Thermal/Statistical Physics Problem

[NIQ]

The problem set can be found here: http://www.physics.utoronto.ca/%7Epoppitz/hw1.pdf I am mainly having a problem with question II Part 2.

Here's what I have so far:

II) 1. Since the probability that a given molecule is in a subvolume $$V$$ is $$\frac{V}{V_0}$$. It follows that the mean number of molecules is proportional to this ratio as well.

$$\frac{<N>}{N_0} = \frac{V}{V_0}$$
$$<N> = \frac{V N_0}{V_0}$$

2.
$$<(N-<N>)^2> = <N^2 - 2 N <N> + <N>^2$$
$$<(N-<N>)^2> = <N^2> - 2 <N>^2 + <N>^2$$
$$<(N-<N>)^2> = <N^2> - <N>^2$$

$$\frac{\sqrt{<(N-<N>)^2>}}{<N>} = \frac{\sqrt{<N^2> - <N>^2}}{<N>}$$

Now from here I can substitute into the regular $$<N>$$ terms but I don't know how I'm supposed to find $$<N^2>$$?? Any help would be greatly appreciated.. thanks!

 

[StatusX]

This problem involves the binomial distribution. You know the probability of one molecule being in V, so you need to caculate the probability that exactly N will be (that is, that N are in V, and N₀-N are not, and don't forget to multiply by the number of ways of choosing these N molecules). Once you have the distribution you can calculate any expectation value you want, or you could just look up the standard deviation of a binomial distribution.