Practice question

**Equipartition theorem** energy ½ k_B T per degree of freedom per molecule, f degrees give U = f/2 k_B T per molecule, f/2 R T per mole, internal energy function of T only for ideal gas, no intermolecular potential. P = (ρ R T)/(M), M = (ρ R T)/(P).M = (1.43 × 8.31 × 273)/(1.01 × 10⁵) = 0.0321 kg/mol ≈ 32.1 g/mol. Substituting values gives 32 g/mol, which matches expected kinetic theory result, confirming mean free path λ = 1/(√2 n π d²), ideal gas law P V = n R T and v_rms = √(3 R T/M) relations.