK ) da sublimaciis ( DA ) wirebi. es imiTaa gamowveuli, rom orTqladqcevisa da sublimaciis dnobis

dros

dros.

kuTri moculobis

magaliTad,

wylis

cvlileba gacilebiT

duRilis 351

temperaturis

metia,

erTi

vidre

gradusiT

Sesacvlelad saWiroa wnevis mxolod 3,610-2 atmosferoTi gazrda, yinulis SemTxvevaSi ki aseTive efeqtis misaRebad saWiroa wnevis 1,3102 atm-Ti gazrda. D

wertili, romelSic ikveTeba dno-

bis, orTqladqcevisa da sublimaciis wirebi, gamosaxavs im wnevasa da temperaturas, romlis drosac mocemuli nivTierebis samive faza (myari, Txevadi da airadi) imyofeba wonasworobaSi. mas s am m a gi w e r t il i ewodeba. aRsaniSnavia, rom sammagi wertili gaaCnia yvela nivTierebas, garda heliumisa*. magali-

sur. 101.1

Tad wylisaTvis p D  4,6 mm vwy.sv. da TD  0,01 C . dnobis, orTqladqcevisa da sublimaciis wirebi pT sibrtyes yofs sam nawilad, romelTagan TiToeulis wertili gamosaxavs nivTierebis myar, Txevadsa

airad

mdgomareobas.

amitom

101.1

suraTze

mocemul

diagramas

m d gom ar e ob is di a gr am a ewodeba. mdgomareobis diagramaze DL wiri aris iseTi nivTierebebis dnobis wiri, romelTa kuTri moculoba dnobis Sedegad izrdeba.

cxadia,

rom

anomaluri

nivTierebebisaTvis

dnobis

wirs

eqneba

uaryofiTi daxra ( DL' wyvetili wiri). mdgomareobis diagrama Zalian mosaxerxebelia imis gamosarkvevad, Tu ra mdgomareobaSi imyofeba mocemuli nivTiereba wnevisa da temperaturis mocemuli mniSvnelobebis dros. magaliTad, wnevisa da temperaturis im pirobebSi, romlebic gamosaxulia 1 wertiliT, nivTiereba imyofeba myar mdgomareobaSi, 2 wertili gamosaxavs myari da Txevadi fazis wonasworobis pirobas, 3 wertilis Sesabamisi wnevisa da temperaturis pirobebSi nivTiereba imyofeba Txevad mdgomareobaSi, 4 wertili gamosaxavs siTxisa da orTqlis wonasworobis pirobas, xolo 5 wertilis Sesabamisi wnevisa da temperaturis pirobebSi nivTiereba imyofeba airad mdgomareobaSi.

p  p D wnevis dros, nivTiereba, imis

mixedviT Tu rogoria misi temperatura, SeiZleba imyofebodes rogorc myar, ise Txevad da airad mdgomareobaSi. Tu

p  p D , nivTierebis Txevadi mdgo-

mareoba SeuZlebelia, gaTbobisas igi pirdapir gadadis airad mdgomareobaSi. *

heliumis es Tavisebureba imiT aixsneba, rom igi Txevad mdgomareobaSi rCeba absolutur nulzec, Tu gare wneva 25 atmosferoze naklebia ($106).

352

XIII Tavi

fizikuri kinetikis elementebi ยง 102. gadatanis movlenebi wina TavebSi vixilavdiT makroskopuli sistemis wonasworul mdgomareobas. axla ganvixiloT arawonasworuli mdgomareoba, rodesac sistemis sxvadasxva nawilSi simkvrive, fenebis gadaadgilebis siCqare an temperatura sxvdasxvaa. fizikis nawils, romelic Seiswavlis arawonasworul sistemaSi mimdinare procesebs, fi zik u ri k ine tik a ewodeba, TviT am procesebs ki xSirad k i ne t ik u rs uwodeben. yoveli sistema wonasworobis darRvevisas miiswrafvis wonasworobis mdgomareobisaken. am process, romelsac ganapirobebs nawilakebis qaosuri moZraoba da damokidebulia maTi urTierTqmedebis xasiaTze, Tan sdevs entropiis zrda da, maSasadame, igi Seuqcevia. amrigad, procesebi, romlebsac Seiswavlis fizikuri kinetika (`kinetikuri procesebi~), Seuqcevia. yoveli kinetikuri procesis dros xdeba raime sididis (masis, muxtis, saSualo kinetikuri energiis an impulsis) gadatana sistemis erTi nawilidan meoreSi. amitom am procesebiT ganpirobebul movlenebs g a d at an is m ov le ne b i ewodeba. gadatanis movlenebia d i fu zi a (masis gadatana), T b o gam t ar ob a (impulsis

(saSualo kinetikuri energiis gadatana), S i n a g a ni gadatana),

e l e q t ro g am ta r ob a

(muxtis

gadatana).

x a xu n i mocemul

TavSi ganvixilavT gadatanis pirvel sam movlenas*. amasTan, vinaidan es movlenebi damokidebulia sistemis

nawilakebis

urTierTqmedebis

xasiaTze,

romelic yvelaze martivia airSi, difuzias, Sinagan xaxunsa da Tbogamtarobas ganvixilavT mxolod airSi. garda amisa, davuSvebT, rom airis wonasworobis mdgomareobidan gadaxra didi ar aris. fizikur kinetikaSi, iseve rogorc wonasworuli sistemis ganxilvisas, SesaZlebelia sakiTxisadmi ori principulad gansxvavebuli midgoma: makroskopuli (fenomenologiuri) da mikroskopuli (molekulur-kinetikuri). gadatanis movlenebs jer ganvixilavT fenomenologiuri TvalsazrisiT da

davwerT

difuziis,

Tbogamtarobisa

Sinagani

xaxunis

empiriul

gantolebebs, romlebic marTebulia nebismieri (myari, Txevadi da airadi) *

eleqtrogamtarobas ganvixilavT kursis meore tomSi.

353

garemosaTvis. Semdeg amave gantolebebs, molekulur-kinetikuri Teoriis saSualebiT, gamoviyvanT airisaTvis. gadatanis movlenebis mkacri molekulur-kinetikuri Teoria metad rTulia.

igi

daiyvaneba

e.w.

b olc m a ni s

k ine tik u ri

ga nt ole b is

miaxloebiT amoxsnaze. es gantoleba, romelic ZiriTadia airis kinetikur TeoriaSi, principSi SesaZleblobas iZleva moinaxos molekulebis koordinatebisa da siCqareebis mixedviT ganawilebis funqcia ara marto wonasworobis mdgomareobaSi, aramed maSinac, roca airSi mimdinareobs sxvadasxva procesi. magram bolcmanis gantolebas Cven ar gamoviyenebT. difuziis, Sinagani xaxunisa da Tbogamtarobis ganxilvisas visargeblebT gacilebiT ufro

martivi meTodiT,

romelic dafuZnebulia molekulis

s a Su a lo

T av i s u f al i g an arb e n is cn e b a ze . aRsaniSnavia, rom es gamartivebuli meTodic iZleva imis saSualebas, rom davadginoT gadatanis movlenebis yvela

ZiriTadi

Tviseba,

oRond

koeficientebis

miRebuli

ricxviTi

mniSvnelobebi mTlad zusti ar iqneba*. dabolos, SevniSnoT, rom sazogadod, gadatanis procesi ar aris stacionaruli, vinaidan droTa ganmavlobaSi sistemis sxvadsxva nawils Soris gansxvaveba mcirdeba da amitom saTanado sididis gadatanas sul ufro da ufro neli tempiT mimdinareobs. procesi stacionaruli SeiZleba iyos mxolod araizolirebul sistemaSi, roca gansxvaveba sistemis sxvadasxva nawils Soris xelovnuradaa daculi. cxadia, drois mcire SualedisaTvis gadatanis procesi yovelTvis SeiZleba ganvixiloT rogorc stacionaruli.

§ 103. molekulebis dajaxebaTa ricxvisa da saSualo Tavisufali ganarbenis gamoTvla T av i s u f al i g an arb e n is cneba pirvelad Semoitana klauziusma im Seusabamobis

asaxsnelad,

romelic

arsebobda

molekulis

moZraobis

siCqaresa da difuziis procesis siCqares Soris. maTlac, 61-e paragrafSi vnaxeT, rom airis molekulis siCqare oTaxis temperaturaze Zalian didia (   500 m wm ), igi daaxloebiT tyviis siCqaris tolia. meore mxriv cnobilia,

es gamowveulia imiT, rom rogorc qvemoT vnaxavT, meTodSi

gamoyenebulia siCqarisa da

Tavisufali ganarbenis saSualo mniSvnelobebi da ar aris gaTvaliswinebuli am sidideebis ganawileba maqsvelis kanonis mixedviT.

354

rom oTaxSi daRvrili sunamos suni gacilebiT nela vrceldeba. es Seusabamoba klauziusma axsna m o le k u l aT a da j a xe b iT . sunamos molekulebi Tavisuflad

(daubrkoleblad)

moZraoben, aramed ganicdian mravaljerad dajaxebebs haeris molekulebTan (da erTmaneTTan). TiToeli dajaxebisas icvleba molekulis siCqaris rogorc sidide, ise mimarTulebac, igi moZraobs texil wirze, ris gamoc mis

sur. 103.1

mier

erTi

garkveuli

mimarTulebiT

gavlili AB manZili mcirea (sur. 103.1). erTi dajaxebidan meoremde molekula moZraobs wrfivad da Tanabrad da gadis manZils, romelsac T av i s u f a li

g an ar b e n i ewodeba (  1,  2 ,  3 ... manZilebi 103.1 suraTze). molekulis

Tavisufali ganarbeni, iseve rogorc misi moZraobis damaxasiaTebeli sxva sidideebi (siCqare, energia, impulsi da sxv.), swrafad da uwesrigod icvleba, amitom SeiZleba ganisazRvros mxolod s a Su al o Tavisufali ganarbeni  , romelic warmoadgens saSualo manZils or momdevno dajaxebas Soris: z

 

aRvniSnoT

molekulis

dajaxebaTa saSualo ricxvi 1 wm-Si

gavlili

manZilis

 i 1

. z saSualo siCqare

 –Ti, xolo erT wamSi

z –iT, maSin (vinaidan siCqare ricxobrivad tolia)

cxadia,

rom

saSualo

Tavisufali

. z gamovTvaloT z . amisaTvis gamartivebis mizniT davuSvaT, rom:

(103.1)

ganarbeni

 



1. molekula dajaxebisas moZraobis mimarTulebas ar icvlis. 2. mocemuli molekula moZraobs, danarCeni ki uZravia. CavTvaloT, rom molekula aris

r –radiusiani birTvi. maSin igi moZraobisas daejaxeba yvela im molekulas, romelTa centrebi daSorebulia sur. 103.1

misi centridan araumetes 2r  d man355

Zilisa (sur. 103.2). radgan molekula 1 wamSi gadis  manZils, amitom igi am droSi

daejaxeba

radiusisa da 

yvela

molekulas,

romelic

moTavsebulia

simaRlis mqone cilindrSi. am cilindris moculobaa

 d 2 , Tu molekulaTa raodenoba moculobis erTeulSi aris n0 , maSin am cilindrSi moTavsebul molekulaTa raodenoba iqneba n0 d 2 . yvela am molekulas daejaxeba mocemuli molekula 1 wamSi, amitom (103.2) z   d 2 n0 . SeiZleba Cveneba, rom Tu gaviTvaliswinebT yvela molekulis moZraobas, maSin (103.3) z  2 d 2 n0 . Tu z –is miRebul mniSvnelobas SevitanT (103.1) formulaSi, saSualo Tavisufali ganarbenis gamosaTvleli formula miiRebs saxes:



, (103.4) 2 d 2 n0 e.i. saSualo Tavisufali ganarbeni miT metia, rac mcirea molekulis d diametri da molekulaTa n0 koncentracia. es gasagebia, vinaidan, rac ufro gaiSviaTebulia airi da rac ufro mcire zomisaa molekulebi, miT ufro Tavisuflad moZraoben isini, urTierTdajaxeba iSviaTia da amis gamo  didia. airis kinetikuri Teoriis ZiriTadi gantolebis Tanaxmad idealuri airis wneva p  n0 kT . amis gamoyenebiT (103.4) tolobis safuZvelze miviRebT:

 1 n02 p 2 ,    2 n01 p1

(103.5)

e.i. molekulis saSualo Tavisufali ganarbeni airis wnevis ukuproporciulia, rac gasagebia, radgan ucvleli temperaturis dros airis wnevis gadideba iwvevs misi moculobis Semcirebas. (103.3) da (103.4) formulebSi Semavali sidideebis miaxloebiTi mniSvnelobebis Casma gvaZlevs z da  sidideebis Sefasebis saSualebas. magaliTad, Tu aviRebT:

d ~ 10 10 m, n0 ~ 3  10 25 m - 3 ,  ~ 103 m/wm, miviRebT: z ~ 10 9 wm -1 da  ~ 10 6 m , e.i. molekula ganicdis wamSi daaxloebiT miliard dajaxebas da dajaxebidan dajaxebamde gadis mkm-is rigis manZils. wnevis SemcirebiT saSualo

356

Tavisufali ganarbeni izrdeba da didi gaiSviaTebis dros (103.5) formulis Sesabamisad ramdenime metrs aRwevs. dabolos SevniSnoT, rom vinaidan saSualo Tavisufali ganarbenis gansazRvris sxvadasxva meTodi arsebobs, (103.4) formulas xSirad iyeneben molekulis efeqturi diametris gamosaTvlelad. rogorc aRvniSneT, molekulebs Soris ganzidvis Zalebis arseboba ganapirobebs imas, rom molekulebis uSualo Sexlas arasdros ara aqvs adgili. molekulaTa dajaxebaSi igulisxmeba maTi miaxloeba iseT manZilze, roca Zlieri ganzidvis Zalis moqmedebiT molekulebi iwyeben erTmaneTisagan daSorebas. es xdeba maSin, rodesac maTi urTierTganzidvis potenciuri energia gadaaWarbebs fardobiTi moZraobis kinetikur energias. umcires d manZils, romelzec ori molekulis centri SeiZleba erTmaneTs miuaxlovdes, ewodeba molekulis efeqturi diametri, xolo

ď ° d 2 â&#x20AC;&#x201C;s

_ efeqturi kveTa. zusti gamoTvlebi

gvaZlevs, rom (103.3) da (103.4) formulebSi. d aris molekulis efeqturi diametri. vinaidan molekulis kinetikuri energia absoluturi temperaturis proporciulia, amitom cxadia, rom molekulis efeqturi diametri damokidebuli iqneba temperaturaze,

T -s gazrdiT d Semcirdeba. amiT aixsneba eqsperimentuli faqti _ saSualo Tavisufali ganarbenis sididis ramdenadme gazrda temperaturis gazrdiT. Cven mier miRebulma Sedegma molekulis Tavisufali ganarbenisa da dajaxebaTa ricxvis gamo SeiZleba gvafiqrebinos, rom saerTod ar aqvs azri airis idealurobaze laparaks. marTlac, Tu molekula wamSi miliardobiT dajaxebas ganicdis, dajaxebaSi ki Cven vgulisxmobT urTierTqmedebas, aseTi xSiri dajaxebebi unda niSnavdes Zlier urTierTqmedebas. magram advili saCvenebelia, rom es ase ar aris. marTlac, molekulebi urTierTqmedeben im drois ganmavlobaSi, vidre isini daSorebulni arian erTmaneTisagan molekulis diametris rigis manZiliT, e.i. 10-10 m-iT. danarCen dros ki dajaxebebs Soris isini moZraoben Tavisuflad, urTierTqmedebis gareSe. vinaidan molekulis Tavisufali ganarbeni 10-6 m rigisaa, rac 104-jer sWarbobs molekulis zomebs, amitom dro, romlis ganmavlobaSic molekulebi ar urTierTqmedeben, amdenjerve sWarbobs maTi urTierTqmedebis dros. aseT pirobebSi, cxadia, Tavisuflad SeiZleba urTierTqmedebis ugulebelyofa. amitomaa, rom iseTi airi, rogoricaa, Jangbadi da azoti, rom araferi vTqvaT wyalbadze da

heliumze,

Cveulebriv

pirobebSi

kargad

akmayofileben

idealurobis pirobas.

molekulis

saSualo

Tavisufali

ganarbenis

eqsperimentuli gansazRvris ramdenime meTodi arsebobs (difuzia, Tbogamtaroba da sxv.). Cven ganvixiloT erTerTi maTgani _ molekuluri konis meTodi, romelic dafuZnebulia molekuluri (atomuri) konis gamoyenebaze. cdis idea SemdegSia. aRebulia minis mili, romlis erTerT

boloze

moTavsebulia

advilad 357

aqroladi

sur. 103.3

liToni (sur. 103.3). milSi moTavsebulia a diafragma da C1 , C2 , C3 ... tixrebi. milSi haeri gaiSviaTebulia imdenad, rom Tavisufali ganarbeni milis sigrZis rigisaa. Tu milis im bolos, sadac

liTonia moTavsebuli,

gavaTbobT, liToni aorTqldeba. aorTqlebuli atomebis nawili gaivlis a diafragmaSi,

imoZravebs

wrfivad

Tanabrad

milis

mopirdapire

boloze mogvcems K ’ danaleqs*. cxadia, rom wnevis aRniSnuli mniSvnelobis dros C1 , C2 , C3 ... tixrebze liTonis atomebi ver moxvdeba. vzardoT milSi TandaTan haeris wneva. wnevis raRac

mniSvnelobaze liTonis atomebi

daileqebian ara milis boloze, aramed C3 tixarze. tixarze atomebis daleqvis faqti miuTiTebs imas, rom Semcirda maTi Tavisufali ganarbeni, igi aC3 –s toli gaxda, am manZilis gavlis Semdeg liTonis atomebi Seejaxa haeris molekulebs, icvala mimarTuleba da moxvda C3 tixarze. wnevis Semdgomi gazrdiT danaleqi miiReba ukve C 2 tixarze, rac imas miuTiTebs, rom am wnevis dros atomebis Tavisufali ganarbeni aC2 –s tolia da a.S. aRsaniSnavia, rom am meTodiT gansazRvruli Tavisufali ganarbeni kargad

Tanxvdeba

(103.4)

formuliT

(molekulur-kinetikuri

TeoriiT)

gamoTvlil saSualo Tavisyfal ganarbens.

§ 104. difuzia vTqvaT, mocemulia airi, romlis simkvrive mis sxvadasxva nawilSi sxvdasxvaa. simartivisaTvis davuSvaT, rom simkvrive icvleba mxolod

RerZis mimarTulebiT (    x  ) da cdis ganmavlobaSi misi gradienti mudmivia (sur. 104.1), e.i. procesi stacionarulia. siTburi moZraobis gamo airis molekulebi gadaadgildebian yvela mimarTulebiT, magram cxadia, rom drois erTeulSi RerZis marTobul

S

farTobSi marcxnidan marjvniv

molekulebis ufro meti raodenoba gadava, vidre marjvnidan marcxniv, e.i. adgili eqneba Warbi M masis gadatanas marcxnidan marjvniv.

erTi mimarTulebiT gavrcelebuli atomebis (molekulebis) erTobliobas, romlebic

erTmaneTTan

urTierTqmedeben,

ewodeba

gamomdinareobs meTodis dasaxeleba.

358

atomuri

(molekuluri)

kona.

aqedan

n iv T ie re b is

g a da t an as ,

g am ow v e u l s

m ole k u l aT a

q a os u r i

m oZ r ao b iT d a f e ne b s So ri s s im k v r iv e T a g a n s xv av e b iT , d i fu z i a ewodeba. 1855 wels germanelma fizikosma fikma empiriulad daadgina, rom difuziis dros S farTobSi gadatanili M masa miT metia, rac metia d simdx

gansxvaveba airis sxvadasxva fenis simkvrives Soris, e.i. rac metia

kvrivis gradienti, S farTobis da t dro, romlis ganmavlobaSic mimdinareobs difuziis procesi:

M   D (104.1)

d  S  t . dx

formulas

(104.1)

ewodeba

difuziis

formula. mas fikis I kanonsac uwodeben. niSani

minusi

miuTiTebs,

rom

masis

gadatana xdeba simkvrivis Semcirebis mimarTulebiT. proporciulobis D koeficients ewodeba difuziis koeficienti. misi fizikuri arsis gasagebad (104.1) formulaSi davuSvaT,

rom

dρ  1, ΔS  1 , dx

t  1 ,

maSin

M  D , e.i. difuziis koeficienti ricxobsur. 104.1

rivad tolia im masisa, romelic gadaitane-

ba drois erTeulSi farTobis erTeulze, rodesac simkvrivis gradienti erTis tolia. SI sistemaSi igi izomeba m2/wm-obiT, xolo CGS sistemaSi sm2/wm-obiT. ganvixiloT

difuziis

movlena

molekulur-kinetikuri

Teoriis

TvalsazrisiT. S farTobidan marcxniv da marjvniv,  Tavisufali ganarbenis tol manZilze, gamovyoT airis ori Txeli fena (I da II) (sur. 104.1). molekulebis koncentracia pirvel fenaSi aris n1 , xolo meoreSi _ n 2 . vinaidan I fena daSorebulia

S

farTobidan



manZiliT, amitom am

fenidan molekulebi S farTobamde mivlen dajaxebis gareSe da, maSasadame, S farTobSi t droSi gaivlis is molekulebi, romlebic amave droSi gamovlen I fenis S1 farTobidan ( S1  S 2  S ). gamovTvaloT molekulebis es raodenoba. t droSi molekula gaivlis  t 359

manZils, sadac 

molekulis

saSualo

siCqarea.

maSasadame,

t

droSi

S1

farTobidan

gamovlen mxolod is molekulebi, romlebic daSorebuli arian misgan araumetes  t manZilisa. avagoT paralelepipedi, romlis fuZea S1 , xolo simaRle  t . am paralelepipedis moculoba iqneba  St , xolo molekulebis raodenoba masSi n1 St . molekulebis moZraobis qaosurobis gamo SeiZleba CaiTvalos (miaxloebiT, vinaidan arsebobs simkvrivis gradienti), rom am raodenobidan 1/3 moZraobs

RerZis gaswvriv, xolo amaTgan

S1 farTobidan

naxevari, e.i. 1/6 _ marcxnidan marjvniv. maSasadame, droSi

gamosuli

molekulebis

raodenoba

aRvniSneT, yvela es molekula gava S

1 n1 St . 6

iqneba

farTobSi, e.i. S

t

rogorc

farTobSi t

droSi marcnidan marjvniv gasuli molekulebis raodenoba N1 

1 n1 St . 6

(104.2)

gadatanili masis sapovnelad molekulebis es raodenoba unda gavamravloT erTi molekulis m masaze: M 1  mN1 

1 mn1 St 6

(104.3)

an, vinaidan mn1   1 aris airis simkvrive I fenaSi, M1 

analogiurad miviRebT,

rom

1  1 St . 6 S

(104.4)

farTobSi

t

droSi marjvnidan

marcxniv gasuli molekulebis raodenoba N2 

1 n2 St , 6

(104.5)

M2 

1  2 St . 6

(104.6)

xolo gadatanili masa

vinaidan

 1  2 , amitom M 1  M 2 da S farTobSi t droSi marcnidan

marjvniv gadatanili Warbi masa 1 M  M 1  M 2   St 1   2  . 6

1   2

aris simkvrivis cvlileba (Semcireba)

2

(102.7)

manZilze. igi tolia

d   simkvrivis gradientisa  sigrZis erTeulze simkvrivis cvlilebisa  anu dx   da 2 sigrZis namravlisa: 360

1   2   niSani

minusi

imis

maCvenebelia,

d  2 . dx rom

x –is

zrdas

 –s

Seesabameba

Semcireba. CavsvaT (104.7) formulaSi:

1 d M      St . 3 dx

(104.8)

(104.8) formulis Sedareba fikis (104.1) empiriul formulasTan gvaZlevs: 1 D   . 3

(104.9)

amrigad, molekulur-kinetikuri Teoria saSualebas gvaZlevs ara marto gamoviyvanoT

difuziis

formula,

aramed

ganvsazRvroT

difuziis

koeficientic _ gamovsaxoT igi mikroskopuli sidideebis _ molekulis saSualo siCqarisa da saSualo Tavisufali ganarbenis saSualebiT. molekulis saSualo siCqare wnevaze damokidebuli ar aris, saSualo Tavisufali ganarbeni ki wnevis ukuproporciulia (§ 103), amitom D ~

1 , e.i. p

wnevis gadidebiT difuziis procesi neldeba. garda amisa,  ~ T , saSualo Tavisufali

ganarbenic

temperaturis

gazrdiT

izrdeba D

izrdeba

temperaturis (ufro

gazrdiT,

swrafad,

vidre

amitom

T ),

anu

difuziis procesi Cqardeba. Cven ganvixileT airis difuzia `Tavis TavSi~, roca mocemuli airis molekulebi vrceldeba (difundirebs)amave airis molekulebs Soris (e.i. xdeba erTi da igive airis molekulebis

urTierTSereva)

TviTdifuziis

eqsperimentuli

process

Seswavla

xSirad

SeiZleba

`Tv iTdif uzi as ~

mxolod

`niSnuli~

uwodeben.

(radioaqtiuri)

atomebis an molekulebis gamoyenebiT, roca mocemuli nivTiereba warmoadgens izotopebis narevs, romelTagan nawili stabiluria, nawili ki _ radioaqtiuri. difuziis (104.8), formula marTebulia urTierTdifuziisaTvis, roca xdeba ori sxvadasxva airis urTierTSereva, oRond, am SemTxvevaSi difuziis koeficients aqvs sxva saxe. gamoTvlebi gviCvenebs, rom

D12  sadac

n1υ2 λ 2  n2 υ1 λ1 3n1  n2 

n1 , υ1, λ1 da n2 ,  2 ,  2 aris Sesabamisad pirveli da meore airis molekulebis

koncentracia, saSualo siCqare Tavisufali ganarbeni.

361

§ 105. Tbogamtaroba davuSvaT, molekulis

airis

sxvadasxva

nawilSi

saSualo

kinetikuri

energiac

temperatura i   kT  2 

da,

maSasadame,

sxvadasxvaa.

amasTan

ganvixiloT umartivesi SemTxveva, roca temperatura icvleba mxolod Ox RerZis gaswvriv T  T x  da cdis dros misi gradienti mudmivia (sur. 105.1). qaosuri

moZraobis

gamo

molekulebi

gadadian

fenidan

fenaSi,

magram

cxadia, rom x RerZis marTobul S farTobSi maT marcxnidan marjvniv meti energia gadaaqvT, vidre marjvnidan marcxniv (Tumca, rogorc qvemoT vnaxavT,

TviT

gadasuli molekulebis

raodenoba erTi

da igivea),

e.i.

adgili aqvs Warbi energiis gadatanas marcxnidan marjvniv. am movlenas, e.i. e ne r gi is

g a da t an as ,

g am ow v e u ls

te m pe r a tu r aT a

s xv aob iT

m ole k u le b is q a os u r i m oZ ra ob iT , T b o g am ta rob a ewodeba. 1820 wels frangma fizikosma furiem (1768-1830) empiriulad daadgina, rom Tbogamtarobis Sedegad gadatanili Q siTbos raodenoba pirdapirproporciulia temperaturis

dT gradientisa, S farTobisa da im t drodx

isa, romlis ganmavlobaSic mimdinareobs Tbogamtarobis procesi: dT (105.1) St . dx (105.1) formulas ewodeba Tbogamtarobis gantoleba. mas furies kanonsac Q   

uwodeben.

niSani

minusi

gantolebaSi

imis

maCvenebelia,

rom

siTbos

raodenobis gadatana xdeba temperaturis Semcirebis mimarTulebiT. proporciulobis k koeficients ewodeba formulidan Cans, rom romelic

Tbogamtarobis koeficienti. (105.1)

ricxvobrivad tolia siTbos im raodenobisa,

gadaitaneba drois erTeulSi

farTobis erTeulSi, rodesac temperaturis

gradienti

erTis

sistemaSi igi izomeba CGS sistemaSi _

tolia.

j –iT, xolo m.wm. K

ergi -iT. sm.wm. K

gamoviyvanoT Tbogamtarobis gantoleba molekulur-kinetikuri Teoriis sa362

sur. 105.1

fuZvelze. S farTobidan marcxniv da marjvniv  Tavisufali ganarbenis tol manZilze gamovyoT ori Txeli I da II fena, romelTa temperaturebia

T1 da T2 (sur. 105.1). pirvel fenaSi molekulis saSualo kinetikuri energia i i kT1 , xolo meoreSi _ kT2 . 2 2

iqneba

(104.2)

formulis

S

Tanaxmad

farTobSi

t

droSi

marcxnidan

marjvniv gasuli molekulebis raodenoba 1 (105.2) n11 St , 6 sadac n1 da 1 molekulaTa koncentracia da saSualo siCqarea I fenaSi. am N1 

molekulebis mier gadatanili energia (siTbos raodenoba) i 1 i Q1  N1  kT1  n11 St  kT1 . 2 6 2 analogiurad, S farTobSi t drois ganmavlobaSi marjvnidan marcxniv

gasuli molekulebis raodenoba N2 

1 n2 2 St , 6

(105.3)

xolo gadatanili energia

molekulaTa

i 1 i Q2  N 2  kT2  n2 2 St  kT2 . 2 6 2 rogorc koncentracia, ise

saSualo

siCqare

damokidebulia temperaturaze, amasTan temperaturis gadidebiT  izrdeba,

n ki mcirdeba. amitom miaxloebiT SeiZleba davuSvaT, rom maTi namravli n

damokidebuli ar aris temparaturaze: n11  n2 2  n . amrigad, N1  N 2

(e.i. difuziis procesi gamoricxulia) da vinaidan T1  T2 , amitom Q1  Q2 . Warbi energia, romelic molekulebs gadaqvT marcxnidan marjvniv Q  Q1  Q2 

1 i n St  k T1  T2  , 6 2

magram T1  T2  

dT  2 , dx

amitom 1 i dT Q   n  k St . 3 2 dx (105.4) formulis Sedareba (105.1) formulasTan gvaZlevs:

1 i k. 3 2 molekulur-kinetikuri

(105.4)

  n  aseT

gamosaxulebas

gvaZlevs

(105.5) Teoria

airis

Tbogamtarobis koeficientisaTvis. mas SeiZleba sxva saxe mivceT, kerZod, 363

gamovsaxoT siTbotevadobis saSualebiT. amisaTvis (105.5) tolobis marjvena mxare

gavamravloT

gavyoT

molekulis

bolcmanis mudmivas mniSvneloba k 

masaze

SevitanoT

R : NA

1 i R , 3 2 NA m magram nm   airis simkvrivea, xolo N A  m   _ molis masa, amitom

  nm 

1 3

  

i R ; 2

i R  CV aris airis moluri siTbotevadoba mudmivi moculobis dros (§64), 2

i R CV   cV iqneba kuTri siTbotevadoba mudmivi moculobis dros, amitom 2  sabolood 1 3

     cV . (105.6) formulaSi 

 ~ p , amitom 

da cV

(105.6)

wnevaze damokidebuli ar aris,  ~

1 p

xolo

namravli da, maSasadame, Tbogamtarobis koeficientic

wnevaze damokidebuli ar aris. es daskvna marTebulia mxolod miaxloebiT, saxeldobr, maRali wnevis dros. dabali wnevis dros, roca  zomis

rigisaa

aRar

aris

damokidebuli

wnevaze,

WurWlis

Tbogamtarobis

koeficienti wnevis SemcirebiT mcirdeba. wnevis Semcirebisas gaiSviaTebuli airis Tbogamtarobis Semcireba gamoiyeneba

TboizolaciisaTvs.

diuaris

WurWels,

romelic

gamoiyeneba

Txevadi airis Sesanaxad, aqvs ormagi kedlebi. kedlebs Soris Seqmnilia maRali vakuumi*, ris gamoc kedlis Tbogamtaroba Zalian mcirea.

vakuumi ewodeba airis iseT gaiSviaTebas, rodesac molekulaTa saSualo Tavisufali

ganarbeni WurWlis xazovani zomis rigisaa. sityvasityviT vakuumi niSnavs `sicarieles~. magram sakmaod maRali vakuumis drosac moculobis erTeulSi uamravi molekula imyofeba. magaliTad,

p  10 6 mm.v.wy.sv. wnevis dros 1 sm3-Si imyofeba 1010 molekula. aseTi wnevis

dros WurWelSi, romlis diametri 10 sm-ia, molekula sxva molekulasTan daujaxeblad ramdenime aseuljer ejaxeba WurWlis kedlebs. am TvalsazrisiT SeiZleba CaiTvalos, rom WurWeli `carielia~.

364

§ 106. Sinagani xaxuni (siblante) davuSvaT, airis simkvrive da temperatura yvelgan erTnairia (e.i. difuziisa da Tbogamtarobis movlenebi gamoricxulia), magram misi sxvadasxva fena moZraobs sxvadasxva u siCqariT, e.i. airis molekulebi, qaosuri moZraobis garda monawileoben mowesrigebul moZraobaSic. davuSvaT, airis dineba warmoebs Oz RerZis gaswvriv, xolo fenis siCqaris cvlileba _ Ox RerZis gaswvriv (sur. 106.1). qaosuri moZraobis gamo molekulebi gadadian fenidan fenaSi, amasTan, meti siCqariT moZravi fenidan naklebi siCqariT moZrav fenaSi maT meti moZraobis raodenoba (impulsi) gadaaqvT, vidre piriqiT, e.i. adgili aqvs Warbis impulsis gadatanas marcxnidan marjvniv. es iwvevs fenebis moZraobis raodenobis cvlilebas, ris gamoc niutonis II kanonis Tanaxmad, TiToeul fenaze imoqmedebs Zala f 

K . t

am Zalas, romelic mimarTulia airis fenebis mxebad, Sinagani xaxunis Zala ewodeba, vinaidan igi moqmedebs airis SigniT da isea mimarTuli,rom ufro nel fenas aCqarebs,

ufro

swraf

fenas

anelebs

(amuxruWebs). TviT movlenas, e.i. im pu ls i s

sur. 106.1

g a d a ta n as , g am ow v e u l s fe ne b i s s i C q areTa

s xv aob iT

m o le k u le b is

q a os u ri

m oZ ra ob iT ,

e w ode b a

S in a g a ni x a xu ni, a nu s ib l an te . niutonma empiriulad daadgina, rom Sinagani xaxunis Zala pirdapirproporciulia

du siCqaris gradientisa da fenebis Sexebis S farTobisa: dx

f  

du S ; dx

(106.1)

(106.1) formulas ewodeba Sinagani xaxunis gantoleba.  proporciulobis koeficients ewodeba Sinagani xaxunis an siblantis koeficienti *. (106.1) formulidan Cans, rom  ricxobrivad tolia fenis erTeulovan farTobze

 -s zogjer uwodeben siblantis d i n a m i k u r koeficients, riTac mas ganasxvaveben  siblantis k i n e m a t i k u r i koeficientisagan   , sadac  airis (an siTxis) simkvrivea.   –s Sebrunebul sidides d e n a d o b i s koeficienti ewodeba. 365 *

moqmedi Sinagani xaxunis Zalisa, rodesac siCqaris gradienti erTis tolia.

n.wm

SI sistemaSi misi erTeulia

, xolo CGS sistemaSi

dn.wm sm 2

. am ukanas-

knels frangi mecnieris puazeilis (1799-1869) pativsacemad ewoda puazi (pz): 1 (106.1)

n.wm m



10 5 dn.wm 4

10 sm

gantoleba

marTebulia

 10

dn.wm sm 2

niutonma

airisaTvisac.

 10 pz .

miiRo

airis

siTxisaTvis.

siblante

igi

gacilebiT

naklebia, vidre siTxisa, magram misi aRmoCena SesaZlebelia mTeli rigi cdebiT. erT-erTi cdis sqema mocemulia 104.2 suraTze: or A da B TanaRerZul cilindrs Soris sivrce avsebulia

gamosakvlevi

centridanuli Camokidebulia

manqanis

airiT.

RerZze,

cilindri

dadgmulia

cilindri

C Zafze, romlis grexvis momenti SeiZleba

gaizomos. Tu A cilindrs swrafad vabrunebT, SevniSnavT, rom garkveuli drois Semdeg B cilindric Semobrundeba

sur. 106.2

imave mimarTulebiT. B cilindris Semobruneba imiT aixsneba, rom

A cilindris brunva gadaeca Semxeb airis fenas,

romelmac Sinagani xaxunis gamo, es brunva gadasca Semdeg fenas da a.S. bolos brunva daiwyo B cilindris Semxebma fenam da TviT B cilindrmac. B cilindris Semobruneba gagrZeldeba manam, sanam Zafis grexvis momenti ar gaawonasworebs cilindrze moqmedi Zalis moments.

miviRoT Sinagani xaxunis gantoleba molekulur-kinetikuri Teoriis safuZvelze. S farTobidan marcxniv da marjvniv  Tavisufali ganarbenis manZilze gamovyoT airis ori Txeli I da II fena, romelTa siCqareebia

u1 da u 2 . u1  u 2 (sur. 106.1). pirvel fenaSi molekulis impulsi iqneba mu1 , meoreSi _ mu2 *. vinaidan airis simkvrive yvelgan erTnairia, Ox RerZis marTobul S farTobSi t droSi marcxnidan marjvniv da marjvnidan marcxniv gasuli molekulebis raodenoba erTnairia da (106.2) formulis Tanaxmad tolia N

1 n St . 6

molekulas garda mowestigebuli moZraobis Sesabamisi

qaosuri moZraobis Sesabamisi m

mu impulsia, cxadia, gaaCnia

impulsic. magram es impulsi Cven ar gvainteresebs,

vinaidan igi yvela molekulisaTvis erTnairia.

366

am molekuleebis mier marcxnidan marjvniv gadatanili impulsi K1  N  mu1 

1 1 n  S  t  mu1    u1St , 6 6

xolo marjvnidan marcxniv gadatanili impulsi 1 K 2  N  mu2    u 2 St . 6

vinaidan

u1  u 2 ,

amitom

K1  K 2

Warbi

impulsi,

romelic

gadaaqvT

molekulebs marcxnidan marjvniv: 1 K  K1  K 2    St u1  u 2  , 6

magram u1  u 2  

du 2 , dx

amitom 1 du K      St . 3 dx

(106.2) formula gamosaxavs S farTobis impulsis cvlilebas t droSi. impulsis

cvlileba drois erTeulSi,

anu

farTobze moqmedi Sinagani

xaxunis Zala f 

K 1 du    S . t 3 dx

(106.3)

(106.1) da (106.3) tolobebis Sedareba gvaZlevs 1 3

   . amrigad, molekulur-kinetikuri Teoria, iseve, rogorc difuziisa da Tbogamtarobis SemTxvevaSi, saSualebas gvaZlevs ara marto gamoviyvanoT Sinagani xaxunis gantoleba, aramed ganvsazRvroT Sinagani xaxunis koeficientic. (104.9) formulis Tanaxmad xolo

~

1    D , amitom   D . magram 3

D~ T ,

1 1 , amitom  ~ , e.i. temperaturis gazrdiT airis siblante T T

mcirdeba. garda amisa, D ~

1 , xolo  ~ p , amitom  wnevaze damokidebuli ar p

aris. es erTi SexedviT paradoqsuli Sedegi advilad aixsneba: wnevis SemcirebiT, marTalia mcirdeba molekulaTa n koncentracia da, maSasadame, S 367

farTobSi gasuli molekulebis N raodenoba, magram samagierod izrdeba TiToeuli molekulis mier gadatanili impulsi, vinaidan p –s SemcirebiT



izrdeba

S

da molekulebi daujaxeblad gadian

farTobSi ufro

daSorebuli fenebidan, romelTa u ufro metia. aqac,

iseve

rogorc

Tbogamtarobis

SemTxvevaSi,



aris

damokidebuli wnevaze mxolod SedarebiT maRali wnevis dros. dabali wnevis dros (ultragaiSviaTebuli airi), roca  WurWlis zomis rigisaa da aRaraa damokidebuli wnevaze,  wnevis SemcirebiT mcirdeba. rogorc aRvniSneT, siTxis siblante gacilebiT metia airisaze. magaliTad, oTaxis temperaturaze wylisaTvis

  0,010

puazs, xolo haerisaTvis

  1,8  10 4

puazs. sib-

lantis mxriv gansakuTrebuli Tvisebebi aqvs Txevad heliums. atmosferuli wnevis dros heliumi Txevaddeba T  4,2 K temperaturaze. miiReba e.w. Txevadi heliumi I, romelsac aqvs metad mcire, magram savsebiT zomvadi siblante, romelic airuli heliumis siblantes ar aRemateba (   2  10

5

pz). T  2,17 K temperaturaze Txevad heliumSi xdeba meore gvaris

fazuri gadasvla - Txevadi heliumi I gadadis Txevad helium II-Si, romlis

  0 . es imas

niSnavs, rom helium II SeiZleba Tavisuflad gaedinos iseT uwvriles xvrelebsa da kapilarebSi, romlebic airisaTvisac ki gauvalia. am Tvisebas z e d e n a d o b a

ewodeba.

amasTan, aRsaniSnavia, rom helium II ar iyineba da inarCunebs zedenadobis Tvisebas TviT absolutur nulamde gacivebis drosac ki*. zedenadobis movlena aRmoaCina sabWoTa fizikosma p.kapicam 1940 w. helium II-is Tvisebebis SeswavlaSi didi Rvawli miuZRvis qarTvel fizikoss elefTer andronikaSvils. heliumis zedenadoba, iseve rogorc meore metad saintereso movlena _ zegamtaroba, aixsneba mxolod kvanturi Teoriis saSualebiT. kerZod, mxolod kvanturma Teoriam axsna, Tu ratom aris heliumi e r T a d e r T i siTxe,

romelic

iyineba

Zalian

dabali

temperaturis

drosac

ki.

temperaturis

SemcirebiT nawilakebis rxevis intensivoba klebulobs da sxeulis Sinagani energia mcirdeba, magram, kvanturi Teoriis Tanaxmad, roca miiswrafvis

ara

nulisaken,

aramed

energiis

T  0 , sxeulis Sinagani energia

garkveuli

mniSvnelobisaken,

romelsac

n u l o v a n i e n e r g i a ewodeba. am energiis `warTmeva~ sxeulisaTvis araviTari saSualebiT ar SeiZleba, igi ganpirobebulia nawilakebis n u l o v a n i r x e v e b i T da miT metia, rac naklebia

nawilakis

masa.

dabal

temperaturaze

nivTiereba

iqneba

myar

Txevad

mdgomareobaSi imazea damokidebuli, Tu ra TamaSobs ganmsazRvrel rols _ nawilakebs Soris mizidva, romelic iwvevs kristaluri mesris Seqmnas, Tu nulovani rxevebi, romelic am Seqmnas ewinaaRmdegeba. heliumSi mizidvis Zalebi Zalian sustia, nulovani rxevebi ki (masis simciris gamo) Zalian intensiuria, ris gamoc ar xdeba kristaluri mesris Seqmna. *

heliumi myar mdgomareobaSi miiReba misi 4,2 K-ze qveviT gacivebisas, Tu wneva 25

atmosferoze metia.

368

S i n a a r s i winasityvaoba . §1. Sesavali .

§2. fizikuri sidideebis erTeulebi. erTeulTa sistemebi .

pirveli nawili me qa ni k i s sa fu Zvl ebi I Tavi. w e r t i l i s k i n e m a t i k a §3. meqanikuri moZraoba. moZraobis kinematikuri gantolebebi .

. 11

$4. wrfivi Tanabari da Tanabarcvladi moZraobebi. maTi gantolebebi da grafikebi

§5. mrudwiruli moZraobis siCqare .

. 12

. 18

§6. mrudwiruli moZraobis aCqareba. aCqarebis mxebi da normaluri mdgenelebi .

I I Ta vi . m y a r i s x e u l i s k i n e m a t i k a §7. myari sxeulis gadataniTi moZraoba .

§8. myari sxeulis brunviTi moZraoba .

. 24

§9. kuTxuri siCqaris veqtori .

. 27

I I I Ta vi . n i v T i e r i w e r t i l i s d i n a m i k a §10. klasikuri meqanikis gamoyenebis sazRvrebi .

§11. niutonis pirveli kanoni. inerciuli sistemebi

§12. niutonis meore kanoni. moZraobis diferencialuri gantolebebi .

§13. niutonis mesame kanoni

$14. fardobiTobis meqanikuri principi. galileis gardaqmnis formulebi . $15. impulsi. impulsis cvlilebisa da mudmivobis kanonebi $16. drekadobis Zala. hukis kanoni . $17. xaxunis Zala .

. 45

$18. msoflio mizidulobis Zala .

$19. simZimis Zala. wona. uwonoba .

I V Ta vi . a r a i n e r c i u l i s i s t e m e b i $20. Sesavali

. 58

$21. wrfivad da TanabaraCqarebulad moZravi arainerciuli sistema .

. 59

$22. Tanabrad mbrunavi arainerciuli sistema

. 61

$23. dedamiwa, rogorc arainerciuli sistema .

. 65

V Ta vi . m u S a o b a d a e n e r g i a $24. muSaoba. cvladi Zalis muSaoba. erTeulebi

$25. simZimis Zalis muSaoba. konservatuli da arakonservatuli Zalebi

$26. simZlavre .

§28. energia. energiis mudmivobis kanoni .

§29. kinetikuri da potenciuri energia .

§30. kavSiri Zalasa da potenciur energias Soris . 369

. .

. 77 .

§31. nawilakebis dajaxeba

VI Ta vi . m y a r i s x e u l i s d i n a m i k a §32. myari sxeulis gadataniTi moZraobis ZiriTadi gantoleba .

§33. myari sxeulis brunviTi moZraobis dinamikis damaxasiaTebeli sidideebi

§34. brunviTi moZraobis dinamikis ZiriTadi gantoleba .

§35. impulsis momentis mudmivobis kanoni .

§36. myari sxeulis wonasworobis pirobebi .

. .

VI I Ta vi . r x e v e b i d a t a l R e b i §37. harmoniuli rxeva .

§38. harmoniulad merxevi sxeulis siCqare, aCqareba da energia

§39. erTi wrfis gaswvris mimarTuli ori harmoniuli rxevis Sekreba §40. urTierTmarTobuli harmoniuli rxevebis Sekreba . §41. milevadi rxeva .

§42. iZulebiTi rxeva. rezonansi

107 .

105

. 100

111 113

. 116

§43. talRuri moZraoba. talRis gantoleba. talRis energia

. 119

§44. hiugensis principi .

124

§45. talRebis interferencia. mdgari talRebi .

. 126

. 131

VI I I Ta vi . si Txi sa da a i ri s meqa ni k a §46. hidroaerostatikis Ziriadi kanonebi .

§47. idealuri siTxe. siTxis stacionaruli dineba. uwyvetobis gantoleba §48. bernulis gantoleba .

§49. realuri siTxis dineba. siblante

. .

139

§50. sxeulebis moZraoba realur siTxeSi. stoqsisa da niutonis formulebi

136

143 . 146

§51. realuri siTxis dineba wriuli ganivkveTis mqone milSi. puazeilis formula .

m e o r e

. 150

. 153

n a w i l i

molekuluri fizikisa da Termodinamikis safuZvlebi airis Tvisebebi §52. Sesavali. molekuluri fizikisa da Termodinamikis sagani da Seswavlis meTodebi. .

I X Ta vi §53. molekulur-kinetikuri Teoriis ZiriTadi debulebani

156

§54. moculoba da wneva. erTeulebi

161

§55. temperatura da misi gazomva .

. .

§56. atomisa da molekulis masa da zoma

. .

. 163

. 169

§57. idealuri airi. idealuri airis cdiseuli kanonebi

. 172

§58. idealuri airis mdgomareobis gantoleba. airis universaluri mudmiva .

177

§59. airis kinetikuri Teoriis ZiriTadi gantoleba

180

§60. molekulebis ganawileba siCqareTa mixedviT maqsvelis kanoni . 370

. .

. 186

§61. molekulaTa saSualo ariTmetikuli siCqaris gamoTvla .

191

§62. maqsvelis siCqareTa ganawilebis kanonis eqsperimentuli Semowmeba. Sternis cda .

§63. idealuri airis Sinagani energia Tavisuflebis xarisxTa ricxvi .

§64. airis siTbotevadoba. airis universaluri mudmivas fizikuri arsi . §65. barometruli formula .

192

. 194

198

. 203

§66. bolcmanis kanoni. maqsvel-bolcmanis ganawileba .

. 206

§67. avogadros ricxvis gansazRvra. perenis cda .

§68. realuri airi. van-der-vaalsis gantoleba .

§69. van-der-vaalsis gantolebis izoTermebi. kritikuli temperatura . §70. realuri airis Sinagani energia. joul-tomsonis efeqti .

. .

207 .

210 .

219

225

X Ta vi . T e r m o d i n a m i k i s s a f u Z v l e b i §71. ZiriTadi cnebebi .

§72. siTbo da muSaoba .

§73. Termodinamikis I kanoni .

§75. adiabaturi procesi. puasonis gantoleba . .

§77. muSaoba sxvadasxva Termodinamikur procesSi .

. 238

240

. 244

246

§78. wriuli procesi (cikli). siTburi manqanis mq koeficienti .

249

§80. karnos cikli .

§81. Termodinamikis II kanoni

. .

§82. klauziusis utoloba. entropia .

237

. 232

228

§79. Seqcevadi da Seuqcevadi procesebi .

§74. Termodinamikis I kanoni idealuri airisaTvis . §76. politropuli procesi

§84. hipoTeza `samyaros siTburi sikvdilis~ Sesaxeb

§83. Termodinamikis II kanonis statistikuri arsi

258

253

264

268

276

. 283

XI Tavi. m y a r i s x e u l i s T v i s e b e b i §85. kristaluri da amorfuli sxeuli

§86. kristalis bunebrivi forma da misi simetria. simetriis klasebi §87. kristaluri mesris fizkiuri tipebi. .

§88. myar sxeulSi nawilakebis siTburi moZraobis xasiaTi §89. myari sxeulis siTburi gafarToeba §90. myari sxeulis siTbotevadoba . §91. kristalis defeqtebi

305

303

. 298

290

285

310 316

X I I T a v i . siTxis Tvisebebi §92. Txevadi mdgomareobis zogadi daxasiaTeba . §93. molekuluri wneva .

321

324

§94. zedapiruli daWimuloba .

326

§95. wneva siTxis mrude zedapirze. laplasis formula . §96. damsvelebeli da aradamsvelebuli siTxe. kapilaroba 371

. .

331 .

333

§97. aorTqleba da kondensacia . §98. duRili .

. .

§99. fazisa da fazuri gardaqmnis cneba

. .

§100. klapeiron-klauziusis gantoleba .

. .

342

. 345

338 .

. .

§101. mdgomareobis diagrama. sammagi wertili .

349

351

XI I I Ta vi . fizikuri kinetikis elementebi §102. gadatanis movlenebi

353

354

§103. molekulebis dajaxebaTa ricxvisa da saSualo Tavisufali ganarbenis gamoTvla §104. difuzia .

. .

§105. Tbogamtaroba .

. .

§106. Sinagani xaxuni (siblante) .

. .

372

. .

. . .

. .

358 362

365