Practice question

**Temperature dependence of RMS speed** v_rms ∝ √T, doubling T increases v_rms by √2≈1.414, e.g., at 300 K v_rms for O₂ ≈483 m/s, at 600 K ≈683 m/s, illustrating kinetic theory relation between temperature and molecular motion, average kinetic energy ½ m v_rms² =3/2 k_B T. P = (ρ R T)/(M), so M = (ρ R T)/(P).M = (0.8 × 8.31 × 300)/(1.01 × 10⁵) = 0.01975 kg/mol = 19.75 g/mol ≈ 20 g/mol . Substituting values gives 20 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