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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 $T_I$​= 150K and $T_{II​}$= 300K by connecting to heat reservoirs. Let $λ_I$​ and $λ_{II}$​ be the mean free paths of the gas particles in the two parts such that $d>>λ_I$​ and $d>>λ_{II}$​. Then $\frac{λ_I}{λ_{II}}$​ is close to:

  1. 0.25
  2. 0.5
  3. 0.7
  4. 1.0

My Attempt: I reasoned that after the system had settled down, pressure in both parts would be same. Using the formula given in textbook(Physics by Resnick/Halliday/Krane), $\lambda=\frac{kT}{\sqrt{2}\pi d^2P}$, I obtained the ration $0.5$. But the answer given is $0.7$.

Now Wikipedia gives me me another formula for mean free path, which does give the answer, but since this formula is not present in the textbook and this is a class 11-12 level problem, I think it is just coincidence.

So how do I solve this problem or am I overthinking this?

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  • $\begingroup$ The formula you've used assumes that a moving molecule collides with another molecule that is at rest, which is not a valid assumption. The one on wikipedia is more accurate because it accounts for the randomised (and isotropic) motion of all molecules of the ideal gas. $\endgroup$
    – Cross
    Commented Dec 31, 2021 at 9:27
  • $\begingroup$ @Cross Sorry. I have added the $\sqrt{2}$ in the formula now, which AFAIK takes into account the relative motion. $\endgroup$
    – Alpha Delta
    Commented Dec 31, 2021 at 10:00
  • $\begingroup$ By any chance did you figure out the answer to this? $\endgroup$
    – Box Box Box Box
    Commented Feb 6, 2022 at 13:27
  • $\begingroup$ @AshishAhuja Sadly no. $\endgroup$
    – Alpha Delta
    Commented Feb 6, 2022 at 14:30
  • $\begingroup$ You have probably already read this but the wiki page you have linked to mentions "These different definitions of the molecular diameter can lead to slightly different values of the mean free path".. and it talks a bit more in detail above this line. I have learnt this same formula, I think the other formula on wiki which involves viscosity is a bit too advanced. May I ask where you found this problem? I guess the problem has to specify the kind of mean free path used, since if the viscosity formula is being used then... (cont.) $\endgroup$
    – Box Box Box Box
    Commented Feb 6, 2022 at 15:29

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