Q.Derive the expression for mean free path of the gas.

Definition. The mean free path, $\lambda$, is the average straight-line distance a gas molecule manages to travel between two successive collisions with other molecules. Even though gas molecules typically move at several hundred metres per second, a scent released across a room takes a noticeable time to reach someone else, precisely because each molecule of that scent is deflected an enormous number of times by collisions with the surrounding air molecules and zig-zags along a very indirect path instead of travelling straight to its destination.

The formula. Modelling a molecule of diameter $d$ as sweeping out an imaginary cylinder of cross-section $\pi d^2$ as it moves, and correctly accounting for the fact that every molecule (not just the one being tracked) is simultaneously in motion (which introduces a factor of $\sqrt2$), the mean free path works out to

where $n$ is the number density, $\rho$ the mass density, and the last form uses the ideal gas relation $n=P/kT$. …