Mean Free Path & Real Gases question from JEE Main 2020 (6 Sep, Shift 2)

Answer: (d)

1. Mean time between collisions: $\tau = \dfrac{\lambda}{\bar v}$ with $\bar v \propto \sqrt{T}$.
2. At a given pressure $P$: $\lambda = \dfrac{kT}{\sqrt{2}\pi d^2P} \propto T$, so $\tau \propto \dfrac{T}{\sqrt{T}} = \sqrt{T}$ (option a).
3. If instead the number density is held fixed (closed container), $\lambda$ is constant and $\tau \propto \dfrac{1}{\sqrt{T}}$ (option d).
4. Key: (d) $\dfrac{1}{\sqrt{T}}$, which assumes fixed number density.