An ideal gas in a closed container is slowly heated. As its temperature increases, which of the following statements are true?
(A) the mean free path of the molecules decreases.
(B) the mean collision time between the molecules decreases.
(C) the mean free path remains unchanged.
(D) the mean collision time remains unchanged.

Solution

Physics sir · 4 steps

A closed container fixes the number density, so the path length stays the same and only the speed changes.

1. Mean free path. The mean free path is λ = 1/(√2 π d² n), where n is the number of molecules per unit volume. In a closed container neither the number of molecules nor the volume changes, so n is fixed and λ stays the same.
2. Check the pressure form. You may also see λ = kT/(√2 π d² P). Heating at fixed volume raises P in proportion to T, so T/P stays constant and λ still does not change.

   τ = λ/v̄, v̄ ∝ √T

3. Mean collision time. The time between collisions is the mean free path divided by the average speed. Heating makes the molecules faster, so they cover the same λ in less time and τ decreases as 1/√T.
4. So. Statement (C) is true because λ is unchanged. Statement (B) is true because τ decreases. The answer is option (c).

The idea to keep. Mean free path depends only on how crowded the molecules are, that is on n. Temperature enters the collision time only through the speed, so hotter means more frequent collisions.

Answer from the NTA final answer key. Solution by PrepWiser.