Practice question

**Kinetic theory mean free path** λ = 1/(√2 π d² n) quantifies collision frequency. With n =1.0×10²⁵ m⁻³, λ=9×10⁻⁷ m, d² =1/(1.414×10²⁵×3.14×9×10⁻⁷)=2.5×10⁻²⁰ m², d≈1.58×10⁻¹⁰ m, typical molecular size ~10⁻¹⁰ m, consistent with gas kinetic theory. v_rms ∝ √(T), (v₂)/(v₁) = √((T₂)/(T₁)).(1000)/(500) = √((T₂)/(250)), 2 = √((T₂)/(250)).Square both sides: 4 = (T₂)/(250), T₂ = 1000 K. Substituting values gives 1000 K, 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.