Physical Chemistry Third Edition

22.2 The Nonelectronic States of Diatomic Molecules 925

The electronic energyEelfor the electronic ground state is usually assigned the

value zero, which makesEelequal to a different positive constant for each excited

level.

EXAMPLE22.4

For a carbon monoxide molecule in a cubical box 0.100 m on a side, compare the spacing

between the ground state and the first excited level for translation, rotation, vibration, and

electronic motion. The equilibrium bond length is 1. 128 × 10 −^10 m, and the vibrational

frequency is 6. 505 × 1013 s−^1 for the ground electronic state. The excitation energy to the

first excited electronic state is 6.036 eV.

Solution

We assume the most common isotopes,^12 C and^16 O, for whichM 4. 469 × 10 −^26 kg and

μ 1. 138 × 10 −^26 kg. For translation,

E 211 −E 111 

h^2

8 Ma^2

(6−3)

(6. 6261 × 10 −^34 J s)(3)

8(4. 469 × 10 −^26 kg)(0.100 m)^2

 3. 68 × 10 −^40 J 2. 30 × 10 −^21 eV

For rotation,

E 1 −E 0  ̄

h^2

2 Ie

(2−0)

h^2

8 π^2 μr^2 e

(2−0)



(

6. 6261 × 10 −^34 Js

) 2

( 2 )

8 π^2

(

1. 138 × 10 −^26 kg

)(

1. 128 × 10 −^10 m

) 2

 7. 68 × 10 −^23 J 4. 79 × 10 −^4 eV

For vibration,

E 1 −E 0 hνe

(

3

2

−

1

2

)

hνe

(6. 6261 × 10 −^34 J s)(6. 505 × 1013 s−^1 ) 4. 310 × 10 −^20 J 0 .269 eV

The rotational level spacing is larger than the translational level spacing by a factor of 10^17 ,

the vibrational energy level spacing is larger than the rotational level spacing by a factor of

about 600, and the electronic level spacing is larger than the vibrational level spacing by a

factor of about 20.

The results of Example 22.4 are typical. The translational levels are much closer

together than the rotational levels, which are quite a bit closer together than the vibra-

tional levels, which are in turn quite a bit closer together than the electronic levels,

which are typically spaced by several electron volts.

The energy level expression of Eq. (22.2-29) is only a first approximation. The

power series expression for the vibrational potential energy functionVwas truncated at

the quadratic term and the internuclear distance was replaced by its equilibrium value.

One additional term of the power series can be kept or an alternate representation of the