Physical Chemistry Third Edition

25.3 The Probability Distribution and the Molecular Partition Function 1061



4(0.02018 kg mol−^1 )(1.00 m)^2 (1. 3807 × 10 −^23 JK−^1 )(300 K)

(6. 022 × 1023 mol−^1 )(6. 6261 × 10 −^34 Js)^2

 1. 2 × 1021

nx 3. 5 × 1010

b.

ε

3

2

kBT(1.5)(1. 3807 × 10 −^23 JK−^1 )(298.15 K) 6. 175 × 10 −^21 J

∆ε

∂ε

∂nx

∆nx

∂

∂x

(

h^2

8 ma^2

(3n^2 x)

)

∆nx

h^2

8 ma^2

6 nx∆nx

2 ε

nx

∆nx

2 ε

nx



3 kBT

nx



3(1. 3807 × 10 −^23 JK−^1 )(300 K)

3. 5 × 1010

 3. 6 × 10 −^31 J

∆εis smaller thanεby a factor of roughly 10−^10.

Exercise 25.14

Replaceβby 1/kBTin Eq. (25.3-15). Find the value of the term in this sum corresponding to

nx 3 .51010. Find the difference between this term and the term corresponding tonx+1 and

show that this difference is small compared with the value of the term.

Since the typical difference between one term in the sum and the next term is very small,

the integral approximation to the translational partition function is quite accurate (to

8 or 9 significant digits).

In the derivation of the most probable distribution there was no restriction to a

particular dilute gas. We showed thatβ 1 /(kBT) for a dilute monatomic gas with

negligible electronic excitation, but this relation must be valid for all dilute gases. With

this identification the probability of a molecular energy level of any dilute gas is

pj

Nj

N



1

z

gje−εj/kBT

(

dilute gas

energy levels

)

(25.3-22)

and the probability distribution for molecular states is

pi

Ni

N



1

z

e−εi/kBT

(

dilute gas

states

)

(25.3-23)

The molecular partition function is given by

z

∑

j

gje−εj/kBT (sum over levels) (25.3-24)

or

z

∑

i

e−εi/kBT (sum over states) (25.3-25)