What is mean free path? Derive the equation for the mean free path.

(N/A) The average distance traveled by gas molecules between two successive collisions is called the mean free path.
The calculation of the mean free path is based on two hypotheses:
$(1)$ Gas molecules are rigid spheres of diameter '$d$'.
$(2)$ Molecules other than the one in motion are considered stationary.
Let the diameter of a gas molecule be $d$ and the average speed of one molecule be $\langle v \rangle$.
Let this molecule collide with any other molecule that comes within a distance $d$ between their centers.
It sweeps a volume $\pi d^{2} \langle v \rangle \Delta t$ in a time interval $\Delta t$.
If $n$ is the number of molecules per unit volume,the molecule undergoes $n \pi d^{2} \langle v \rangle \Delta t$ collisions in the time interval $\Delta t$.
Thus,the rate of collision is $n \pi d^{2} \langle v \rangle$.
The time interval between two successive collisions is:
$\tau = \frac{1}{n \pi \langle v \rangle d^{2}}$
The average distance between two successive collisions is called the mean free path,denoted by $\bar{l}$.
$\therefore \bar{l} = \langle v \rangle \tau$
$\therefore \bar{l} = \frac{1}{n \pi d^{2}}$