Many gases contain molecules of several atoms tightly bound to each other.
Clearly our expression for the mean squared translational velocity must
work for such gases as well
because it relies only on the gas being ideal. The internal
energy however now contains contributions not only from translational kinetic
energy but also from rotational and vibrational kinetic energy.
There is an important theorem of statistical physics called the equipartition
theorem which states that in thermal equilibrium each
microscopic degree of freedom has
an amount 
of thermal energy associated with it.
Denoting by s the number of degrees of freedom, this means that
Correspondingly the constant volume specific heat reads:
Thus the magnitude of the specific heat is governed by the total amount of
microscopic degrees of freedom, Ns, of the material in question. Examples of microscopic
degrees of freedom are
, and 
.