Probability of Molecules in Adjacent Gas Containers | Statistical Mechanics

[LocationX]

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:
$$P=\frac{V_1}{V_1+V_2}$$

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

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

Is this correct?

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

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

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

 

[Jhero]

Your calculation for the probability of one molecule being in V1 is correct. However, your calculation for the probability of k molecules being in V1 is not quite right. The Poisson distribution is used for the probability of a certain number of events occurring in a given time or space, assuming that the events occur randomly and independently. In this case, the number of molecules in V1 is not a random event, as it depends on the initial conditions and the size of V2. Instead, we can use the binomial distribution to calculate the probability of k molecules being in V1.

The probability of k molecules being in V1 can be calculated as:
P(k)=\binom{N}{k} \left( \frac{V_1}{V_1+V_2} \right)^k \left(1-\frac{V_1}{V_1+V_2}\right)^{N-k}

This is because we have N molecules in total, and we need to choose k of them to be in V1, which has a probability of V1/(V1+V2). The remaining N-k molecules will be in V2 with a probability of 1-V1/(V1+V2).

Your next step to find the probability when V2 -> infinity and N->infinity makes sense. In this case, we can use the law of large numbers to approximate the binomial distribution with a Poisson distribution. This is because as N and V2 become large, the probability of a molecule being in V1 becomes smaller and smaller, making the events more and more rare. Therefore, we can use the Poisson distribution to approximate the binomial distribution.

Your final calculation for the probability of k molecules being in V1 in the limit of V2 -> infinity and N->infinity is correct. This is a common result in statistical mechanics, where we use the Poisson distribution to describe the distribution of particles in a system with a large number of particles.

I hope this helps clarify your calculations. Keep up the good work!