Geoscience Reference
In-Depth Information
We have seen that when an equilibrium is established between several components that
react together, the change in Gibbs' free energy
G
of the reaction is zero. By definition,
the change in G is equal to:
G
=
H
−
T
S
(C.39)
Let us consider two equilibrium states of the system that are infinitely close and separated
by temperature and pressure intervals d
T
and d
P
. We can write that between these states
the
G
of the reaction remains unchanged:
d
G
=−
S
d
T
+
V
d
P
(C.40)
giving Clapeyron's equation:
d
P
d
T
=
S
H
V
=
(C.41)
T
V
where the second equation results from the condition that, at equilibrium,
0. This
equation is extremely important in the Earth sciences as it is used for connecting the slope
of phase change or mineralogical reaction in the pressure-temperature space familiar to
geologists with two measurable quantities: the change in density (and hence in molar
volume) and the latent heat absorbed in the course of transformation.
The phase diagrams, which are representations of the stability field of the different
phases that may appear within a system of given composition, are often approached qual-
itatively, but are also amenable to thermodynamic treatment. Let us consider a simplified
granitic system, the quartz-albite (qz-ab) binary system, and ask which phases (quartz,
albite, liquid) are stable at ambient pressure for a given proportion of albite/quartz and at
a given temperature. If we consider that the latent heats of fusion of quartz and albite vary
little with temperature,
(C.38)
, applied to the solubility of quartz and albite, becomes after
integration:
G
=
1
T
−
ln
x
qz
=−
H
qz
(
T
,
P
)
R
1
T
f
(C.42)
qz
1
T
−
ln
x
ab
=−
H
ab
(
T
,
P
)
R
1
T
f
ab
(C.43)
where the integration condition has been introduced so that, when the molar fraction
x
of
each component is equal to unity, the temperature must be equal to the melting temperature
T
f
ab
and
T
f
qz
of the pure component. These two equations define the curve of the liquidus
of the quartz-albite system (
Fig. C.2
). They cross at the eutectic point E. For each compo-
sition, the system has a stable branch at high temperature and a metastable branch at lower
temperature.
If a liquid initially of composition
x
A
is cooled to temperature
T
A
, it allows some of the
quartz to crystallize. Its composition evolves toward the eutectic point E where the albite
begins to precipitate. If too little albite crystallizes, the system moves on to the metastable
branch of quartz, on which it cannot remain without violating the minimum energy princi-
ple. If too much albite crystallizes, the system moves on to the metastable branch of albite,
