Thermodynamics and Chemistry

CHAPTER 8 PHASE TRANSITIONS AND EQUILIBRIA OF PURE SUBSTANCES

8.4 COEXISTENCECURVES 214

300

T=

K

400

0:2

p=bar

1

0

=

kJ mol

^1

10

T

p

gas

liquid

Figure 8.12 Top: chemical potential surfaces of the liquid and gas phases of H 2 O;

the two phases are at equilibrium along the intersection (heavy curve). (The vertical

scale forhas an arbitrary zero.) Bottom: projection of the intersection onto thep–T

plane, generating the coexistence curve. (Based on data in Ref. [ 70 ].)

8.4.1 Chemical potential surfaces

We may treat the chemical potentialof a pure substance in a single phase as a function
of the independent variablesT andp, and represent the function by a three-dimensional
surface. Since the condition for equilibrium between two phases of a pure substance is that
both phases have the sameT,p, and, equilibrium in a two-phase system can exist only
along the intersection of the surfaces of the two phases as illustrated in Fig.8.12.
The shape of the surface for each phase is determined by the partial derivatives of the
chemical potential with respect to temperature and pressure as given by Eqs.7.8.3and7.8.4:

@
@T



p

DSm



@

@p



T

DVm (8.4.1)

Let us explore howvaries withT at constantpfor the different physical states of a
substance. The stable phase at each temperature is the one of lowest, since transfer of a
substance from a higher to a lowerat constantTandpis spontaneous.
From the relation.@=@T /pD Sm, we see that at constantpthe slope ofversus
T is negative since molar entropy is always positive. Furthermore, the magnitude of the
slope increases on going from solid to liquid and from liquid to gas, because the molar