Physical Chemistry Third Edition

23.7 Raman Spectroscopy 989

1 2

2 1

2

1 2

1

2 1

2 2

2 2

1 1

YY

YYAg

n 1

B1g

n 5

B1g

n 6

B1u

n 7

B2g

n 8

B3u

n 12

B3u

n 11

B2u

n 10

B2u

n 9

Ag

n 2

Ag

n 3

Au

n 4

Figure 23.18 The Vibrational Normal Modes of Ethylene, C 2 H 4 .The arrows show the

direction of motion of each nuclei in one-half of the period. The length of each arrow is

proportional to the amplitude of motion of the nucleus. From G. Herzberg,MolecularSpectra

andMolecularStructure,Vol. II,InfraredandRamanSpectra of Polyatomic Molecules,Van

Nostrand Reinhold, New York, 1945, p. 107.

A linear polyatomic molecule has two equal moments of inertia,IBandIC. In the

rigid rotor approximation, the energy levels are given by Eq. (22.4-7)

EJ

h ̄^2

2 IB

J(J+1) (23.7-9)

Equation (23.7-8) can be used for a linear polyatomic molecule if ̃Beis replaced by

̃B h

8 π^2 IBc

(23.7-10)

EXAMPLE23.14

Figure 23.19 shows the rotational Raman spectrum of carbon dioxide. From the splitting

between the lines, 3.09 cm−^1 , calculate the equilibrium bond lengths.

Solution

Since the carbon nucleus at the center of mass and since the two bond lengths are equal to

each other the two equal moments of inertia are

IBIC 2 mOre^2

wherereis the bond length andmOis the oxygen nuclear mass, 2. 656 × 10 −^26 kg. The

parameter ̃Bis given by

̃B h

8 π^2 IBc



h

8 π^22 mOr^2 ec