Thermodynamics and Chemistry

CHAPTER 8 PHASE TRANSITIONS AND EQUILIBRIA OF PURE SUBSTANCES

8.4 COEXISTENCECURVES 215

bc

bc

bc

A

B

solid C

liquid

gas,

0:03

bar

gas,

0:003

bar

250 260 270 280 290 300 310

0

1

2

3

4

5

T=K

(a)

=

kJ mol

^1

bc

bc bc

bc

A

B

C triple point

solid liquid gas

250 260 270 280 290 300 310

0

0:01

0:02

0:03

0:04

T=K

(b)

p=

bar

Figure 8.13 Phase stability of H 2 O.a

(a) Chemical potentials of different physical states as functions of temperature. (The

scale forhas an arbitrary zero.) Chemical potentials of the gas are shown at0:03bar

and0:003bar. The effect of pressure on the curves for the solid and liquid is negligi-

ble. AtpD0:03bar, solid and liquid coexist atT D273:16K (point A) and liquid

and gas coexist atTD297:23K (point B). AtpD0:003bar, solid and gas coexist at

TD264:77K (point C).

(b) Pressure–temperature phase diagram with points corresponding to those in (a).

aBased on data in Refs. [ 70 ] and [ 87 ].

entropies of sublimation and vaporization are positive. This difference in slope is illustrated
by the curves for H 2 O in Fig.8.13(a). The triple-point pressure of H 2 O is0:0062bar. At
a pressure of0:03bar, greater than the triple-point pressure, the curves for solid and liquid
intersect at a melting point (point A) and the curves for liquid and gas intersect at a boiling
point (point B).
From.@=@p/T DVm, we see that a pressure reduction at constant temperature low-
ers the chemical potential of a phase. The result of a pressure reduction from0:03bar to
0:003bar (below the triple-point pressure of H 2 O) is a downward shift of each of the curves
of Fig.8.13(a) by a distance proportional to the molar volume of the phase. The shifts of
the solid and liquid curves are too small to see (Åis only0:002kJ mol^1 ). Because
the gas has a large molar volume, the gas curve shifts substantially to a position where it
intersects with the solid curve at a sublimation point (point C). At0:003bar, or any other
pressure below the triple-point pressure, only a solid–gas equilibrium is possible for H 2 O.
The liquid phase is not stable at any pressure below the triple-point pressure, as shown by
the pressure–temperature phase diagram of H 2 O in Fig.8.13(b).

8.4.2 The Clapeyron equation

If we start with two coexisting phases,íandì, of a pure substance and change the tem-
perature of both phases equally without changing the pressure, the phases will no longer be