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(B) In the steady state,the rate of diffusion of gas molecules through the hole must be the same from both sides. For a hole where $d \gg \lambda$,the

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$0.5$

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(B) In the steady state,the rate of diffusion of gas molecules through the hole must be the same from both sides. For a hole where $d \gg \lambda$,the flow is governed by hydrodynamic conditions,implying the pressures in both parts must be equal,i.e.,$P_{I} = P_{II}$.The mean free path $\lambda$ of a gas molecule is given by $\lambda = \frac{k_{B}T}{\sqrt{2}\pi d_{m}^{2}P}$,where $d_{m}$ is the molecular diameter.From this expression,we see that $\lambda \propto \frac{T}{P}$.Since $P_{I} = P_{II}$ in the steady state for this condition,the ratio of the mean free paths is:$\frac{\lambda_{I}}{\lambda_{II}} = \frac{T_{I} / P_{I}}{T_{II} / P_{II}} = \frac{T_{I}}{T_{II}}$Substituting the given temperatures:$\frac{\lambda_{I}}{\lambda_{II}} = \frac{150}{300} = 0.5$Therefore,the correct option is $(b)$.

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