Value of x/(e x - 1). 
109 
1919-20.] Molecular Energy in Gases. 
various modes of vibration take up a more and more important part of the 
total energy as the temperature rises. And measurements of the radiation 
show that in this progressive development of E'", vibrations of long period 
begin to contribute substantially while the gas is still comparatively cold, 
whereas those of shorter period begin to contribute only when the gas is 
more strongly heated. 
How does this progression occur ? If we accept the Quantum Theory, 
the answer is at once given by Planck’s formula 
E„'" = — — RT, 
e* - 1 
where E/" is that part of the average energy of vibration, per molecule, 
which is contributed by a mode of vibration whose frequency is v, and x is 
TEMPERATURE 
hvfRT. In a gas whose molecules are capable of more than one mode of 
vibration the whole vibrational energy E'" would be the sum of as many 
terms, in the above form, as there are modes. For any given frequency the 
factor x/(e x — 1) tends towards the value 1 as T is indefinitely increased, but 
is insignificantly small when T is low. Plotted in relation to T it gives 
a curve, shown in the figure, which closely resembles the typical curve 
of magnetisation of a ferromagnetic substance under a steadily increasing 
magnetic force, with its very gradual beginning, its subsequent rapid rise, 
and its final asymptotic approach to a limiting value. 
So little is known as to the mechanism of vibration in the molecules 
of a gas that one may be less disinclined to apply the Quantum Theory 
to the vibrations of the molecules than to their rotations, especially because 
