Physical Chemistry Third Edition

918 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

Solution

a.

3 kBT

2



3

2

(1. 3807 × 10 −^23 JK−^1 )(298 K) 6. 17 × 10 −^21 J

h^2

8 Ma^2

(

3 n^2 x

)

nx

[

8(1. 674 × 10 −^27 kg)(0.100 m)^2 (6. 17 × 10 −^21 J)

3

(

6. 6261 × 10 −^34 Js

) 2

] 1 / 2

[6. 27 × 1017 ]^1 /^2  7. 92 × 108

b. ∆E

h^2

8 Ma^2

[

(nx+ 1 )^2 −n^2 x

]



(6. 6261 × 10 −^34 Js)^2

8

(

1. 674 × 10 −^27 kg

)

( 0 .100 m)^2

( 2 nx+ 1 )

(3. 278 × 10 −^39 J)(2(7. 92 × 108 )+1)

 5. 19 × 10 −^30 J 3. 24 × 10 −^11 eV

Ratio of∆EtoE

5. 19 × 10 −^30 J

6. 17 × 10 −^21 J

 8. 41 × 10 −^10

Ratio of∆Eto the electronic energy

3. 24 × 10 −^11 eV

10 .2eV

 3. 18 × 10 −^12

PROBLEMS

Section 22.1: The Translational States of Atoms

22.1 Repeat the calculation of Example 22.1 for a box 1.00 m
on a side. Compare your answer with that of Example 22.1
and comment on any qualitative difference.

22.2 Find the values of the three translational quantum numbers
(assumed equal) of a xenon atom in a cubical box 0.200 m
on a side if the translational energy is equal to 8.315 eV,
the excitation energy to the first excited electronic
level.

22.3 If a gaseous CH 4 molecule has a translational energy equal
to 3kBT/2 at 298.15 K, what are the values of the three

translational quantum numbers if the molecule is contained

in a cubical box 0.100 m on a side? Assume the three

quantum numbers to be equal to each other.

22.4 Find the values of the three translational quantum numbers

(assumed equal) of a xenon atom in a cubical box 20.00 m

on a side if the translational energy is equal to 8.315 eV,

the excitation energy to the first excited electronic level.

22.5 a.For a xenon atom in a cubical box with side 0.100 m,

find the values of the translational quantum numbers

(assumed equal) if the energy is equal to 3kBT/2at

298 K. Compare your values with those of

Example 22.2.