Molar Specific Heat of Helium Gas

A gas of helium atoms is about the simplest substance you can picture. Since helium does not form molecules, the gas simply consists of individual atoms moving around and bouncing off of each other. If the temperature of the gas is T, then the average thermal kinetic energy of the atoms is 3/2 kT.
If you have a mole of helium gas at a temperature T, then the thermal energy of the atoms should be the average energy of 1 atom, 3/2 kT, times the number N_A atoms in a mole. Thus we easily estimate that the thermal energy E_He of a mole of helium atoms is

Using the fact that N_Ak = R, the gas constant, we get

If we raise the temperature one degree, from T to (T + 1), the thermal energy goes from 3/2 RT to 3/2 R(T + 1), an increase of 3/2 R. Thus the molar specific heat, which we will call C_V , is

......(1)

As we mentioned, we get the right answer. Equation 1 is in agreement with experiment.
The subscript V on the symbol C_V is there to remind us to measure the specific heat at constant volume. If you add heat to a gas, and at the same time allow the gas to expand, some of the energy goes into the work required to expand the volume, pushing the surrounding gas aside. For now we will leave the subscript V on C_V to remind us not to let the volume increase.