Physical Chemistry Third Edition

216 5 Phase Equilibrium

it is the equilibrium phase. Above the coexistence temperature, the vapor curve lies

lower, so that the vapor is the stable phase, and thesuperheatedliquid is metastable.

Below the normal boiling temperature, the liquid is stable and thesupercooled vapor

is metastable. Near the coexistence temperature a small change in temperature can

change the equilibrium state of the substance from gas to liquid or from liquid to gas

as it moves from one curve to the other.

We can also consider the fact that the molar Gibbs energy is given by

GmHm−TSm (5.4-2)

At constant temperature, we can minimizeGmeither by loweringHmor by raisingSm.

The second term is more important at high temperature than at low temperature since

it is proportional toT. The phase of higher molar entropy is the more stable phase at

high temperature and the phase of lower molar enthalpy is the more stable phase at low

temperature.

Figure 5.6 shows schematically the molar Gibbs energy of liquid and gaseous water

as a function of pressure at constant temperature. The two curves intersect at the equi-

librium pressure for the phase transition at this temperature. The slope of the tangent

to the curve is given by Eq. (4.2-21):

(

∂Gm

∂P

)

T

Vm (5.4-3)

Liquid

Liquid

Vapor

Vapor

G

m

1 atm

P

Figure 5.6 The Molar Gibbs En-
ergy of Water as a Function of
Pressure Near the Liquid–Vapor
Phase Transition (Schematic).

The molar volume of the vapor is greater than that of the liquid phase, so that the

tangent to the vapor curve has a more positive slope than that of the liquid curve. At

any pressure, the lower curve represents the equilibrium phase. If the temperature is

373 .15 K, the liquid is the stable phase at a pressure greater than 1.000 atm, but at a

pressure less than 1.000 atm the vapor is the stable phase.

It is also possible to construct schematic graphs like those of Figures 5.5 and 5.6 for

solid–liquid, solid–solid, and solid–vapor phase transitions.

Exercise 5.11

Sketch rough graphs representing the molar Gibbs energy of water as a function of the tem-

perature and as a function of the pressure in the vicinity of the solid–liquid phase transition.

Liquid water has a smaller molar volume than solid water but a larger molar entropy.

Classification of Phase Transitions

Phase transitions are classified according to the partial derivatives of the Gibbs energy.

Ordinary phase transitions such as vaporizations, freezings, and so on, are calledfirst-

orderphasetransitions, which means that at least one of the first derivatives (∂Gm/∂T)P

or (∂Gm/∂P)Tis discontinuous at the phase transition. In most first-order transitions,

both of these derivatives are discontinuous. From Chapter 4 we know that (∂Gm/∂T)P

is equal to−Smand that (∂Gm/∂P)Tis equal toVm. Figure 5.7 shows schematically

the molar volume as a function of pressure as it would appear for a solid–liquid or a

solid–solid transition. Figure 5.8 shows schematically the molar entropy as a function

of temperature as it would appear for a liquid–vapor transition.

Vm

P

Figure 5.7 The Molar Volume as
a Function of Pressure at a First-
Order Phase Transition (Sche-
matic).

Sm

T

Figure 5.8 The Molar Entropy as
a Function of Temperature at
a First-Order Phase Transition
(Schematic).