Physical Chemistry Third Edition

1138 27 Equilibrium Statistical Mechanics. III. Ensembles

wherehis Planck’s constant. For a dilute monatomic gas without electronic excitation,

the quantum canonical partition function is related to the classical function as follows:

Zqm

zNqm

N!



(

2 πmkBT

h^2

) 3 N/ 2

VN

N!



Zcl

h^3 NN!



1

N!

(z

cl

h^3

)N

(27.4-21)

The divisorN! occurs because of the indistinguishability of the particles, which

is not recognized in classical mechanics. The divisors of Planck’s constant indicate a

relationship between a volume in phase space and a quantum state. We illustrate this

relationship in the two-dimensional phase space of a particle in a one-dimensional box.

Figure 27.3 shows several trajectories in this phase space. The first trajectory is for the

motion of a classical particle that happens to have an energy equal toε 1 , the energy

eigenvalue of a quantum mechanical particle of the same mass in the same box. The

trajectory has two parts, one for motion from left to right, and one for the motion from

right to left. The second trajectory is for the motion of a classical particle with energy

equal toε 2 , and so on.

n 55

n 51

n 52

n 53

n 54

n 51

n 55

n 54

n 53

n 52

0

px

a x

Figure 27.3 Several Trajectories in
a Two-Dimensional Phase Space
for a Particle in a Hard One-
Dimensional Box.

We now show that the area between any two adjacent trajectories is equal to

Planck’s constant. Forn′, a given value of the quantum number, the energy is

εcl

p^2

2 m

εqmεn′

h^2 n′^2

8 ma^2

(27.4-22)

Solving for|p|, we obtain

|p|

hn′

2 a

(27.4-23)

The area between the trajectory forn′and that forn′+1is

Area

(

h(n′+1)

2 a

−

hn′

2 a

)

(2a)h (27.4-24)

We multiply by 2ainstead of byabecause the trajectory has two parts, one for positivep

and one for negativep. The relation of Eq. (27.4-24) is an example of a general relation.

For a 2-dimensional phase space, an area equal tohcorresponds to one quantum state.

For a 6-dimensional phase space, a volume equal toh^3 corresponds to one quantum

state and for a 6N-dimensional phase space a volume equal toh^3 Ncorresponds to one

quantum state.

The quantum rotational partition function for a diatomic or linear polyatomic mole-

cule is equal to the classical version divided byh^2 and also divided by the symmetry

numberσ, equal to 1 for a heteronuclear diatomic molecule and equal to 2 for a

homonuclear diatomic molecule.

zrot,qm

8 π^2 IekBT

σh



zrot,cl

σh^2

(

dilute diatomic or

linear polyatomic gas

)

(27.4-25)

The division by the symmetry number corrects for the indistinguishability of the

nuclei of a homonuclear molecule, which is not recognized in classical mechan-

ics. The divisor ofh^2 comes from the relationship of a quantum state with a vol-

ume in the 4-dimensional phase space ofθ,φ, and their conjugate momenta, similar

to the relationship between a quantum state and an area ofhin a 2-dimensional

phase space.