Degrees of Freedom

Degrees of freedom (DoF) f: independent ways a molecule can store energy.

Monatomic gas (He, Ar, Ne): 3 translational ⇒ f = 3.

Diatomic at room T (N₂, O₂): 3 translation + 2 rotation = 5. (Vibrations frozen out at room T.)

Diatomic at high T: vibrations contribute 2 more ⇒ f = 7.

Polyatomic non-linear (H₂O, NH₃): 3 translation + 3 rotation = 6 (room T). Vibrations add more at high T.

Equipartition theorem: each quadratic DoF gets ½k_BT per molecule.

Specific heat per mole: C_v = (f/2)R, C_p = (f/2 + 1)R. γ = C_p/C_v = 1 + 2/f.

Monatomic γ = 5/3. Diatomic (room T) γ = 7/5 = 1.4. Polyatomic γ = 4/3 (approximately).