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Statistical mechanics

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In adjacent containers of volume V1 and V2, there contains a gas of N molecules. The gas is free to move between the containers through a small hole in their common wall.

What is the probability to find k molecules in the V1?

Probability of one molecule to be in V1 is given by:
[tex]P=\frac{V_1}{V_1+V_2} [/tex]

Using the Poisson distribution, I think the probability that k molecules are in V1 should be:

[tex]p_{poisson}(k)=\frac{a^k}{k!} e^{-a} [/tex]
[tex]p(k)=\frac{\left( \frac{N V_1}{V_1+V_2} \right) ^k}{k!} e^{-\frac{N V_1}{V_1+V_2}} [/tex]

Is this correct?

My next step is to find the probability when V2 -> infinity, and N->infinity such that [tex]N/V=\rho[/tex] is finite.

For this, I get:
[tex]p(k)=\frac{\left( \rho \right) ^k}{k!} e^{-\rho} [/tex]

Does this work make any logical sense at all? Comments are appreciated, thanks for the help!
 

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