General Theory of the Internal and Thermal Constituents of Thermodynamic Properties of Real Gases

www.bacpl.org/j/pcc/ Physical Chemistry Communications, Volume 2 Issue 1, April 2015

General Theory of the Internal and Thermal Constituents of Thermodynamic Properties of Real Gases Nedostup V. A.V. Bogatsky Physical‐Chemical Institute National Academy of Sciences of Ukraine, Lustdorfskaya Doroga 86, Odessa 65080, Ukraine vinedostup@yandex.ua

Abstract Representations of thermodynamic properties (e. g. pressure, Helmholtz Free energy, Gibbs Free energy, enthalpy, internal energy) as a sum of two constituents: internal and thermal, were reviewed. General form of thermodynamic relationships between internal and thermal components of caloric properties was derived. It is shown that existence of ideal curves on thermodynamic surface is result of a mutual indemnification of internal and thermal contribution in thermodynamic properties.

Dependence of the behavior of thermal and internal components on features of intermolecular interactions is considered.

Keywords Thermodynamic Properties; Equation of State; Thermodynamic Surface; Ideal Curve

I. Introduction At first time pressure was described as a sum of two components – internal (Pi) and thermal (Pt).

P = Pt+ Pi (1) in Gern’s equation of state[1] where the correction term for pressure caused by intermolecular interaction was introduced. In 8 years Van der Waals [2] proposed a volume dependence of this value, giving a theoretical background which is relevant even nowadays. Back in those days, scientists considered internal pressure to be solely dependent on volume. Famous philosopher During stated:‘ it is just as inconsistent to think that internal pressure is dependent on temperature as to think that temperature impacts gravity[3].In modern studies internal pressure of the liquid is linked to the liquid’s structure, phase transitions heat and other characteristics defined by intermolecular interactions [4]. However, internal and thermic components of caloric properties have not been discussed until now.

II. Pressure. Ideal Gas Curve

 U   P  Definition of the internal pressure P     as well as definition of thermal pressure P  T   , is derived V   T  T V from thermodynamic relationship  U   P    T   P (2)  V T  T V Multicomponent empiric equations of state as well as theoretical equations of state, presented mostly by distribution functions, do not allow expression of internal and thermal pressure component in explicit forms. There is a precise presentation of equation (1) in Van der Waals equation of state [2] and its modifications.

Most prominent result of internal and thermal components existence is a presence of so‐called ideal curves on thermodynamic surfaces of the real gas. Ideal curves are loci of points in which properties of real gas are the same as those properties of the ideal gas at same temperature and density. It also applies to the curve where absolute value of real gas part of thermal pressure is equal to the value of internal pressure, which is also called ideal gas

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curve (there are other names – ZENO, Bachinsky curve etc. )[5, 6, 7], where PV = RT or

PV   Pt  Pi V  RT  0 (3) Hereinafter  means real gas (configurational) part of the property which is defined by intermolecular interaction.

Van der Waals equation, which can predict liquid‐gas phase transition, spinodal, theory of similarity, existence and the success of the basic properties of ideal curve and can be explained by strict distinction between attraction and repulsion contribution to thermodynamic properties and representation of the attraction contribution by mean molecular field. The last suggestion draws criticism of theorists because it implies that molecules are equally distributed in space. However, this suggestion does not contradict distribution function theory because attraction almost does not affect distribution functions and they are the same formetals, molecular and ionic liquids [8].

Location of ZENO curve on thermodynamic surface and its configurations are related to regularity of thermal and internal pressure.

Van derWaals model became a basis for Burshtein [9] work where he introduced the fundamental relationship between the ideal gas curve and both internal and thermal pressure.

 

PV i PV t  1 (4) RT

In (4).‐temperature on ideal gas curve. Also:

   PV  RT  PV t 1   (5) RT    RT  PV   PV i 1    (6)  Expression (5) and (6) represent equivalent structural forms of equation of state with ideal gas curve inserted into it.

From (5) and (6) equations thermal and internal components relationship with pressure for isochors small curvature crossing the curve of ideal gas at the point, where TIGC=T(ρ) follows .

Pt  T

P T  TIGC (  ) (7)

Pi  T (  )

P T  TIGC (  ) (8)

 P  It is important to notice that equation (7) is a differential ratio T   , written in finite differences for isochors  T V count off the curve of ideal gas. Relationship (7) and (8) allow developing a demonstrative scheme for the analysis of internal and thermal components contribution on configurational pressure on isochor. It is well‐known that each isochors intersects ideal gas curve at TIGC(ρ). Configuration of isochors is simpler than configuration of other isolines on thermodynamic surface: in area of dense liquid it differs little from straight line, in the area of rarefied gas it is defined by curvature of the second virial coefficient P= RTρ+RTB(T)ρ2 (9) Further reasoning requires presenting isochor in coordinates P (T) as line of small deviation from straight line.

Equation (7) and (8) are quite precise allowing analyzing ratio of pressure components impact in various isochor points. Equation ideal gas curve is in according with [5].   T  TS  1   0

   (10)

As for linear relationship of TZ (  ) , it was discovered for the first time in 1908 by Bachinsky [10] as a result of experimental data comparison with results of Van der Waals equation prediction TB =a/Rb, ρO=1/b.

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Expressions (7) and (8) show that for points on isochor, which are above ideal gas curve ΔP>0, ΔPt>Pi, and for points ΔP≤ ΔP P, on the curveZ=1, ΔP<0 ΔP=P. Van der Waals equation dimension less by ideal curve ΔZ=0 parameters (Δπ=ΔP/RTВρ,  

 ). 0  

t 2   2 (11) 1 

Consequently, along the Z=1 line the following condition is carried out:

    (12)       1   2

In this case internal pressure depends solely on density (volume). This relationship is satisfactory for the particular area of thermodynamic surface, where isochors is a slight sloppy line and, therefore, can substitute derivatives included in equation (2) by ratios of definite residues (7) and (8).

From fundamental equation (4) it is easy to see that if Pi ~  2 , Pt ~ g   , i.e. thermal pressure could be adequately described by hard spheres model, then temperature dependence on density on ideal gas curve TZ    ~

1

g   (13)

A linear dependence of TZ    is linked to rectilinearity of reverse value of g   . It was proved by a test, which was carried out for different values of g (σ), both experimental [11,12] and model [13], has shown that in the range of reduced densities from 0.15 to 0.7 with an error of no more than 3% the inverse value of the radial distribution function at the contact point is rectilinear. The rectilinearity of the ideal gas curve observed in a large range of densities [6] causes restrictions on the relationship of thermal and internal pressure, defined by equation (4)..

1 The rectilinearity of the ideal gas curve Replacement Van der Waals radial distribution function g(σ)~ 1  b observed in a large range of densities [6] causes restrictions on the relationship of thermal and internal pressure, defined by equation (4).. Replacement Van der Waals radial distribution function g(σ) by theoretical forms does not change a linear dependence on temperature of the density (10). Converse is also true: the introduction of the exact dependence g (σ) in the thermal pressure of the Van der Waals (9), as long as ideal gas curve stays rectilinear leads to the same dependence for the internal pressure. Pi ~ aρ2. Therefore, various modifications of the Van der Waals equation, which were carried out regardless the relations (4), namely equations of Redlich‐Kwong, Peng‐ Robinson and many others that describe the data in the field of phase equilibria with high accuracy, cannot reproduce basic properties of the ideal gas curves [14].

Expressions (7) and (8) can be converted using the virial equation of state for pressure at low density isochore which contains only second virial coefficient (9). We omit simple calculations in order to present the final expression for the internal and thermal components of the second virial coefficient.

Bi 

TB B (14) T  TB

Bt 

TB (15) T  TB

Similar to what was mentioned before, equation (15) is given as finite differences of differential ratio, which is commonly used to estimate the second virial coefficient, which is determined by repulsion potential.

III. Caloric properties  P 

From equation (2) ratio P   T  can be derived. Similar to this ratio is the following ratio:  t  1     T V

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 A   T     A i (16)  1   T V

where  Ai is the configurational part of thermodynamic properties derivative and Ai is its internal component of property.

The same is right for internal energy equation form:  U  T   1  T

    U i (17)  V

The same is right for Helmholtz Free energy equation form:  F  T   1  T

    F i U (18)  U

i. e. internal component of Helmholtz Free energy is the configurational part of internal energy. As for Gibbs Free energy the ratio goes as follows:    T   1  T

   U  (19)   U   i  V  U  PV  V T  V

This equation obviously leads to internal component of enthalpy expression:  H  T   1  T

   H   U   P  (20)   H  T  i   U  Pv  T   T   V  U i  PV  T V  V T  T V  V

Therefore, thermic component are, respectively:

 U  U t  T   v  T Cv (21)  T   F  Ft  T   v  T S (22)  T  t  T S  PV (23) t H t  U t  PV (24) t

Similar to (4) for the curve ΔF=0 we introduce auxiliary function θΔF which has the same meaning as (4), i.e. T    function on the curve ΔF=0

F  

F (25) F RT

i. e. at ΔF=0, ΔFi= ΔFt,  F  RTF    , where RTF    is a temperature dependence on density on ideal curve for Helmholtz free energy. Auxiliary function allows deriving the expressions for thermic and internal components of ΔF:

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RT F  F  Ft  T  (26)    T RT   F  V Fi  U  

 F .F (27) RT    F

where, ΔFt is the essence of equation (26) given in finite differences, Similar to Z=1(4) curve and ΔF=0 (24) curve the ideal enthalpy curve can be described by:

 H  

H i H (28) RT

which implies that the numerator contains function proportional toρ, and in the denominator of the thermal pressure, proportional to g (σ ), then (13) is valid for an ideal enthalpy curve, where dependence of eTΔH (ρ) is also rectilinear. It is confirmed in the works [15, 16, 17] with experimental data involving a large number of real gases and gas mixtures of constant composition and results of computational research modeling systems.

Also, just like before, we write the expression for the thermal and the internal components of the enthalpy and internal energy using the relationship (4) for corresponding ideal curve T H  P  H t  T  (29)  .V  T  TH (  )  T V TH (ρ).H  U  H i  U   (30)  U  T  TH (  )  V T

For the curve ΔU=0presented at very high temperatures T <TJ (for nitrogen TJ = 2500K) the required ratio between the thermal and internal component of internal energy is fulfilments and it is similar to (4) U t 

T U T  TU (  ) (31)

U i  

TU (  )U (32) T  TU (  )

It must be mentioned that equations for thermal and internal components could be used for creation of conditions existence too for ideal curves of derived properties such as Boyle’s curve, Joule‐Thompson curve(inversion curve) etc.

IV. Cold Isotherm At T = 0, the thermal components of properties go to zero, and the configuration properties go to internal properties. Imagine a continuous extrapolation of the thermodynamic surface of a real gas homogeneous system at the T = 0 disregard leaps in properties caused by phase transitions. Technically such extrapolation is feasible because according to [18] for overcooled liquid there is no spinodal and condition of the system on the so‐called ʺColdʺ isotherm is metastable liquid supercooled to 0K, and it is determined only by the intermolecular interaction. dU x  Px Properties on this curve are assigned the index «x» ‐ Ux (ρ) and Px(ρ), where  dVx

Calculations have shown that real gases dependence of the thermal properties on density is similar to the dependence of the corresponding thermodynamic properties of hard spheres on the density: the properties increase monotonically with density growth. This also applies to the thermal pressure and, of course, to the  F  thermal component of the Helmholtz free energy ‐ T    T S ‐ where a monotonic increase is  T V predetermined by the content of entropy, etc.

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Internal components show the properties of extremes, changes of sign, because they are much more sensitive to the details of the intermolecular interaction. For example, according to the acoustic measurements [19], molecular dynamics experiments [20] and calculations based on the cell theory [21, 9] internal pressure has maxima that increase with decreasing temperature and shift towards higher densities.   2U  The presence of maxima Pi is linked to the presence of ideal curve  2   0 on the thermodynamic surface.  V T Furthermore, at the area of higher parameter there is a curve of the internal pressure inversion (curve Joule)  U  where     Pi  0 . Prior to that curve the internal pressure is negative, as is expanding.  V T

All ideal curves reflecting above‐mentioned peculiarities of the thermal and internal components of thermodynamic properties, in their extension at T = 0 are parts of corresponding points of cold isotherm. Curve ΔU d 2U x  P  = 0 and ΔF = 0 are included at the point Ux = 0, Pi = 0, the point Px = 0, the point Px =0,  i   0 ,and  0 . dVx2  V T

Thus, the ideal curves are continuation of zeros of cold isotherm which in turn is a consequence of zeros of intermolecular interaction [22].

V. Conclusion This study allowed deriving equations for calculations of caloric properties thermal and internal components. It was shown that thermal and internal components of internal energy, enthalpy, Gibbs and Helmholtz potentials and other is related to corresponding ideal curves on thermodynamic surface.

The proposed reasoning provides useful advice in the first place for the development of the so‐called wide‐range equation of state covering the entire area of thermodynamic surface from T = 0 to ρ = 0 and the high temperatures and pressures.

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[15] V.I.Nedostup,E.P,Galkevich ,Features of the behavior of real gases ideal curves. Russian Themophys.High Temperatures 14 (5) (1976),1094‐1097.

[16] V.I.Nedostup, Asymptotic properties ideal curves on a thermodynamic surface. High Temperatures 51(2013)72‐78.

[17] E.M.Apfelbaum,V.S.Vorobev,Regarding the universality of some consequence of the Van der Vaals equation in the supercritical domain Jour.Phys.Chem B 117 (2013) 7750‐7755.

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[19] M Dzida Heat capacity and internal pressures of cyclopentanol at pressures up to 100 Mpa determined by acoustic method.

ActaPhys. Polonica‐A 114(2008)75‐80.

[20] E.Gonorshadi,A.Morsali.,S.Mansoori ,Molecular dynamic study on role attractive and repulsive forces in internal energy and internal pressures and structures of dence fluids. Chem.Phys. 331(2007) 332‐338.

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V.I.Nedostup –was born in 1939 in Odessa ( former USSR,now Ukraine).In 1961 received master degree in a field marine energy.1967 he defended thesis for degree Ph.D.on study of the thermodynamic properties of gas mixtures .1983 received degree of doctor of science for a new method of calculation and prediction thermodynamic properties gases and liquids.In 1986 he received the title of professor in chemical engineering.Since 1972 he have been working in the Academy of science and last 25 years depute director of Physico‐chemical institute.V.I.Nedostup‐ author of several monographs and large number articles and reports on the thermodynamic properties substances.Hememder of editorial board of 3 journals.He has a State awards and honorary academic titles.

List of symbols: P F Ф H U T V Z B R a,b

Greek letters ρ Δ Θ

Subscript i t ΔZ,ΔF,ΔФ,ΔH,ΔU B o

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pressure free energy of Helmgoltz free energy of Gibbs entalpy internal energy temperature volume compressibility factor second virial coefficient gas constant parameters equation of Van der Waals

density configuration part of property temperature function on ideal curve

internal part of property thermal part of property reference to ideal curves. temperature of Boyle density at 0K