Physical Chemistry Third Edition

1130 27 Equilibrium Statistical Mechanics. III. Ensembles

We have used the statistical thermodynamics definition of the chemical potential as a

derivative with respect to the number of molecules instead of the amount in moles. We

have replaced this derivative by a quotient of finite differences as in Eq. (26.1-24), and

have attached a subscript to the canonical partition function to indicate the number of

particles in the system.

HU+PVkBT^2

(

∂ln(Z)

∂T

)

V,N

+VkBT

(

∂ln(Z)

∂V

)

T,N

(27.2-8e)

PROBLEMS

Section 27.2: Thermodynamic Functions in the Canonical
Ensemble

27.6 A definition of the statistical entropy that can be used for
the canonical ensemble is:

Sst−kB

∑

k

pkln(pk)+S 0

where the sum is over all microstates of the system.

a.Apply this definition to the case that all of thepkvalues

are equal to 1/Ω(as is the case in the microcanonical

ensemble) and recover the same formula as in

Eq. (26.1-1).

b.Apply this definition to the probability distribution of

the canonical ensemble and recover the same formula

for the entropy as in Eq. (27.2-7).

27.3 The Dilute Gas in the Canonical Ensemble

The formulas for the thermodynamic functions in the previous section apply to any

kind of system. They can be applied to a dilute gas by using Eq. (27.1-27) to express

the canonical partition function in terms of the molecular partition function.

〈E〉NkBT^2

(

∂ln(zN/N!)

∂T

)

V,N

NkBT^2

(

∂ln(z)

∂T

)

V

(dilute gas) (27.3-1)

Equation (27.3-1) is identical with Eq. (25.3-26). For a dilute gas the pressure is

PkBT

(

∂ln(Z)

∂V

)

T,N

kBT

(

∂ln(zN)

∂V

)

T,N

−kBT

(

∂ln(N!)

∂V

)

T,N

NkBT

(

∂ln(z)

∂V

)

T

− 0 NkBT

(

∂ln(z)

∂V

)

T

(27.3-2)

This equation is identical with Eq. (26.1-11). The formulas for the other thermodynamic

functions can be derived in the same way, and are identical with those in Chapter 26.

Exercise 27.2

Show that the formulas in Eq. (27.2-8) lead to the same formulas for the thermodynamic functions

of a dilute gas in terms of the molecular partition function as in Section 26.1.