Concise Physical Chemistry

c09 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

126 THE PHASE RULE

H 2 O(gas)

H 2 O(liquid)

H 2 O(gas)

FIGURE 9.1 Pure water in one phase (left) and two phases (right). At some temperature,

the phases coexist in equilibrium.

exist wherewith we can determineEandCpfrom a knowledge ofpandTeven

though we may not know exactly what these equations are. We could even write

p=f(Cp,E). We are not restricted as to what the independent variables are, so long

as there are precisely two of them.

The number of degrees of freedom for a pure, one-phase system is easy. Identifying

somephases is easy as well.^3 We can see the phase separation in mixtures of carbon

tetrachloride and water because they have different refractive indices. The two phases,

one as the top layer and one as the bottom layer, look different. The trick is to express

the behavior of a completely general system, which need not be pure, and may consist

of many chemical entities distributed over many phases.

If we choose water vapor as our example of a pure phase, examine the Gibbs free

energy as our state function, and take pressure and temperature as our independent

variablesμ=f(p,T), the system as defined has a wide range of freedom. Within

reasonable limits, we can changepandTarbitrarily. The state of the system can

be represented as a point on a two-dimensional plot ofpvs.T(orpvs.V,asin

Chapter 1). Any point represents the system in some state.

Sooner or later though, we are bound to exceed reasonable limits; and the water,

even in a closed container that previously contained only water vapor, is bound to

condense to produce some liquid water in a state of equilibrium with gaseous water

(Fig. 9.1)

We treat this physical equilibrium just as we do a chemical equilibrium. The

vaporization equilibrium is

H 2 O(l)←→H 2 O(g)

The equilibrium constant is

Keq=

pH 2 O(g)

aH 2 O(l)

=pH 2 O(g)

where pure water is in its standard state, resulting in a denominator ofaH 2 O(l)≡ 1. 0

in the normal equilibrium expression. Anequilibriumbetween pure liquid and pure

(^3) Identification of some phases may not be easy. Solids can exist as phases that differ only in more subtle
ways such as heat capacity or molar volume.