Physical Chemistry Third Edition

27.2 Thermodynamic Functions in the Canonical Ensemble 1129

where we have applied the chain rule (see Appendix B). We can also write Eq. (27.1-22)

for the pressure in terms of the temperature:

〈P〉kBT

(

∂ln(Z)

∂V

)

T

(general system) (27.2-2)

In order to obtain formulas for other thermodynamic variables, we must have a

formula for the entropy. We begin with the thermodynamic relation in Eq. (4.2-3),

which holds for reversible changes in any closed system:

dUTdS−PdV (27.2-3)

We divide this equation byTand write it in a different form:

dS

1

T

dU+

P

T

dVd

(

U

T

)

+

U

T^2

dT+

P

T

dV (27.2-4)

Using Eqs. (27.2-1) and (27.2-2) forUandP, we obtain

dSd

(

U

T

)

+kB

(

∂ln(Z)

∂T

)

V

dT+kB

(

∂ln(Z)

∂V

)

T

dV (27.2-5)

For a closed system (fixedN),Zis a function ofTandV. Therefore

dSd

(

U

T

)

+kBdln(Z) (closed system) (27.2-6)

An indefinite integration leads to

S

U

T

+kBln(Z)+S 0 (27.2-7)

whereS 0 is a constant of integration that we can set equal to zero whenever it is

convenient. An alternative derivation of Eq. (27.2-7) is found in Problem 27.6.

We can now write formulas for the other thermodynamic functions of a general

system in terms of the canonical partition function.

AU−TS−kBTln(Z) (27.2-8a)

CV

(

∂U

∂T

)

V,N

kBT^2

(

∂^2 ln(Z)

∂T^2

)

V,N

+ 2 kBT

(

∂ln(Z)

∂T

)

V,N

(27.2-8b)

GA+PVkBTln(Z)+VkBT

(

∂ln(Z)

∂V

)

T,N

(27.2-8c)

μ

(

∂A

∂N

)

T,V

−kBTln

(

ZN

ZN− 1

)

(27.2-8d)