Physical Chemistry , 1st ed.

equilibrium—must have conditions corresponding to that line). The num-

ber of degrees of freedom has dropped because the number of phases in

your system has increased.

Suppose you know that you have three phases of H 2 O in your system at

equilibrium. You don’t have to specify any degrees of freedom because there

is only oneset of conditions in which that will occur: for H 2 O, those condi-

tions are 273.16 K and 6.11 mbar. (See Figure 6.5: there is only one point

on that phase diagram where solid, liquid, and gas exist in equilibrium, and

that is the triple point.) There is a relationship between the number of

phases occurring at equilibrium and the number of degrees of freedom nec-

essary to specify the point in the phase diagram that describes the state of

the system.

In the 1870s, J. Willard Gibbs (for whom Gibbs free energy is named) de-

duced the simple relationship between the number of degrees of freedom and

the number of phases. For a single-component system,

degrees of freedom  3 P (6.17)

where Prepresents the number of phases present at equilibrium. Equation 6.17

is a simplified version of what is known as the Gibbs phase rule.In this rendi-

tion, it assumes that one of the state variables of the system, usually the vol-

ume, can be determined from the others (via an equation of state). You should

verify that this simple equation provides the correct number of degrees of free-

dom for each situation described above.

6.7 Natural Variables and Chemical Potential

We have implied previously that the conditions of the phase equilibrium de-

pend on the state variables of the system, namely volume, temperature, pres-

sure, and amount. Usually we deal with changes in systems as temperature and

pressure vary. It would therefore be useful to know how the chemical potential

varies with respect to temperature and pressure. That is, we want to know

(/ T) and (/ p). The chemical potential is the change in the Gibbs free

energy with respect to amount. For a pure substance, the total Gibbs free en-

ergy of a system is

Gn

where nis the number of moles of the material having chemical potential .

[This expression comes directly from the definition of, which is ( G/ n)T,p.]

From the relationship between Gand presented in Chapter 4, and know-

ing how Gitself varies with Tand p(given in equations 4.24 and 4.25), we

can get

^



T

(^) 
p,n

S (6.18)

and

^



p

(^) 
T,n

V (6.19)

The natural variable equation for dis

dSdTVdp (6.20)

This is similar to the natural variable equation for G. We can also write the

derivatives from equations 6.18 and 6.19 in terms of the changein chemical

6.7 Natural Variables and Chemical Potential 159

Temperature

Pressure

Solid

Gas

Liquid

Figure 6.14 If all you know about a system is
that H 2 O is solid, then any set of pressure and
temperature conditions in the shaded area would
be possible conditions of the system. You will
need to specify two degrees of freedom to describe
your system.