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 $\lambda_1, \lambda_2 \ldots \ldots \lambda_n$ 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+\ldots+\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 $=\frac{n}{V}=\frac{P}{R T}$
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 $N=\frac{P N_A}{R T}$

$
{ }_{\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 $\lambda \propto m$