1
Note 6.
Equilibrium II
: Applications of free energy concepts
6.1 Phase Diagram: Clapeyron equation
We’ve talked about phase equilibria in a couple of different ways (i.e. in terms of free
energy (pg 5 in Note 5) and in terms of chemical potentials (in pg 14 in Note 5, if you
consider A and B are two phases of a substance). The Clapeyron equation connects
P
,
V
,
T
and
H
in a new way, which is often useful for thinking about phase transitions.
Consider two phases, a liquid and its vapor in equilibrium at temperature
T
and
pressure
P
. If we slightly change
T
and
P
, we get the new values
T + dT
and
P + dP
.
Since
dG = V dP
−
S dT
we can write the change in free energies for both the liquid and gas phases:
dG
l
= V
l
dP - S
l
dT
dG
g
= V
g
dP - S
g
dT
Under the new conditions (i.e. temperature
T + dT
and pressure
P + dP
), we can
calculate equilibrium by
dG
l
= dG
g
. Equating these two we get
V
l
dP - S
l
dT = V
g
dP - S
g
dT
and by rearranging, we can put this in the form
vap
vap
l
g
l
g
V
S
V
V
S
S
dT
dP
Δ
Δ
≡
−
−
=
Since at
Δ
G
vap
=
G
g
- G
l
= 0
the two phases are in equilibrium, we can write
Δ
G
vap
=
Δ
H
vap
–
T
trans
Δ
S
vap
= 0
→
trans
vap
vap
T
H
S
Δ
=
Δ
Note that this is equivalent to say that the chemical potentials for gas and liquid are the
same i.e.
dG
l
= dG
g
→
μ
(liquid) =
μ
(gas).
Taking this and putting it into our previous formula, we get
vap
trans
vap
vap
vap
V
T
H
V
S
dT
dP
Δ
Δ
=
Δ
Δ
=

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