• word_count: 200

Each independent mode of molecular energy storage is a degree of freedom (DOF). The equipartition theorem states: each DOF contributes $\frac{1}{2}k_BT$ of energy per molecule.

Monatomic gases (He, Ne, Ar): 3 translational DOF, $U = \frac{3}{2}nRT$, $C_v = \frac{3}{2}R$, $\gamma = 5/3$. Diatomic gases at moderate temperature (N$_2$, O$_2$, H$_2$): 3 translational + 2 rotational = 5 DOF, $U = \frac{5}{2}nRT$, $C_v = \frac{5}{2}R$, $\gamma = 7/5$. At very high temperatures, vibrational modes activate: each vibrational mode adds 2 DOF (kinetic + potential), giving $f = 7$ for diatomic, $C_v = \frac{7}{2}R$, $\gamma = 9/7$.

Non-linear polyatomic molecules (H$_2$O, CH$_4$): 3 translational + 3 rotational = 6 DOF at moderate temperature, $\gamma = 4/3$. Linear polyatomic (CO$_2$): 3 + 2 = 5 DOF, same as diatomic.

The general formula $\gamma = (f+2)/f = 1 + 2/f$ shows that $\gamma$ decreases as complexity (DOF) increases. For gas mixtures: $C_{v,\text{mix}} = \sum n_i C_{v,i}/\sum n_i$ and $\gamma_{\text{mix}} = C_{p,\text{mix}}/C_{v,\text{mix}}$.