Concise Physical Chemistry

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

290 EXPERIMENTAL DETERMINATION OF MOLECULAR STRUCTURE

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FIGURE 18.2 Parabolic potential wells for the harmonic oscillator. The narrow well has the

larger force constant. Only one set of quantum levels is shown.

you start, the absorption spectrum for the harmonic oscillator consists of only one

line with frequencyν=(E/h)s−^1 becauseE=hν.

The simple harmonic oscillator is a fairly good model for a hydrogen atom sus-

pended from a heavy carbon atom framework (Fig. 18.1). From C–H resonance

frequencies of about 2900–3000 cm−^1 , one finds the force constant to be about

500 N m−^1. Typical experimental values for molecular force constants vary consid-

erably, from about 100 N m−^1 to about 800 N m−^1.

The reader with practical knowledge of infrared spectra will find a discrepancy

between the complicated structure of a real IR band spectrum and the single line

predicted by Hooke’s law. This is the result of many factors, including the failure

of Hooke’s law, energy coupling among chemical bonds, and many other motions

(bending, torsion, etc.) that are possible in a real polyatomic molecule. Nevertheless,

the presence of a strong peak near 2900–3000 cm−^1 is a good indicator of a C–H

stretch lurking somewhere in the molecule. Other characteristic frequencies are used

in qualitative IR “fingerprint” analysis.

18.3 DIATOMIC MOLECULES

For diatomic molecules attached by a chemical bond, the picture is very similar to

the harmonic oscillator of one mass. The atoms vibrate harmonically relative to one

another with a natural frequency determined by their mass and the strength of the

electronic spring connecting them. One replaces the mass of the simple harmonic

oscillator with thereduced massof the diatomic moleculeμ=m 1 m 2 /m 1 +m 2 ,

wherem 1 andm 2 are the atomic masses, and proceeds with the calculation. The

problem has indeed beenreducedfrom one of two masses vibrating relative to one

another to one of a single fictitious massμvibrating relative to a central point.

18.4 THE QUANTUM RIGID ROTOR

A small mass on a circular orbit in a fixed plane shows quantum phenomena. The

energy level spacing shows a pattern that is similar to the energy levels of the particle