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Mayer's relation $C_p - C_v = R$ holds for one mole of any ideal gas. The physical origin: at constant pressure, additional energy $R$ per mole per kelvin goes into expansion work ($P\Delta V = nR\Delta T$). The ratio $\gamma = C_p/C_v = (f+2)/f$ depends on the degrees of freedom $f$.

For monatomic gases (He, Ne, Ar): $f = 3$ (translational only), $C_v = 3R/2$, $C_p = 5R/2$, $\gamma = 5/3 \approx 1.67$. For diatomic gases at moderate temperatures (N$_2$, O$_2$, H$_2$): $f = 5$ (3 translational + 2 rotational), $C_v = 5R/2$, $C_p = 7R/2$, $\gamma = 7/5 = 1.4$. For polyatomic gases (CO$_2$, H$_2$O): $f = 6$, $C_v = 3R$, $C_p = 4R$, $\gamma = 4/3 \approx 1.33$.

At very high temperatures, vibrational modes activate, adding 2 more degrees of freedom per vibrational mode, increasing $f$ and decreasing $\gamma$. The polytropic process $PV^n = C$ generalises all standard processes: $n = 0$ (isobaric), $n = 1$ (isothermal), $n = \gamma$ (adiabatic), $n = \infty$ (isochoric). The molar heat capacity for a polytropic process is $C = C_v(\gamma - n)/(1 - n)$.