• id: JTHERM-02-N06
• title: Predicting C_v, C_p, and gamma from Degrees of Freedom
• tags: specific-heat, Cv, Cp, gamma

Internal energy per mole: $U = \frac{f}{2}RT$. At constant volume: $C_v = dU/dT = \frac{f}{2}R$. Using Mayer's relation: $C_p = C_v + R = \frac{f+2}{2}R$. The ratio: $\gamma = C_p/C_v = (f+2)/f$. Monatomic ($f = 3$): $\gamma = 5/3$. Diatomic ($f = 5$): $\gamma = 7/5$. Polyatomic ($f = 6$): $\gamma = 4/3$. As $f$ increases, $\gamma$ decreases toward 1. The kinetic theory successfully predicts these values, which match experimental measurements at moderate temperatures. Discrepancies at extreme temperatures arise from quantum effects (freezing/activating of vibrational modes).