The Clapeyron Equation
If we confine ourselves to conditions where two phases, say vapor and liquid, are
in equilibrium, we effectively limit ourselves to a onedimensional world defined
by the line separating the liquid region from the vapor region.
Under those
conditions, the Maxwell relation
(3.124)
æ
è
¹
P
¹
T
ö
ø
V
=
æ
è
¹
S
¹
V
ö
ø
T
,
(3.124) can be modified as follows:
(5.8)
dP
dT
=
dS
dV
j
S
m
,
v
−
S
m
,
l
V
m
,
v
−
V
m
,
l
=
,
vap
S
m
,
vap
V
m
Substituting the definition of entropy change for a phase change, we obtain
(5.9)
dP
dT
=
,
vap
H
m
T
,
vap
V
m
.
This is called the
Clapeyron equation
.
Similar equations can be written for the solidvapor boundary (enthalpy of
sublimation!) and the solidliquid boundary (enthalpy of fusion!).
Note that, since
enthalpy is a state function,
∆
sub
H
m
=
∆
fus
H
m
+
∆
vap
H
m
.
(5.10)
The ClausiusClapeyron Equation:
Eq. (5.9) may be written as
(5.11)
dP
dT
=
,
vap
H
m
T
æ
è
V
m
,
g
−
V
m
,
l
ö
ø
l
,
vap
H
m
TV
m
,
g
j
,
vap
H
m
RT
2
P
,
where we have assumed that the vapor behaves ideally to get the last equality.
This equation may be rearranged and integrated to give
or
(5.14)
ln
P
u
=
−
,
vap
H
m
RT
+
C
,
(5.16)
ln
æ
è
P
2
P
1
ö
ø
=
,
vap
H
m
R
æ
è
1
T
1
−
1
T
2
ö
ø
.
The ClausiusClapeyron equation is applicable to solidvapor equilibrium as well.
The pressure plotted in the phase diagrams and used in these relationships is the
vapor pressure
of the substance under consideration, not
the
total
pressure
.
Eq.
(5.14) gives the
P

T
relationship required to plot the lines in the phase diagrams.
Therefore, Eq. (5.14) for the liquidvapor and the solidvapor equilibria can be
used to find the triple point for a onecomponent system.
Examples 5.2, 5.3,
Problems 5.1²5.4, 5.7²5.10, 5.12, 5.15, 5.18, 5.20.
2