Specific Heat of an Ideal Gas

As previously noted, the specific heat of any substance is amount of heating required per unit mass to raise the temperature of the substance by one degree. For a gas one must clarify whether the the volume or the pressure is held constant as the temperature increases -- the specific heat differs between these two cases because in the latter situation the added energy from the heating is split between the production of internal energy and the production of work as the gas expands.

At constant volume all heating goes into increasing the internal energy, so $\Delta Q = \Delta E$ from the first law of thermodynamics. From equation (24.13) we find that $\Delta E = (3/2) N k_B \Delta T$. If the molecules making up the gas have mass $M$, then the mass of the gas is $NM$. Thus, the specific heat at constant volume of an ideal gas is

$$C_V = \frac{1}{NM} \frac{3 N k_B}{2} = \frac{3 k_B}{2M} ~~~ \mbox{(specific heat at const vol)}$$ (25.20)

As noted above, when heat is added to a gas in such a way that the pressure is kept constant as a result of allowing the gas to expand, the added heat $\Delta Q$ is split between the increase in internal energy $\Delta E$ and the work done by the gas in the expansion $\Delta W = p\Delta V$ such that $\Delta Q = \Delta E + p\Delta V$. In a constant pressure process the ideal gas law (24.19) predicts that $p\Delta V = Nk_B \Delta T$. Using this and the previous equation for $\Delta E$ results in the specific heat of an ideal gas at constant pressure:

$$C_P = \frac{1}{NM} \left( \frac{3 N k_B}{2} + N k_B \right) = \frac{5 k_B}{2M} ~~~ \mbox{(specific heat at const pres}) .$$ (25.21)