Question

A container is divided into two equal parts I and II by a partition with a small hole of diameter $d$. The two partitions are filled with same ideal gas, but held at temperatures $T_{1}=150 \,K$ and $T_{ II }=300\, K$ by connecting to heat reservoirs. Let $\lambda_{I}$ and $\lambda_{I I}$ be the mean free paths of the gas particles in the two parts, such that $d>>\lambda_{ I }$ and $d>>\lambda_{ II }$. Then, the $\lambda_{ I } / \lambda_{ II }$ is close to

• A. 0.25
• B. 0.5
• C. 0.7
• D. 1.0

Answer 1

Answer: (C)

In steady state, rate of diffusion of gases must be same from both sides.
$\Rightarrow r_{1}=r_{2}$
$\frac{\Rightarrow P_{I}}{\sqrt{T_{I}}}=\frac{P_{II}}{\sqrt{T_{II}}}$
Now, mean free path of a gas molecule is
$\lambda=\frac{k_{B}T}{\sqrt{2}\pi d^{2.}P}$
or $\lambda\propto \frac{T}{P}$
So $\frac{\lambda_{I}}{\lambda II}=\frac{T_{I}P_{I}}{T_{II}P_{II}}$
$=\frac{T_{I}}{T_{II}}\times\frac{P_{II}}{P_{I}}=\frac{T_{I}}{T_{II}}\times\frac{\sqrt{T_{II}}}{\sqrt{T_{I}}}$
$ \Rightarrow \frac{\lambda_{I}}{\lambda_{II}}=\frac{\sqrt{T_{I}}}{\sqrt{T_{II}}}=\sqrt{\frac{150}{300}}$
$\Rightarrow \frac{\lambda_{I}}{\lambda_{II}}=0.7$