An ideal gas is enclosed in a cylinder at pressure of 2 atm and temperature 300K.  The mean time between two successive collisions is $4\sqrt{2}\times {10}^{-8}$s.  If the pressure is doubled and temperature is increased to 600K, the mean time between two successive collisions will be close to

The mean time between collisions becomes about 3.0 × 10⁻⁸ s (roughly 0.707 times the original 4.2 × 10⁻⁸ s).

For an ideal gas with fixed molecular size, the collision frequency is proportional to number density times average speed. Number density n scales as P/T (ideal gas law), and average molecular speed scales as √T (kinetic theory). Therefore, collision frequency z ∝ (P/T)·√T = P/√T. The mean time between collisions is τ = 1/z, so τ ∝ √T / P.

Step-by-step scaling

1. Given: Initial pressure P₁ = 2 atm, temperature T₁ = 300 K, mean time τ₁ = 4.2 × 10⁻⁸ s.
2. Changed to: P₂ = 4 atm (pressure doubled), T₂ = 600 K (temperature doubled).

3. Use ratio:

   τ₂ / τ₁ = (√T₂ / P₂) ÷ (√T₁ / P₁) = (P₁ / P₂) · √(T₂ / T₁)
   = (2 / 4) · √(600 / 300) = (1/2) · √2 ≈ 0.707.
4. Compute τ₂: τ₂ ≈ 0.707 × 4.2 × 10⁻⁸ s ≈ 3.0 × 10⁻⁸ s.