Physical Chemistry Third Edition

928 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

The terms are commonly given in reciprocal centimeters, sometimes calledwave

numbers. When you use these values, you must remember to convert to reciprocal

meters or to use the speed of light in centimeters per second. The parameters in

this equation are marked with a tilde (∼) to distinguish them from the parameters

in Eq. (22.2-45). These parameters are equal to the other parameters divided byc, the

speed of light. Table A.22 in Appendix A gives the values of parameters in cm−^1 for

several diatomic molecules. Some reference books use the same letters without a tilde

to represent the parameters in Eq. (22.2-50) and the Greek letterωis sometimes used

instead of ̃ν.

EXAMPLE22.6

a.Using the expression of Eq. (22.2-50), find the wavelength and frequency of the light

absorbed when carbon monoxide molecules make the transition from theν0,J 0

state to theν1,J1 state.

b.Find the wavelength for the same transition, neglecting the terms in ̃αandD ̃.

c.Find the wavelength for the same transition, neglecting the terms in ̃νexe, ̃α, andD ̃.

Solution

a.From Table A.22,

̃νe 2169 .8cm−^1

̃Be 1 .931 cm−^1

̃De 0. 0612 × 10 −^4 cm−^1

̃αe 0 .0175 cm−^1

̃νexe 13 .29 cm−^1

EνJ

hc

 ̃νe

(

ν+

1

2

)

+ ̃νexe

(

ν+

1

2

) 2

+ ̃BeJ(J+ 1 )− ̃DeJ^2 (J+ 1 )^2

− ̃αe

(

ν+

1

2

)

J(J+ 1 )

1

λ



∆E

hc



E 11

hc

−

E 00

hc

 ̃νe− 2 ̃νexe+ 2 ̃Be− 4 D ̃e− 3 ̃αe

 2169 .8cm−^1 −2(13.29 cm−^1 )+2(1.931 cm−^1 )−4(0. 0612 × 10 −^4 cm−^1 )

−3(0.0175 cm−^1 )

 2147 .0cm−^1

λ 4. 6576 × 10 −^4 cm 4. 6576 μm

ν

c

λ



2. 9979 × 1010 cm s−^1

4. 6576 × 10 −^4 cm

 6. 4365 × 1013 s−^1

b.^1

λ



∆E

hc



E 11

hc

−

E 00

hc

 ̃νe− 2 ̃νexe+ 2 ̃Be

 2169 .8cm−^1 −2(13.29 cm−^1 )+2(1.931 cm−^1 ) 2147 .1cm−^1 (22.2-51)

λ 4. 6575 × 10 −^4 cm 4. 6575 μm

ν

c

λ



2. 9979 × 1010 cm s−^1

4. 6576 × 10 −^4 cm

 6. 4365 × 1013 s−^1