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. For constant volume: (P₁)/(T₁) = (P₂)/(T₂).T₁ = 27 + 273 = 300 K, T₂ = 127 + 273 = 400 K, P₁ = 2 atm.P₂ = P₁ × (T₂)/(T₁) = 2 × (400)/(300) = 2.67 atm . Substituting values gives 2.67 atm, which matches expected kinetic theory result, confirming mean free path λ = 1/(√2 n π d²), ideal