Physical Chemistry Third Edition

1060 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

EXAMPLE25.3

Find the value ofztrfor argon atoms confined in a box of volume 25.0 L (0.0250 m^3 )at

298.15 K.

Solution

m

0 .039948 kg mol−^1

6. 02214 × 1023 mol−^1

 6. 63 × 10 −^26 kg

ztr

(

(2π)(6. 63 × 10 −^26 kg)(1. 3807 × 10 −^23 JK−^1 )(298.15 K)

(6. 6261 × 10 −^34 Js)^2

) 3 / 2

(0.0250 m^3 )

 6. 11 × 1030

The large magnitude of the translational partition function in Example 25.3 is typical.

There are a great many translational states that are effectively accessible to an atom

or molecule at room temperature. The probability of any one state is very small. A

state of zero energy would have a probability of 1/(6. 11 × 1030 ) 1. 64 × 10 −^31.

The probability of a state of higher energy has an even smaller value.

Exercise 25.13

a.Find the value ofztrfor helium atoms confined in a box of volume 25.0 L at 298.15 K.

Compare this value with that of argon in Example 25.3 and comment on the difference.

b.Repeat the calculation for xenon atoms.

We can now assess the accuracy of replacing the sum given by Eq. (25.3-16) by the

integral shown in Eq. (25.3-17).

EXAMPLE25.4

a.Assuming thatnx,ny, andnzare equal to each other and that our gas is confined in a

cubical box 1.00 m on a side, find the value ofnxthat will make the energy of a neon

atom equal to 3kBT/2 at 300 K.

b.Find the change in energy ifnxis replaced bynx+1 and show that this change is small

compared with 3kBT/2 at 300 K.

Solution

a.

3

2

kBT

h^2

8 ma^2

(3n^2 x)

n^2 x

4 ma^2 kBT

h^2