Geology Reference
In-Depth Information
From the differential of U the following (energetic) intensive variables are
defined:
T
o
U
oS
absolute temperature
V
;
n
i
p
o
U
oV
pressure
S
;
n
i
ð
electro
Þ
chemical potential
l
i
o
U
on
i
of the ith component
S
;
V
;
n
j
6¼
i
The intensive variables enable one to deal with the coupling of the system to its
environment. Using them the relation (
2.2
) can be rewritten in the Gibbs form
dU
¼
T dS
pdV
þ
X
i
l
i
dn
i
ð
2
:
3a
Þ
or in the Euler form
U
¼
TS
pV
þ
X
i
l
i
n
i
ð
2
:
3b
Þ
If follows from (
2.3
) that the above criterion of equilibrium can be restated in
terms of minimizing the internal energy at constant S
;
V
;
n
i
.
In practice it is often convenient to use one or both the intensive variables T , p
as independent variables instead of the respective conjugate extensive variables S,
V used in writing (
2.2
). Such transformation leads to the definition of the addi-
tional functions,
Helmholtz
ð
free
Þ
energy
A
AT
;
V
;
n
i
ð
Þ ¼
U
TS
enthalpy
H
HS
;
p
;
n
i
ð
Þ ¼
U
þ
pV
Gibbs
ð
free
Þ
energy
G
GT
;
p
;
n
i
ð
Þ ¼
U
þ
pV
TS
;
to equivalent forms of the fundamental relations (
2.3
),
dA
¼
SdT
pdV
þ
X
l
i
dn
i
ð
2
:
4
Þ
dH
¼
T dS
þ
V dp
þ
X
l
i
dn
i
ð
2
:
5
Þ
dG
¼
SdT
þ
V dp
þ
X
l
i
dn
i
ð
2
:
6
Þ
and to corresponding extremum conditions for equilibrium under constraint of the
specified independent variables (for example, the Gibbs energy G is a minimum at
equilibrium when T, p and n
i
are the independent variables). There are many
mathematical relations between the various parameters or functions of state so far
