Specific Heat Capacities

The relation between the two principal molar specific heats of an ideal gas is Mayer's formula, $C_P-C_V=R$, where $R$ is the universal gas constant. The physical reason is that heating a gas at constant pressure requires extra energy beyond raising internal energy, because the gas also expands and performs work against the surroundings. At constant volume no such work occurs, so all heat raises internal energy and $C_V$ is smaller. The difference is exactly the per-mole expansion work, which equals $R$. The ratio of specific heats $\gamma =C_P/C_V$ is a dimensionless quotient, not a difference, so it is incorrect. Half the gas constant has no basis in the derivation. The claim that the difference is zero for a monatomic gas is false, since Mayer's relation holds for every ideal gas irrespective of atomicity, including monatomic gases where $C_V=\frac{3}{2}R$ and $C_P=\frac{5}{2}R$. A dimensional check confirms consistency: both specific heats and $R$ carry units of joules per mole per kelvin.