Practice question

**Maxwell-Boltzmann distribution** gives distribution of speeds, v_rms = √(3kT/m), most probable v_mp = √(2kT/m), average v_avg = √(8kT/πm), all ∝ √T, ratio v_rms:v_avg:v_mp =1.732:1.596:1.414, temperature raises all speeds proportionally. l = (1)/(√(2) n π d²), d² = (1)/(√(2) n π l).d² = (1)/(1.414 × 3.0 × 10²⁵) × 3.14 × 1.8 × 10⁻⁷ = (1)/(2.4 × 10⁻¹⁹) ≈ 4.17 × 10⁻²⁰.d = √(4.17 × 10⁻²⁰) ≈ 2.04 × 10⁻¹⁰ m. Substituting values gives 2.0 × 10⁻¹⁰ m, 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.