Question

Choose the correct option.

The mean free path λ of molecules is given by where n is the number of molecules per unit volume and d is the diameter of the molecules. 

If ‘n’ is the number of molecules per unit volume and ‘d’ is the diameter of the molecules, the mean free path ‘λ’ of molecules is ______.

Options

• `sqrt(2/(pind^2))`

• `1/(pind^2)`

• `1/(sqrt2pind^2)`

• `1/sqrt(2pind^2)`

• `sqrt(2/(pind))`

• `1/(2pind^2)`

• `1/sqrt(2pind)`

Answer 1

`underlinebb(1/(sqrt2pind^2))`

Explanation:

Express the relation for a mean free path.

`lambda = D/N_c`  .....(i)

Here D is the distance travelled and N_c is the number of collisions. 

Express the relation for distance with time and average velocity.

D = txv'

Express the volume of a cylinder.

V = `pid^2 xx vt`

Express the number of molecules per unit volume.

`n = N/V`

Express the number of collisions.

`N_c = N/V pid^2 xx vt`

Substitute nv' for D and `N/V pid^2 xx vt` for Nc in equation (i).

`lambda = (v"'"t)/(pid^2vt  N/V)`  ......(ii)

Here, v' is an average velocity and v is the relative velocity.

Express the relation between average velocity and relative velocity.

`v = sqrt(2)v"'"`

Substitute `sqrt(2)v"'"` for v' in equation (ii) to find the wavelength.

`lambda = 1/(sqrt(2)pid^2  N/V)`

`lambda = 1/(sqrt(2)pid^2n)`