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Gas in two containers connected by a small hole

A container is divided into 2 parts by a partition containing a hole of radius r. Helium gas in the two parts is held at temperature T1 = 75K and T2 = 300K respectively. After the system reaches a steady state, the mean free paths on each side are lambda1 and lambda2. What is the ratio lambda1/lambda2 when

(a) r>>lambda1 and r>>lambda2,

(b) r<<lambda1 and r<<lambda2.

Solution Preview

If r is much larger than the mean free path then molecules close to the hole will suffer many collisions when they move a distance equal to the hole radius. If on the other side of the hole there is a vacuum, then the molecules that are near the hole will suffer more collisions from molecules in the direction away from the hole than from the direction of the hole. So, a molecule that initially passes alongside the hole and would pass the hole if it didn't suffer any collision, would typically suffer many collisions as it traverses the length of the hole and most of these collisions have the effect of pushing the molecule toward the hole. So, such a molecule will typically move through the hole and go to the other part of the container.

So, we see that the average velocity of molecules near the hole is not zero; it has a component in the direction of the hole. If the hole weren't there, then the gas molecules would have an isotropic ...

Solution Summary

A detailed explanation of the solution is given. Systems reaching a steady state are determined. The ratio of lambda1/lambda2 is determined.

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