Physical Chemistry Third Edition

1142 27 Equilibrium Statistical Mechanics. III. Ensembles

where we assume that the probability distribution is normalized:

∫

f(p,q)dpdq 1 (27.5-5)

The Other Thermodynamic Functions

Once we have expressions forUandS, we can obtain formulas for the other thermo-

dynamic functions. The Helmholtz energy is given by

AU−TS−kBTln(Zcl) (27.5-6)

and the pressure is given by the thermodynamic relation in Eq. (26.1-8)

P−

(

∂A

∂V

)

T,n

kBT

(

∂ln(Zcl)

∂V

)

T,N

(27.5-7)

Expressions for the enthalpy, the Gibbs energy, the heat capacity, and the chemical

potential can be obtained from these formulas.

Exercise 27.5

Write formulas forH,G,CV, andμfor the classical canonical ensemble. Compare your formulas

with those in Eq. (27.2-8).

The Thermodynamic Functions of a Dilute Gas

We have obtained formulas for the thermodynamic functions of a general system in

terms of the classical canonical partition function.

The Energy of a Dilute Gas

If the system is a dilute gas we can express the energy in terms of the classical molecular

partition function, using Eq. (27.4-18):

U〈E〉N〈ε〉NkBT^2

(

∂ln(zcl)

∂T

)

V

(27.5-8)

Using the formula forzclin Eq. (27.4-12), the energy of a monatomic dilute gas is

U

3 NkBT

2

(

monatomic dilute gas

without electronic excitation

)

(27.5-9)

The indistinguishability of the molecules and the relation between quantum states and

phase space volume do not affect the energy expression.

The classicaltheorem of equipartition of energystates that if a molecular vari-

able occurs in the classical energy in a quadratic form (to the second power) the

contribution to the ensemble average system energy corresponding to that variable is

equal to NkBT/2. Equation (27.5-9) conforms to this theorem. We can verify