In a dilute gas at pressure P and temperature T,

Question:

In a dilute gas at pressure $P$ and temperature $T$, the mean time between successive collisions of a molecule varies with $T$ is :

1. (1) $T$

2. (2) $\frac{1}{\sqrt{T}}$

3. (3) $\frac{1}{T}$

4. (4) $\sqrt{T}$

Correct Option: , 2

Solution:

(2) Mean free path, $\lambda=\frac{1}{\sqrt{2} \pi n d^{2}}$

where, $d=$ diameter of the molecule

$n=$ number of molecules per unit volume

But, mean time of collision, $\tau=\frac{\lambda}{v_{\mathrm{rms}}}$

But $v_{\mathrm{rms}}=\sqrt{\frac{3 k T}{R}}$

$\therefore \tau=\frac{\lambda}{\sqrt{\frac{3 k T}{m}}} \Rightarrow t \propto \frac{1}{\sqrt{T}}$