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QUANTITATIVE EFFECTS OF PRESSURE AND TEMPERATURE CHANGE 197

boundaries of a phase diagram. For example:
O
dp H m (l→s)
for the liquid → solid line =
dT T V m (l→s)
O
dp H m (v→s)
for the vapour → solid line = s
dT
T V m (v→s)
v
O
dp H m (v→s)
for the vapour → liquid line =
l
dT T V m (v→s)
v
Approximations to the Clapeyron equation
We need to exercise a little caution with our terminology: we performed the calculation
in Worked Example 5.1 with Equation (5.1) as written, but we should have written p
6
rather than dp because 10 Pa is a very large change in pressure.
Similarly, the resultant change in temperature should have been The ‘d’ in Equation
written as T rather than dT , although 0.07 K is not large. To (5.1) means an infini-
tesimal change,
accommodate these larger changes in p and T , we ought to be
whereas the ‘ ’symbol
rewrite Equation (5.1) in the related form:
here means a large,
macroscopic, change.
p H m O
≈
T T V m
We are permitted to assume that dp is directly proportional to dT because H and
V are regarded as constants, although even a casual inspection of a phase diagram
shows how curved the solid–gas and liquid–gas phase boundaries are. Such curvature
clearly indicates that the Clapeyron equation fails to work except
over extremely limited ranges of p and T . Why? The Clapeyron equa-
We assumed in Justification Box 5.1 that H O is indepen- tion fails to work for
(melt)
dent of temperature and pressure, which is not quite true, although phase changes involv-
the dependence is usually sufficiently slight that we can legiti- ing gases, except
mately ignore it. For accurate work, we need to recall the Kirchhoff over extremely limited
equation (Equation (3.19)) to correct for changes in H. ranges of p and T.
Also, we saw on p. 23 how Boyle’s Law relates the volume of
a gas to changes in the applied pressure. Similar expressions apply
for liquids and solids (although such phases are usually much less It is preferable to
compressible than gases). Furthermore, we assumed in the deriva- analyse the equilib-
tion of Equation (5.1) that V m does not depend on the pressure ria of gases in terms
changes, which implies that the volumes of liquid and solid phases of the related Clau-
each change by an identical amount during compression. This sius–Clapeyron equa-
approximation is only good when (1) the pressure change is not tion; see Equation (5.5).
extreme, and (2) we are considering equilibria for the solid–liquid