Physical Chemistry Third Edition

25.2 The Probability Distribution for a Dilute Gas 1047

25.4 Consider a system of four distinguishable rigid rotating
diatomic molecules with a total energy of 20hB, whereB
is the rotational constant.
a.Make a list of the possible system states analogous to
that of Table 25.1. Don’t forget the degeneracies.
b.Find the average distribution of molecular levels,
analogous to that of Eq. (25.1-7).
c.Find the most probable distribution.

25.5 The standard deviation of a statistical sample ofN

members,w 1 ,w 2 ,...,wN, is defined by

s

[

1

N− 1

∑N

i 1

(wi−〈w〉)^2

] 1 / 2

Calculate the standard deviation of the energy of an

oscillator in our model system.

25.2 The Probability Distribution for a Dilute Gas

We now want to consider macroscopic systems of many molecules. We focus on the

simplest macroscopic system, a dilute gas of a single substance. We assume that the

macroscopic state of our dilute gas is specified by the internal energyU, the volumeV,

and the amount of the single substancen. We consider only system microstates cor-

responding to the energy eigenvalueEequal toUand corresponding to the correct

volume and the correct number of molecules.

In a dilute gas the molecules are on the average far enough from each other that

the intermolecular forces are negligible, as can be seen in Problem 25.9. In this case

the system Hamiltonian operator can be written as a sum of Hamiltonian operators for

independent molecules:

Ĥsys

∑N

i 1

Ĥ(i) (25.2-1)

whereĤ(i) is the molecular Hamiltonian operator for molecule numberi, and where

Nis the number of molecules in the system. The label (i) stands for the coordinates

of all of the nuclei and electrons in molecular numberi. There are no terms in which

the coordinates of particles in two or more molecules occur because the intermolecular

forces have been neglected. In addition to the dilute gas, there are crystal models due

to Einstein and to Debye in which Eq. (25.2-1) applies. We discuss these models in a

later chapter.

The system Hamiltonian of Eq. (25.2-1) corresponds to a Schrödinger equation that

can be solved by the separation of variables, a method that we have used a number of

times. It gives energy eigenvalues that are a sum of molecular energy eigenvalues

Ek

∑N

i 1

εki (25.2-2)

and energy eigenfunctions that are products of molecular energy eigenfunctions

Ψk

∏N

i 1

ψki(i) (25.2-3)

where the subscriptkiis an abbreviation for the values of the quantum numbers that

are needed to specify the molecular state of molecule numberi, given that the system

state isΨk. We assume that the molecule energy eigenvaluesεkiand molecule wave

functionsψkiare known to a usable approximation.