Heterogeneous systems and reaction equilibria
B. Mishra
PART – 4
HETEROGENOUS SYSTEMS & REACTION EQUILIBRIA INVOLVING SOLIDS
Let us consider a mineralogical reaction involving solid phases aA +bB
cC +dD
(1)
For each solid phase, an equation of state of equilibrium can be written for C : GCP ,T = GC0, P ,T + RT ln X C + RT ln γ C for A : G AP ,T = G A0, P ,T + RT ln X A + RT ln γ A
Then we can write the free energy change of the reaction (ΔGrP ,T ) as follows: P ,T P ,T ΔGrP ,T = ∑ G products − ∑ Greac tan ts
(
) (
⇒ ΔGrP ,T = cGcP ,T + dG DP ,T − aG AP ,T + bG BP ,T
) (2)
(X C ) (X D ) ( X A ) A ( X B )B C
⇒ ΔGrP ,T = ΔGr0, P ,T + RT ln
D
(γ C ) (γ D ) (γ A ) A (γ B )B C
+ RT ln
D
(3) where ΔGr0, P ,T is the free energy change of the pure phases at equilibrium P, T and defined by eqn. (24) in part – 1. Now, let’s recall eqn. (16) in part -1.
G P ,T = H P0T0 − TS P0T0
T P ⎡T ⎛ CP ⎞ ⎤ + ⎢ ∫ C P dT − T ∫ ⎜ ⎟dT ⎥ + ∫ V .dP T ⎝ ⎠ ⎥⎦ P0 T0 ⎣⎢T0
If the terms in the square bracket is expressed as Cp* then we can write the following expression from eqn. (3) by replacing G P ,T (for pure phase) with G 0, P ,T .
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Heterogeneous systems and reaction equilibria
B. Mishra
P
G 0, P ,T = H 0,1, 298 − TS 0,1, 298 + C P* + ∫ V .dP 1
(4) If we further assume that V 0.P ,T ≅ V 0, P0 ,T0 i.e., the change in volume of a phase from 0
( P 0 , T ) to ( P, T ) is negligible, then eqn. (4) reduces to G 0, P ,T = H 0,1, 298 − TS 0,1, 298 + C P* + V 0,1, 298 ( P − 1)
(5)
Substituting eqn. (2) for each phase in eqn. (3) we get 0 = ΔH r0,1, 298 − TΔS r0,1, 298 + ΔC P* + ΔV 0,1, 298 ( P − 1) + RT ln
( X C )C ( X D )D ( X A ) A ( X B )B
+ RT ln
(γ C )C (γ D )D (γ A ) A (γ B )B (6)
(
) (
where ΔH r0,1, 298 = cH C0,1, 298 + dH D0,1, 298 − aH A0,1, 298 + bH B0,1, 298 and similarly for ΔS r
0 ,1, 298
)
, ΔVr0,1, 298 and ΔCP* .
Cation Exchange Reactions: Geothermometry
Exchange reactions involving cations are of the type
Fe−α + Mg − β = Fe− β + Mg −α where α and β are the phases forming solid solutions of the exchangeable cations Fe and Mg. For such reactions we can write
(
) ( ) − (C
ΔC P* = C P* , Fe − β + C P* , Mg −α − C P* , Fe −α + C P* , Mg − β
(
or , ΔC P* = C P* , Mg −α − C P* , Fe −α
* P , Mg − β
)
− C P* , Fe −α
)
Typically, (C P* , Mg − C P* , Fe ) in α and β are very small positive quantities [since (C P* , Mg −α > C P* , Fe −α ) ]. Hence, ΔC P* is further reduced to an even smaller value and is neglected i.e., ΔC P* = 0
(7)
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Heterogeneous systems and reaction equilibria
B. Mishra
Further, if the Fe-Mg mixing in α and β phases are assumed to be ideal, i.e., γ Fe in α and
[
]
α α β β β and γ Mg in α and β =1.0, then eqn. (6) reduces to γ Fe , γ Mg , γ Fe , γ Mg =1
0 = Δ H r0 ,1, 298 − T Δ S r0 ,1, 298 + Δ V r ⇒ T Δ S r0 ,1, 298 − RT ln
(
⇒ T ΔS
⇒T =
0 ,1, 298 r
0 ,1, 298
( P − 1) + RT ln
(X )(X ) = Δ H (X )(X ) β
α
Fe
Mg
α
β
Fe
Mg
α −β
)
0 ,1, 298 r
(X )(X ) (X )(X ) β
α
Fe
Mg
α
β
Fe
Mg
+ Δ V r0 ,1, 298
− R ln k D Fe − Mg = Δ H r0 ,1, 298 + Δ V r0 ,1, 298 ( P − 1)
Δ H r0 ,1, 298 + Δ V r0 ,1, 298 ( P − 1) Δ S r0 ,1, 298 − R ln k Dα Fe− β− Mg
(8)
whereK D =
(X )(X ) (X )(X ) β
α
Fe
Mg
α
β
Fe
Mg
In eqn. (8), ΔH 0 , ΔS 0 , ΔV 0 for phases at standard states are known and therefore T = f (Kd, P).But since change in volume in this type of reactions is very small, temperature is truly a function of composition of the participating phases in the cation exchange reaction. α α β β . X Mg . X Mg g mex, ss = WFeα − Mg . X Fe + WFeβ − Mg . X Fe α α β β (1 − X Mg ) + WFeβ − Mg . X Mg (1 − X Fe ) = WFeα − Mg . X Mg
[
(
α α = WFeα − Mg X Mg − X Mg
) ]+ W 2
β
Fe − Mg
[X
β Mg
(
β − X Mg
)] 2
However, if the solid solution phases [( Fe − Mg ) α and ( Fe − Mg ) β ] are non-ideal, then the geothermometric expression evolves into more complex forms because the activity coefficient (γi) terms come into play.
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Heterogeneous systems and reaction equilibria
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Say, for example, if we assume that Fe-Mg mixing in α and β behave non-ideally and that the g mex, ss be expressed as α α β β g mex, ss = WFeα − Mg X Fe X Mg + WFeβ − Mg X Fe X Mg
[
(
α α = WFeα − Mg X Mg − X Mg
Then,
) ]+ W 2
β
Fe − Mg
[X
β Mg
(
β − X Mg
)] 2
(9)
α α )2 RT ln γ Mg = WFeα − Mg (1 − X Mg α α RT ln γ Fe = WFeα − Mg ( X Mg )2
β β Similar expressions can be written for RT ln γ Fe andRT ln γ Mg terms.
Further, RT ln
α γ Mg α α = RT ln γ Mg − RT ln γ Fe α γ Fe
(
α = WFeα − Mg 1 − X Mg
[
)
2
(
α − WFeα − Mg X Mg
)
2
(
α α α = WFeα − Mg 1 + ( X Mg − X Mg ) 2 − 2 X Mg
( (X
α = WFeα − Mg 1 − 2 X Mg
= WFeα − Mg ∴ RT ln
α Mg
)
α α + X Fe − 2 X Mg
)] 2
)
α γ Mg α α = WFeα − Mg (X Fe − X Mg ) α γ Fe
(10)
⎛ γ Mg Similarly RT ln⎜⎜ ⎝ γ Fe
β
⎞ β α ⎟⎟ = WFeβ − Mg X Fe − X Mg ⎠
(
)
Now from eqn. (6), the equilibrium condition for the exchange reaction can be expressed as: ΔGrP ,T = 0 = ΔH r0,1, 298 − TΔS r0,1.298 + ΔVr0,1, 298 ( P − 1) + RT ln k D + RT ln k γ
4
(11)
Heterogeneous systems and reaction equilibria
B. Mishra α
⎛γ ⎛γ ⎞ ∴ 0 = ΔH r − T ΔSr + ΔVr ( P − 1) + RT ln k D + RT ln ⎜ Mg ⎟ + RT ln ⎜ Fe ⎜γ ⎝ γ Fe ⎠ ⎝ Mg
⎞ ⎟⎟ ⎠
β
β α α ⇒ T ( ΔSr − R ln k Dα ,,Mg − X Mg ) = ΔH r + ΔVr ( P − 1) + WFeα − Mg ( X Fe ) + WFeβ −Mg ( X Mgβ − X Feβ )
⇒T =
α α ΔH r + ΔVr ( P − 1) + WFeα − Mg ( X Fe − X Mg ) + WFeβ −Mg ( X Mgβ − X Feβ ) β ΔSr − R ln k Dα ,,Fe − Mg
(12) Comparing eqns.(8) and (12), it is evident that the difference lies on the last two terms in the numerator, that arise due to non-ideal mixing as represented in eqn. (12).Please note that if one of the phases (say β ) displays, ideal mixing, then the last terms in the numerator will disappear (as γ iβ = 1.0 ).
Again, if the excess free energy of mixing in some solid solution may not be symmetric, and therefore, may take more complex form such as g mex, ss = X A . X B [WBA X A + W AB X B ]
(13)
(if WBA = W AB ,then eqn. (13) reduces to the expression for symmetric solid solution. In these cases, RT ln γ A and RT ln γ B will accordingly be expressed (through eqn. (14), part-2.) differently and need to be accommodated into the geothermometric expression (eqn.(12)). 1 1 Fe3 Al 2 Si3 O12 (alm) + KMg 3 AlSi3O10 (OH ) 2 ( phl ) = 3 3 Bt-Grt: 1 1 Mg 3 Al 2 Si3O12 ( pyr ) + KMg 3 AlSi3 O10 (OH ) 2 (ann) 3 3 1 1 Opx-Grt: MgSiO3 (en) + Fe3 Al 2 Si3 O12 (alm) = FeSiO3 ( fs ) + Mg 3 Al 2 Si3O12 ( pyr ) 3 3 Cpx-Grt:
1 1 CaMgSi2 O6 (di) + Fe3 Al 2 Si3O12 (alm) = CaFeSi2 O6 (hed ) + Mg 3 Al 2 Si3 O12 ( pyr ) 3 3
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Heterogeneous systems and reaction equilibria
B. Mishra
Cordierite-Grt: 2 Fe3 Al 2 Si3 O12 (alm) + 3Mg 2 Al 4 Si5 O18 ( Mg − cord ) = 2Mg 3 Al 2 Si3O12 ( pyr ) + 3Mg 2 Al 4 Si5 O18 ( Fe − cord ) Hbl-Grt: 4Mg 3 Al 2 Si3 O12 + 3NaCa 2 Fe4 Al3 Si6 O22 (OH ) 2 = 4 Fe3 Al 2 Si3 O12 + 3NaCa 2 Mg 4 Al3 Si6 O22 (OH ) 2
Net Transfer Reactions: Geobarometry
The cation exchange reactions, with thermometric significance, are characterized by insignificant volume change (ΔVr →0). Hence,
dP ΔS r = → ∞ (Fig. 1) dT ΔVr
Fig.1 Thermodynamic representation of Ideal thermometer and barometer.
In contrast, the net transfer reactions have high values of ΔVr (ΔVr >>>ΔSr). Therefore, dP ΔS r = → 0 (Fig. 1) dT ΔVr
Some examples include 3CaAl 2 Si2 O8 = Ca3 Al 2 Si3 O12 + 2 Al 2 SiO5 + SiO2 ( A)
3CaAl 2 Si2 O8 + 6 FeTiO3 + 3SiO2 = Ca3 Al 2 Si3O12 + 2 FeAl 2 Si3O12 + 6TiO2 ( B)
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Heterogeneous systems and reaction equilibria
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Following eqn. (6) the condition of equilibrium at P, T for these reactions is: 0 = ΔH
0 ,1, 298 A
0 ,1, 298 B
− TΔS
0 ,1, 298 A
− TΔS
0 ,1, 298 B
+ ΔC
* P, A
+ ΔV
0 ,1, 298 A
+ ΔC
* P,B
+ ΔV
0 ,1, 298
(a )(a ) (a ) ( P − 1) + RT ln (a ) Gt gross
Sill / kya 2 Al 2 SiO5
qtz SiO2
Plag 3 an
(14a)
and 0 = ΔH
(a )(a ) + (a ) ( P − 1) + RT ln (a ) + (a ) + (a ) Gt 2 alm
Gt gross
Plag 3 an
Ilm FeTiO3
rut 6 TiO2
6 qtz SiO2
(14b) Gt Gt Gt 3 a gr = ( X Ca .γ Ca ) ( formixig in 3 sites )
( = (X
) )(mixing in one site) ) .γ
Gt Gt Gt a alm = X Fe .γ Ca ( formixing in 3 sites ) Pl where a an
3
Pl Ca
(
Pl .γ Ca
ilm a FeTiO = X FeTiO3 3
and a
qtz SiO2
= 1, a
Ru TiO2
FeTiO3
/ kya = 1, a AlSill2 SiO = 1 .0 5
(since practically pure phases if present) The corresponding geobarometric expressions are P (bars ) = 1 −
P = 1−
1 ΔV A0,1, 298
1 ΔVB0,1, 298
( ) ( )
Gt ⎡ a gr ⎢ΔH A0,1, 298 − TΔS A0,1, 298 + ΔC P* , A + RT ln Plag ⎢⎣ a an
⎤ / kya qtz + 2 RT ln a AlSill2 SiO + RT ln a SiO2 ⎥ 5 ⎥⎦ (15a)
3
( )( ) ) ( )(
Gt Gt 2 ⎡ a gr a alm 0 ,1, 298 0 ,1, 298 * ⎢ΔH B − TΔS B + ΔC P , B + RT ln Pl 3 ilm ⎢ a an a FeTiO 3 ⎣
6
+ 6 RT ln a
Ru TiO2
− 3RT ln a
(15b)
Since the P values are fairly large i.e., of the order of few thousands, the unity term (1) in eqns. (15a) and (15b) can be ignored. For both the eqns. (15a) and (15b), if T is known and the ΔH , ΔS , ΔV and a-x relationships of solid solution ( γ i ) terms are available then, P is uniquely evaluated. Again in both the reaction (A and B) as written, the RHS represents high pressure assemblages and therefore V product << Vreac tan t
7
qtz SiO2
⎤ ⎥ ⎥ ⎦
Heterogeneous systems and reaction equilibria
B. Mishra
(i.e., ΔV A0,1, 298 , ΔVB0,1, 298 are both negative)
Now for example in a realistic assemblage pertaining to reaction (B), quartz is absent. i.e., a
qtz SiO2
< 1.0 , it follows that RT ln a
qtz SiO2
is negative and hence −
qtz 3RT ln aSiO 2
ΔVB0,1,298
(in 15b) is
negative. (since, ΔVB is numerically negative). Clearly, then the computed P values for Qtz-absent assemblage will be lesser compared to P-value calculated in quartz- present ( a SiO2 = 1.0 ) assemblage. On the other hand, if rutile is absent in the assemblage, then pressure will be overestimated.
Net transfer equilibria: some geobarometers
1. Grt-Plag-Qtz-Al2SiO5 (GASP) 3CaAl2Si2O8 = Ca3Al2Si3O12 + 2Al2SiO5 + SiO2 2. Grt-Plag-Mus-Bt Fe end member: Fe3Al2Si3O12 + Ca3Al2Si3O12 + KAl3Si3O10 (OH)2 = 3CaAl2Si2O8 + Grt Ms Pl KFe3AlSi3O10(OH)2 Bt
Mg end member: Mg3Al2Si3O12 + Ca3Al2Si3O12 + KAl3Si3O10 (OH)2 = 3CaAl2Si2O8 Grt Ms Pl + KMg3AlSi3O10(OH)2 Bt 3. Grt-Plag - Hbl- Qtz 3Ca2(Mg, Fe)5Si8O22(OH)2 + 6CaAl2Si2O8 = Hbl Pl 3Ca2 (Mg, Fe)4Al2Si7O22(OH)2 + 2Ca3Al2Si3O12 + (Mg, Fe)3Al2Si3O12 + 6SiO2 Hbl Grt 4. Grt-Plag -Ru- Ilm-Qtz (GRIPS) 8
Heterogeneous systems and reaction equilibria Ca3Al2Si3O12 + 2Fe3Al2Si3O12 + 6TiO2 = 6FeTiO3 + 3CaAl2Si2O8 +3SiO2 Grt Ru Ilm Pl
5. Grt-Ru- Ilm- Al2SiO5 -Qtz (GRAIL) Fe3Al2Si3O12 + 3TiO2 = 3FeTiO3 + Al2SiO5 + 2SiO2 Grt Ru Ilm
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B. Mishra