Question

Which relation is not true about the mean free path of a gas?

• A. It is directly proportional to the temperature at a given density of the gas
• B. It is inversely proportional to the pressure at a given temperature for a fixed amount of a gas
• C. It is directly proportional to the volume of a fixed amount of a gas
• D. It is constant for an ideal gas if the number of gas molecules per litre of volume is fixed

Answer 1

The correct option is A It is directly proportional to the temperature at a given density of the gas

$\lambda =\frac{1}{\sqrt{2}\pi d^2{n}_v}=\frac{V}{\sqrt{2}\pi d^{2}n{N}_A}=\frac{RT}{\sqrt{2}\pi d^2{N}_{A}P}$
We get the above relations by using $n_v=\frac{n{N}_A}{V}$, $V=\frac{nRT}{P}$
From the above equations, option b, c, and d are correct right away.
If the density is fixed then the ratio $\frac{P}{T}$ is fixed as $d=\frac{PM}{RT}$. So, if $T$ increases, $P$ increases as the ratio $\frac{P}{T}$ has to remain constant. In a nutshell, if the density is constant, $\lambda$ is constant.