Geoscience Reference
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giving upon integration:
G
=
nRT
ln
P
+
g
(
T
,
n
)
(C.15)
where
g
(
T
,
n
) is the Gibbs' free energy of
n
moles of gas at unit pressure. Let us now
consider a system made up of several components (e.g. Na
+
,Cl
−
, and H
2
O for a salt
solution). The share of the total free enthalpy
G
of the system that can be assigned to each
component
i
having
n
i
moles in the system is
μ
i
, which is obtained by writing:
G
=
μ
i
n
i
(C.16)
i
where the sum relates to all the components of the system;
μ
i
is known as the chemical
potential of component
i
in the system. It is found, by taking the derivative of this relation
with respect to each variable, that
μ
i
is simply the derivative of
G
relative to
n
i
, when
T
,
P
, and the number of moles of components other than
i
are kept constant. This yields the
general expression of
G
for a system whose temperature, pressure, and composition are
specified:
d
G
=−
S
d
T
+
V
d
P
+
μ
i
n
i
(C.17)
i
If we consider a mixture of ideal gases, i.e. gas molecules of species other than
i
which
do not interact with each other, the equation of state can be written as:
PV
=
(
n
i
)
RT
(C.18)
i
where
n
is replaced by the sum of the numbers of moles of each species. The partial
pressure
P
i
of gas
i
is defined as (
n
i
/
n
)
P
, and the Gibbs' free energy equation becomes:
n
i
RT
ln
P
i
+
μ
i
(
T
)
0
=
G
(C.19)
i
0
where
μ
i
(
T
) is the chemical potential in the standard state
P
i
=
1 atm. By reference to
(C.14)
, the chemical potential
μ
i
of gas
i
in the mixture of gases is defined as:
0
μ
i
=
μ
i
(
T
)
+
RT
ln
P
i
(C.20)
Let us now consider the following reaction in the gaseous state:
CH
4
+
2O
2
⇔
CO
2
+
2H
2
O
(C.21)
As the reaction progresses, the compounds are created and destroyed in proportions
dictated by the stoichiometric coefficients
ν
:
d
n
CH
4
d
n
O
2
d
n
CO
2
d
n
H
2
O
ν
CH
4
=
ν
O
2
=
ν
CO
2
=
ν
H
2
O
=
ξ
d
(C.22)
where
measures the
progress of the reaction and varies between 0 and 1. The reaction will be at equilibrium
when the free energy of the reaction products (right-hand side) is equal to that of the reac-
tants (left-hand side), i.e. when the transfer of matter from one side to the other occurs
ν
CH
4
=−
1,
ν
O
2
=−
2,
ν
CO
2
=+
1, and
ν
H
2
O
=+
2. The parameter
ξ