Physical Chemistry Third Edition

25.1 The Quantum Statistical Mechanics of a Simple Model System 1045

Table 25.2 Average, Most Probable, and Boltzmann Proba-

bility Distributions for the Vibrational States of Four Harmonic

Oscillators

Value ofv pv(Average) pv(Most Probable) pv(Boltzmann)

0 0.42857 0.3750 0.50000

1 0.28571 0.3750 0.25000

2 0.17143 0.1250 0.12500

3 0.08571 0.1250 0.06250

4 0.02857 0 0.03125

5 0 0 0.01562

6 0 0 0.00781

0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

24

v

68

p(average)

p(most probable)

Probability

p(Boltzmann)

Figure 25.1 The Average Distribution and the Most Probable Distribution for the

Vibrational Energy of Four Harmonic Oscillators.

EXAMPLE25.1

Obtain the Boltzmann probability distribution for our model system that corresponds to an

average energy per oscillator ofhν.

Solution

The Boltzmann distribution of Eq. (22.5-1) is not normalized. The normalized Boltzmann

probability distribution is given by

(Probability of a statei)pi

1

z

e−εi/kBT (25.1-10)

where the divisorzis chosen to give normalization:

z

∑∞

i 0

e−εi/kBT (25.1-11)

Note that the Boltzmann probability distribution is not restricted to the states that we have

listed, so that the upper limit of the sum is infinite. The formula forzis the sum of a geometric

progression, a well-known sum:

∑∞

v 0

av

1

1 −a

(25.1-12)