Geoscience Reference
In-Depth Information
the description of the hydrothermal waves. Thus, an emphasis is made on some of
the basic physicochemical aspects of the hydrothermal growth of crystals.
During the crystallization process from saturated solutions, the mol number of
the solute compounds changes and, therefore, the free energy is represented as a
function of not only the temperature and pressure but also the mol numbers:
G
f
ð
T
;
P
;
n
1
;
n
2
;
...
;
n
i
Þ
ð
4
:
1
Þ
5
dG
5
ðδ
G
=δ
T
Þ
p
;
n
dT
1
ðδ
G
=δ
P
Þ
T
;
n
dP
1
ðδ
G
=δ
n
1
Þ
P
;
T
;
n
dn
1
1
ðδ
G
=δ
n
1
Þ
P
;
T
;
n
dn
2
ð
4
:
2
Þ
where
n
i
5
constant mol number of all components,
n
j
5
constant number of components except one.
The variation in the free energy can be expressed in terms of the chemical
potential:
ðδ
=δ
n
i
Þ
P
;
T
;
n
5
μ
i
ð
:
Þ
G
4
3
When P and T are kept constant,
Eq. (4.2)
becomes:
dG
P
;
T
5
μ
i
dn
1
1
μ
i
dn
2
1
?;
or dG
P
;
T
5
ðΣ
dn
i
Þ
P
;
T
ð
4
:
4
Þ
The integration of
Eq. (4.4)
gives:
G
P
;
T
5
Σ
i
n
i
ð
4
:
5
Þ
Differentiation of
Eq. (4.5)
gives:
dG
P
;
T
5
n
1
d
μ
1
1
n
2
d
μ
2
1
?
1
n
i
d
μ
i
1
μ
i
d
μ
1
1
μ
i
d
μ
2
1
?
1
μ
i
d
μ
i
5
Σ
n
i
d
μ
i
1
Σμ
i
dn
i
ð
4
:
6
Þ
By equating
Eqs (4.4) and (4.6)
we get:
Σ
n
i
d
μ
i
5
0
ð
4
:
7
Þ
Equation (4.7)
is considered as the overall equilibrium condition in the system
with variable mol numbers under constant P and T.
Let us consider a hydrothermal system containing n
A
mol of solid A and par-
tially soluble (n
1A
mol) in n
B
mol solvent. Then the free energy of the system corre-
sponds to
Eq. (4.5)
:
G
n
A
μ
A
1
n
lA
μ
lA
1
n
B
μ
B
ð
4
:
8
Þ
5
