Geoscience Reference
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with no change in the energy of the system. This condition is written by differentiating the
expression for Gibbs' free energy as:
μ CH 4 d n CH 4 + μ O 2 d n O 2 + μ CO 2 d n CO 2 + μ H 2 O d n H 2 O =
0
(C.23)
or alternatively:
μ CH 4 ν CH 4 + μ O 2 ν O 2 + μ CO 2 ν CO 2 + μ H 2 O ν H 2 O d
ξ =
G d
ξ =
0
(C.24)
a condition which can only generally be observed when the content of the parentheses of
the left-hand side cancels out (
= μ i ν i
G
=
0). By replacing the chemical potentials
by their expression (C.14 ) , we obtain:
RT (
2) ln P H 2 O =
1) ln P CH 4 +
(
2) ln P O 2 +
(
+
1) ln P CO 2 +
(
+
(
H 2 O
0
0
0
0
1)
μ
CH 4 +
(
2)
μ
O 2 +
(
+
1)
μ
CO 2 +
(
+
2)
μ
(C.25)
the relation of which can be compacted using the properties of the logarithms to the form
of the “mass action law:”
P CO 2 P H 2 O
P CH 4 P O 2
=−
G 0 ( T )
RT
ln
=
ln K ( T , P )
(C.26)
In this equation, K ( T , P ) is the equilibrium constant of the reaction and
G 0 ( T ) the varia-
tion in Gibbs' free energy when all the components are in the standard state. Two essential
equations accompany the mass action law and control the variation of the constant K with
temperature and pressure. They are a consequence of the equations demonstrated above:
ln K
P =−
H 0 ( T , P )
R
(C.27)
/
(1
T )
ln K
T =−
V 0 ( T , P )
RT
(C.28)
P
This formalism established for ideal gases can be generalized to real gases by defining a
parameter that satisfies the same equations as partial pressure; this is the gas fugacity.
It can also be transposed to liquid and solid solutions by replacing partial pressures by
molar fractions x i
n . For example, the substitution of rubidium (Rb) for potassium
(K) between feldspar and mica can be described by the reaction:
Rb feld +
=
n i /
K mica
Rb mica +
K feld
(C.29)
The chemical potential of rubidium in feldspar is written:
feld
Rb
feld,0
Rb
RT ln x feld
Rb
μ
= μ
( T , P )
+
(C.30)
with similar expressions for other potentials. It will be noticed that the reference chemical
potential
feld,0
Rb
1 is now defined for a unit molar concentration ( x feld
Rb
1) and that it is
dependent on temperature and pressure. The mass action law of equilibrium is written:
ln x mica
μ
=
=
x feld
K
ln (Rb
/
K) mica
=−
G 0 ( T , P )
RT
Rb
=
K) feld =
ln K ( T , P )
(C.31)
x feld
Rb
x mica
K
(Rb
/
 
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