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 $T_1=150K$ and $T_{II}=300K$ by connecting to heat reservoirs. Let ${\lambda }_I$ and ${\lambda }_{II}$ 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 $\frac{{\lambda }_I}{\lambda II}=\frac{T_I{P}_I}{T_{II}{P}_{II}}$
$=\frac{T_I}{T_{II}}\times \frac{P_{II}}{P_I}=\frac{T_I}{T_{II}}\times \frac{\sqrt{T_{II}}}{\sqrt{T_I}}$
$\Rightarrow \frac{{\lambda }_I}{{\lambda }_{II}}=\frac{\sqrt{T_I}}{\sqrt{T_{II}}}=\sqrt{\frac{150}{300}}$
$\Rightarrow \frac{{\lambda }_I}{{\lambda }_{II}}=0.7$