Mean Free Path - Practice Questions & MCQ

Mean Free Path - 

On the basis of kinetic theory of gases, it is assumed that the molecules of a gas are continuously colliding against each other. So, the distance travelled by a gas molecule between any two successive collisions is known as free path.

There are assumption for this theory that during two successive collisions, a molecule of a gas moves in a straight line with constant velocity. Now, let us discuss the formula of mean free path - 

Let ${\lambda }_1,{\lambda }_2\dots \dots {\lambda }_n$ be the distance travelled by a gas molecule during $\mathbf{n}$ collisions respectively, then the mean free path of a gas molecule is defined as -

$\lambda =\frac{\text{ Total distance travelled by a gas molecule between successive collisions }}{\text{ Total number of collisions }}$

Here, $\lambda$ is the mean free path.

It can also be written as -

$\lambda =\frac{{\lambda }_1+{\lambda }_2+{\lambda }_3+\dots +{\lambda }_n}{n}$

Now, let us take d = Diameter of the molecule,

$\mathrm{N}=\text{ Number of molecules per unit volume. }$

Also, we know that, $\mathrm{PV}=\mathrm{nRT}$
So, Number of moles per unit volume $=\frac{n}{V}=\frac{P}{RT}$
Also we know that number of molecules per unit mole $=N_A=6.023\times {10}^{23}$
So, the number of molecules in ' $n$ ' moles $={\mathrm{nN}}_{\mathrm{A}}$
So the number of molecules per unit volume is $N=\frac{P{N}_A}{RT}$

${}_{\text{So, }}\lambda =\frac{\mathrm{RT}}{\sqrt{2}\pi {\mathrm{d}}^2{\mathrm{PN}}_{\mathrm{A}}}=\frac{\mathrm{kT}}{\sqrt{2}\pi {\mathrm{d}}^2\mathrm{P}}$
 

If all the other molecules are not at rest then, $\lambda =\frac{1}{\sqrt{2}\pi N{d}^2}=\frac{\mathrm{RT}}{\sqrt{2}\lambda {\mathrm{d}}^2{\mathrm{PN}}_{\mathrm{A}}}=\frac{\mathrm{kT}}{\sqrt{2}\pi {\mathrm{d}}^2\mathrm{P}}$

Now, if
$\lambda =\frac{1}{\sqrt{2}\pi N{d}^2}$ and $\mathrm{m}=$ mass of each molecule then we can write - $\lambda =\frac{1}{\sqrt{2}\pi N{d}^2}=\frac{m}{\sqrt{2}\pi (\mathrm{mN})d^2}=\frac{m}{\sqrt{2}\pi d^2\rho }$

So,
$\lambda \propto \frac{1}{\rho }$ and $\lambda \propto m$