Physical Chemistry Third Edition

27.1 The Canonical Ensemble 1127

where we now denote a molecule state by the single indexk. This molecular partition

function is the same as the molecular partition function defined in Chapter 25.

For a monatomic gas without electronic excitation Eq. (25.3-18) gives the formula

for the molecular partition function:

zztr

(

2 πm

h^2 β

) 3 / 2

V (27.1-29)

From Eq. (27.1-22), the ensemble average pressure of a monatomic dilute gas is

〈P〉

1

β

(

∂ln(Z)

∂V

)

β,N



N

β

(

∂ln(z)

∂V

)

β,N



N

β

dln(V)

dV



N

βV

(27.1-30)

We assume that our dilute gas obeys the ideal gas equation of state so that

〈P〉

nRT

V



NkBT

V

(27.1-31)

wherekBis Boltzmann’s constant, equal to 1. 3807 × 10 −^23 JK−^1 , and whereTis the

absolute temperature. This requires that

β

1

kBT

(27.1-32)

The molecular partition function of a dilute monatomic gas without electron excitation

becomes the same function as in Eq. (25.3-21):

zztr

(

2 πmkBT

h^2

) 3 / 2

V (dilute monatomic gas) (27.1-33)

The canonical probability distribution becomes

pi

1

Z

e−Ei/kBT (27.1-34)

and the canonical partition function becomes

Z

∑

i

e−Ei/kBT (27.1-35)

Although we have established only thatβ 1 /(kBT) for a dilute gas, we assert that

βcannot have a different meaning for different systems and that Eqs. (27.1-34) and

(27.1-35) are valid for any kind of a system.

Since the probability of any system microstate is proportional toe−Ei/kBT, the

canonical partition function is a measure of the total number of system microstates

effectively available to the system, relative to a probability of 1 for a state of zero

energy. The canonical partition function has a very large value for a macroscopic

system.

EXAMPLE27.1

Estimate the natural logarithm of the canonical partition function for 1.000 mol of helium gas

in a volume of 25.0 L at 298.15 K.