The rms speed of a gas is 350 m/s at 175 K. At what temperature will the rms speed be 700 m/s?
**Collision frequency** Z = √2 n π d² v_avg, n number density, d molecular diameter, v_avg average speed, proportional to n and v_avg, mean free path λ = v_avg/Z =1/(√2 n π d²), inversely proportional to n, so λ ∝1/P at constant T because n ∝ P, collision frequency increases with pressure, λ decreases. v_rms ∝ √(T), (v₂)/(v₁) = √((T₂)/(T₁)).(700)/(350) = √((T₂)/(175)), 2 = √((T₂)/(175)).Square both sides: 4 = (T₂)/(175), T₂ = 700 K. Substituting values gives 700 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
Ref: NCERT > Physics Book > Behaviour of Perfect Gas and Kinetic Theory > Collision Frequency and Mean Free Path Variation