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The natural vibration frequency of a hydrogen molecule is 1.26xx10^(14) Hz. Find the zero-point energy of the vibrations of the molecule. Can the vibrational degrees of freedom in the molecule be excited at 600 K?

Answer»


Solution :The zero-point ENERGY is `epsi_(0)=1//2hv`. The energy of the first excited state is `epsi_(1)=3/2hv`, so the excitation energy is `Deltaepsi=epsi_(1)-epsi_(0)=hv`. The vibrational degrees of freedom will no longer be excited when the energy of thermal motion becomes less than the excitation energy. The usual CRITERION is `3/2kTlehv`, which yields the minimum temperature
`T_("vib")=2hv//3k=4xx10^(3)K`
This does not agree with experiment, since the lines of the vibrational spectrum of hydrogen molecules are observed at lower temperatures.
This is because of the Maxwellian molecular speed distribution (see 25.2), which shows that a gas contains molecules whose speeds are far in EXCESS of the average. For instance, about 2% of the molecules have speeds three TIMES greater than the average. The energy of such molecules is more than 9 times the average kinetic energy.


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