The kinetic energy of the gas particles is also referred to as the internal energy \(U\) of the gas. The numerator in the fraction in energy equation of the previous section is generally derived from the number \(f\) of degrees of freedom of the particles in the gas.
Since in the case of pure translation we only need to consider three spatial directions, we obtain \(f = 3\). At higher energies, however, additional degrees of freedom, such as particle rotation or vibrations, may come into play, increasing \(f\).
In thermal equilibrium, the total internal energy of the gas is distributed equally among all degrees of freedom according to the equipartition theorem. Thus, in general, the internal energy of a gas is:
$$
U = \tfrac{1}{2} f N k_B T
$$
Of course, the internal energy can also be calculated in terms of the amount of substance.
If we want to specify the internal energy per mole, then we can write:
$$
U = \tfrac{1}{2} f R T
$$
If the internal energy of a gas is increased by a certain amount of heat, then simply \(\Delta U = \Delta Q\). Using the definition of heat capacity \(\Delta Q = C \Delta T\), we obtain, analogously, the molar specific heat for a gas at constant volume (an isochoric process):
$$
c_\mathrm{V} = \tfrac{1}{2} f R
$$
This value always refers to one mole of substance and depends only on the number of degrees of freedom.