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. v_rms ∝ (1)/(√(m)), v_N₂v_Ar = √(m_Ar)m_N₂.v_N₂430 = √((39.9)/(28)) ≈ √(1.425) ≈ 1.193.v_N₂ = 430 × 1.193 ≈ 513 m/s. Substituting values gives 513 m/s, 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.