Concise Physical Chemistry

c18 JWBS043-Rogers September 13, 2010 11:29 Printer Name: Yet to Come

300 EXPERIMENTAL DETERMINATION OF MOLECULAR STRUCTURE

Low Resolution High Resolution

Frequency

FIGURE 18.10 Schematic diagram of a vibration–rotation band. The low-resolution vibra-

tional band contains many rotational lines that may be discernable at high resolution. The

separation between rotational lines is given the term 2B.

2. The force constant is extracted from the classical expression for simple har-
   monic motion:

ν=

1

2 π

√

kf

m

Given the experimental measurement of ̄ν=2142 cm−^1 = 2. 142 × 105 m−^1 ,

the frequency is

ν=cν ̄=

(

2. 998 × 108 ms−^1

)

,

(

2. 142 × 105 m−^1

)

= 6. 422 × 1013 s−^1

From the harmonic oscillator approximation

ν=

1

2 π

√

kf

m

we get

kf

m

= 4 π^2 ν^2 = 1. 628 × 1029

Substituting the reduced mass of^12 C^16 Oform, we obtain

kf=μ×

(

4 π^2 ν^20

)

=

(

1. 138 × 10 −^26

)

×

(

1. 628 × 1029

)

=1853 N m−^1

3. The bond length is found from the moment of inertia of the molecule modeled
   as a rigid rotor. The energy spacing is

2 B=

2 ̄h^2

2 I

=

h^2

4 π^2 I