Geoscience Reference
In-Depth Information
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
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
/