Physical Chemistry Third Edition

1046 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

where we must require thata<1. Sincehν/kBT>0,e−hν/kBT<1 and our sum is:

z

∑∞

v 0

e−vhν/kBT

∑∞

v 0

(

e−hν/kBT

)v



1

1 −e−hν/kBT

(25.1-13)

so that the normalized Boltzmann probability distribution is

pv(1−e−hν/kBT)e−vhν/kBT (25.1-14)

We will show later in the chapter that the formula for the mean energy is given by

〈ε〉

hν

ehv/kBT−^1

(25.1-15)

In order to have〈ε〉equal tohνswe must have a value of the temperature given by

ehν/kBT− 1 1orkhνBTln(2) (25.1-16)

which gives the normalized Boltzmann distribution for our model system of four harmonic

oscillators:

pv(Boltzmann)

1

2

e−vln(2)

1

2

(

e−ln(2)

)v



1

2

2 −v (25.1-17)

Exercise 25.6

Sum the values for the Boltzmann distribution in Table 25.2 forv0tov6 and see how

close the result is to 1.000. The error in a partial sum of a geometric progression is equal to the

magnitude of the last term included. Does your sum conform to this?

PROBLEMS

Section 25.1: The Quantum Statistical Mechanics

of a Simple Model System

25.1 a. Find the total number of system vibrational states for a
system of three harmonic oscillators with a total energy
of 3hν.
b.Find the average molecular probability distribution for
the vibrational states of the system of part a, as was
done in Section 25.1.
c.Find the most probable probability distribution for the
vibrational states of the system of part a, as was done in
Section 25.1.

25.2 a.List the system states for a system of four harmonic
oscillators with a total energy of 3hν.

b.Find the average molecular probability distribution for

the vibrational states of the system of part a.

c.Find the most probable probability distribution for the

vibrational states of the system of part a.

25.3 a.Find the total number of system vibrational states for a

system of two harmonic oscillators with a total energy

of 4hν.

b.Find the average molecular probability distribution for

the vibrational states of the system of part a.

c.Find the most probable probability distribution for the

vibrational states of the system of part a.