Mayer's Relation

Setup. Consider one mole of an ideal gas enclosed in a cylinder by a light, frictionless, airtight piston, with pressure $P$, volume $V$ and temperature $T$.

Heating at constant volume. If the gas is heated so its temperature rises by $dT$ while the piston is held fixed (constant volume), no work is done (the piston does not move), so all the heat supplied, $dQ_1$, goes entirely into raising the internal energy:

where $C_V$ is the molar specific heat at constant volume.

Heating at constant pressure. If instead the gas is heated to the same temperature rise $dT$ at constant pressure, its volume increases by some $dV$, and the piston moves outward doing work $dW = P\,dV$ on the surroundings. The heat supplied in this case, $dQ_2$, must therefore both raise the internal energy and supply this mechanical work:

where $C_P$ is the molar specific heat at constant pressure.

Combining. Since the internal energy of an ideal gas depends on temperature alone (Section 3.7), the same $dE = C_V\,dT$ applies in both cases. Substituting into the constant-pressure relation:

For one mole at constant pressure, $PV = RT \Rightarrow P\,dV = R\,dT$, so

This is Mayer's relation. (If heat is measured in calories and work in joules, it is modified to $C_P - C_V = R/J$, where $J$ is the mechanical equivalent of heat -- see Example 3.3.) …