bei48482_FM

284 Chapter Eight

and its moment of inertia Iis

Im R^2 (1.14 10 ^26 kg)(1.13 10 ^10 m)^2

1.46 10 ^46 kgm^2

The lowest rotational energy level corresponds to J1, and for this level in CO

EJ 1 

7.61 10 ^23 J4.76 10 ^4 eV

This is not a lot of energy, and at room temperature, when kT 2.6  10 ^2 eV, nearly all the

molecules in a sample of CO are in excited rotational states.

(b) The angular velocity of the CO molecule when J1 is

 

3.23 1011 rad/s

Rotational Spectra

Rotational spectra arise from transitions between rotational energy states. Only mole-

cules that have electric dipole moments can absorb or emit electromagnetic photons

in such transitions. For this reason nonpolar diatomic molecules such as H 2 and sym-

metric polyatomic molecules such as CO 2 (O“C“O) and CH 4 (Fig. 8.13) do not

exhibit rotational spectra. Transitions between rotational states in molecules like H 2 ,

CO 2 , and CH 4 can take place during collisions, however.

Even in molecules with permanent dipole moments, not all transitions between ro-

tational states involve radiation. As in the case of atomic spectra, certain selection rules

summarize the conditions for a radiative transition between rotational states to be pos-

sible. For a rigid diatomic molecule the selection rule for rotational transitions is

Selection rule J 1 (8.10)

In practice, rotational spectra are always obtained in absorption, so that each tran-

sition that is found involves a change from some initial state of quantum number Jto

the next higher state of quantum number J1. In the case of a rigid molecule, the

frequency of the absorbed photon is

J→J 1 

Rotational spectra  (J1) (8.11)

where Iis the moment of inertia for end-over-end rotations. The spectrum of a rigid

molecule therefore consists of equally spaced lines, as in Fig. 8.17. The frequency

of each line can be measured, and the transition it corresponds to can often be

found from the sequence of lines. From these data the moment of inertia of the

molecule can be calculated. Alternatively, the frequencies of any two successive lines

may be used to determine Iif the lowest-frequency lines in a particular spectral

sequence are not recorded.



2 I

EJ 1 EJ



h

E



h

(2)(7.61 10 ^23 J)



1 .46 10 ^46 kgm^2

2 E



I

(1.054 10 ^34 Js)^2



1.46 10 ^46 kgm^2

2



I

J(J1) 2



2 I

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