NEET 2020 Physics Question 155 Solution

The mean free path ($$\lambda$$) is the average distance traveled by a gas molecule between two successive collisions with other gas molecules.

Suppose a molecule of diameter $$d$$ moves with speed $$v$$ while all other molecules are assumed stationary. A collision occurs whenever the center of another molecule lies within a distance $$d$$. In time $$\Delta t$$, it sweeps a collision cylinder of volume: $$V_{\text{coll}} = \pi d^2 (v \Delta t)$$ With number density $$n$$, the number of collisions is $$N_{\text{coll}} = n \pi d^2 v \Delta t$$. The mean free path is the total distance divided by number of collisions: $$\lambda = \frac{v \Delta t}{N_{\text{coll}}} = \frac{1}{n \pi d^2}$$

In reality, all molecules are in random thermal motion. The effective collision frequency depends on the average relative speed $$\langle v_{\text{rel}} \rangle$$. For a Maxwell-Boltzmann velocity distribution: $$\langle v_{\text{rel}} \rangle = \sqrt{2} \langle v \rangle$$ Substituting this relative velocity into the collision rate introduces a factor of $$\sqrt{2}$$ into the denominator: $$\lambda = \frac{1}{\sqrt{2} n \pi d^2}$$

NCERT states verbatim in the Kinetic Theory chapter: "$$\lambda = \frac{1}{\sqrt{2} n \pi d^2}$$". This corresponds uniquely to Option (2).