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From these considerations it is clear that for a
monotonic gas (like He) no vibrational or rotational energy terms exist, but,
like all gaseous molecules, energy may be contained in translational motion.
Thus the molar energy of a monoatomic gas is simply 3RT/2. The constant
volume heat capacity of a monotonic gas is therefore
For diatomic or polyatomic molecules,
Contributions from electronic states to the total
internal energy have been neglected under the assumption that at room
temperature electronic transitions out of the ground state are unlikely. On
the other hand, the population of excited vibrational states depends strongly
on temperature and thus the various vibrational modes can be at least
partially active. In general, if a vibration involves a heavy atom or
possesses a smaller force constant, then the normal mode will be more active
and make a greater individual contribution to the heat capacity. For example,
the frequencies of bending modes tend to be much lower than those of
stretching modes. Since in the case of most diatomics there are only
stretching modes, the vibrational contribution to CV will be
very small. Indeed, N2 would have its equipartition value for CV
only above about 4000 K. In contrast many polyatomic molecules, especially
those containing heavy atoms,will at room temperature have significant partial
vibrational contributions to the heat capacity.
At ordinary temperatures many of the excited state
rotational levels are thermally accessible and hence the rotational
contribution is in accord with the equipartition theorem of classical
mechanics.
Thus, it is possible to calculate definite values for
CV and hence, through the ideal gas equation of state, the
ratio  for
ideal monatomic and polyatomic gases using the above expressions. Statistical
thermodynamics provides even better results, for which the partially 'frozen'
vibrational contribution to the heat capacity may be evaluated exactly (you
guess
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