Physical Chemistry Third Edition

1140 27 Equilibrium Statistical Mechanics. III. Ensembles

b.At 298.15 K

kBT

hν



(1. 3807 × 10 −^23 JK−^1 )(298.15 K)

(6. 6261 × 10 −^34 J s)(1. 6780 × 1013 s−^1 )

 0. 3702

At 1000.0 K

kBT

hν



(1. 3807 × 10 −^23 JK−^1 )(1000.0K)

(6. 6261 × 10 −^34 J s)(1. 6780 × 1013 s−^1 )

 1. 2418

At 298.15 K,

zvib

kBT /hν

 2 .896. At 1000.0 K,

zvib

kBT /hν

 1 .4561.

Exercise 27.4

Repeat the calculation of the previous example for Br 2.

The classical vibrational partition function divided by Planck’s constant will almost

never be an adequate approximation to the quantum vibrational partition function. The

electronic partition function also cannot be related to a classical version. We define a

composite partition functions as follows for a monatomic substance:

z

ztr,cl

h^3

zqm,el (dilute monatomic gas) (27.4-29)

For a diatomic or linear polyatomic dilute gas we define the composite partition

function:

z

ztr,cl

h^3

zrot,cl

σh^2

zvib,qmzel,qm

(

dilute diatomic or

linear polyatomic gas

)

(27.4-30)

For a nonlinear polyatomic dilute gas we define the composite partition function

z

ztr,cl

h^3

zrot,cl

σh^3

zvib,qmzel,qm

(

dilute nonlinear polyatomic gas

)

(27.4-31)

The composite partition functions are identical with the quantum versions of

Chapter 25. This illustrates the fact that classical statistical mechanics does not provide

any advantage in treating a dilute gas.

PROBLEMS

Section 27.4: Classical Statistical Mechanics

27.14
a.Sketch a diagram in a two-dimensional phase space
for a harmonic oscillator that is analogous to
Figure 27.3 for a particle in a hard box. Show the
trajectories for energies equal to thev0,v1,
andv2 quantum states.
b.Calculate the area between any two such trajectories.

27.15Construct a representation of the trajectory in a

2-dimensional phase space for an object of mass 1.000 kg

falling vertically in a vacuum near the surface of the

earth.

27.16a.A phase space can represent a coordinate and its

conjugate momentum even if the coordinate is not a

Cartesian coordinate. Refer to Appendix E and see

what the conjugate momentum is for the angleφin