Question
- Define mean free path.
- Derive an expression for mean free path of a gas molecule.
✓
Answer
- The mean free path of a gas molecule is defined as the average distance travelled by a molecule between two successive collisions.
According to figure, if a molecule covers free path $\lambda_1,\lambda_2,\lambda_3, \ ....$ after successive collisions, then its mean free path is given by
$\lambda=\frac{\lambda_1+\lambda_2+\lambda_3...}{\text{(total number of collisions)}}$
- Expression for mean free path: Let d be the diameter of each molecule of the gas, then a particular molecule will suffer collision with any molecule that comes within a distance d between centers of two molecules.
If $\bar{\text{v}}$ is average speed of molecule, then from figure, the volume swept by the molecule in small time $\Delta\text{t}$ in which any molecule will collide with it
$=\pi\text{d}^2\langle\text{v}\rangle\Delta\text{t}$
If n is number of molecules per unit volume of the gas, then number of collision suffered by the molecule in time $\Delta\text{t}$
$=\pi\text{d}^2\langle\text{v}\rangle\Delta\text{t}\times\text{n}$
So, number of collisions per second
$=\frac{\pi\text{d}^2\langle\text{v}\rangle\Delta\text{t}\times\text{n}}{\Delta\text{t}}=\text{n}\pi\text{d}^2\langle\text{v}\rangle$
$\therefore$ Average time between two successive collisions
i.e. $\tau=\frac{1}{\text{n}\pi\text{d}^2\langle\text{v}\rangle}$
$\therefore$ Mean free path = average distance between two successive collision
$\Rightarrow\lambda=\tau\times\text{mean velocity}$
$=\frac{1}{\text{n}\pi\text{d}^2\langle\text{v}\rangle}\times\bar{\text{v}}=\frac{1}{\text{n}\pi\text{d}^2}$
Mean free path, $\lambda=\frac{1}{\text{n}\pi\text{d}^2}$
where, d = diameter of each molecule
and n = number of molecules per unit volume
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