Physical Chemistry Third Edition

1054 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

Equations (25.2-24) and (25.2-25) are for a gas with dilute occupation, in which the

distinction between fermions and bosons is unimportant. If Eq. (25.2-10) or (25.2-12)

is used in the expression forWinstead of the expression in Eq. (25.2-11), the results

for noninteracting fermions and bosons are

Nj

gjeαe−βεj

1 +eαe−βεj

(fermions) (25.2-26)

Nj

gjeαe−βεj

1 −eαe−βεj

(bosons) (25.2-27)

Exercise 25.11

Show that Eqs. (25.2-26) and (25.2-27) are correct.

For sufficiently large energies, the second terms in the denominators of both the expres-

sions in Eqs. (25.2-26) and (25.2-27) become negligible, and both expressions approach

Njgjeαe−βεj (25.2-28)

which is the same as Eq. (25.2-25).

PROBLEMS

Section 25.2: The Probability Distribution for a Dilute Gas

25.6 It would be impossible to write down the antisymmetrized
wave function of a system containing more than a few
fermions. Estimate the number of terms in an
antisymmetrized wave function for a system containing a
number of fermions equal to Avogadro’s constant, using
Stirling’s approximation.

25.7 The number of distinct ways to choose states forNj
fermions from a level with degeneracygjis given by
Eq. (25.2-10).
a.Find the percent differences between the result of this
formula and the result of Eq. (25.2-11) forgj 1000
andNj5.
b.Repeat the calculation forgj1000000 andNj5.

25.8 The number of distinct ways to choose states forNjbosons
from a level with degeneracygjis given by Eq. (25.2-12).
a.Find the percent differences between the result of this
formula and the result of Eq. (25.2-11) forgj 1000
andNj5.
b.Repeat the calculation forgj1000000 andNj5.

25.9 In order for the dilute-gas approximation to hold, the
potential energy of interaction of the molecules must be

negligible. The potential energy of two molecules can be

represented by the Lennard–Jones formula

uLJ(r) 4 ε

[(

σ

r

) 12

−

(

σ

r

) 6 ]

(25.2-29)

whereris the distance between the centers of the

molecules and whereεandσare parameters that have

different values for each gas. For helium,σ 2. 56 ×

10 −^10 m andε 1. 41 × 10 −^22 J. Estimate the potential

energy of 1.00 mol of argon gas at 1.00 atm and 298.15 K

as follows: Calculate the volume per molecule, and assume

that each molecule occupies a cubical volume. Estimate

the average nearest-neighbor distance as the distance from

the center of one cube to the center of the next cube.

Evaluate the potential energy of a pair of nearest-neighbor

molecules. Assume that each molecule is surrounded by

twelve nearest-neighbor molecules and neglect interactions

with more distant molecules. Remember that each

intermolecular potential energy is shared by two

molecules. Compare the potential energy with the kinetic

energy, which is given by gas kinetic theory as 3nRT /2,

wherenis the amount of gas in moles.

25.10In order for the dilute-gas approximation to hold, the

potential energy of interaction of the molecules must be