Concise Physical Chemistry

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

THE QUANTUM RIGID ROTOR 291

in a box (Chapter 16) except that the quantum numbernis replaced byJandmtimes

the length of the constraining geometryl^2 is replaced by the moment of inertia of the

rotating massI=mr^2

E=

J^2 h ̄^2

2 I

Because the mass can rotate in either direction (or stand still atJ=0), the allowed

quantum numbers areJ= 0 ,± 1 ,± 2 ,....

When diatomic molecules rotating about their center of mass (balance point) are

considered, the reduced mass of the diatomic molecule is used in the same way as it

was for the vibrating two mass problem. Diatomic molecules are not constrained to

move in a fixed plane, rather, their plane of rotation can tilt at angles from 0 toπ.

This added degree of freedom changes the energy level spacing to

E=J(J+1)

h ̄^2

2 I

The energy levels diverge with increase in the quantum numberJ= 0 ,± 1 ,± 2 ,...,

giving the valuesJ(J+1)= 0 , 2 , 6 , 12 ,....The problem is again reduced to one

of a fictitious mass, this time with a moment of inertiaI=μr^2 , rotating anywhere

on the surface of a sphere. For absorption of aresonance frequencyto occur, the

rotational state must be changed by an increase in energy from one level to the next

J→J+1 with

EJ+ 1 −EJ= 2 , 4 , 6 , 8 ,...joules

The spacing between resonance frequencies is 4− 2 =2, 6− 4 =2, 8− 6 =2,...

joules, by which we predict an absorption spectrum consisting of a series of lines of

different frequencies separated by an energy of 2( ̄h^2 / 2 I) (Fig. 18.3). Measuring ̄ν

leads toI, which gives the bond lengthrthroughI=μr^2.

Vibrational and rotational spectroscopic transitions can occur simultaneously,

leading to increased complexity of the experimental spectrum. Conversely, some

2

4

6

E

FIGURE 18.3 Energy levels within a simple rotor.