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Thermal/Statistical Physics Problem

  1. Jan 26, 2006 #1


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    The problem set can be found here: http://www.physics.utoronto.ca/%7Epoppitz/hw1.pdf [Broken] 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 [tex]V[/tex] is [tex]\frac{V}{V_0}[/tex]. It follows that the mean number of molecules is proportional to this ratio as well.

    [tex]\frac{<N>}{N_0} = \frac{V}{V_0}[/tex]
    [tex]<N> = \frac{V N_0}{V_0}[/tex]

    [tex]<(N-<N>)^2> = <N^2 - 2 N <N> + <N>^2[/tex]
    [tex]<(N-<N>)^2> = <N^2> - 2 <N>^2 + <N>^2[/tex]
    [tex]<(N-<N>)^2> = <N^2> - <N>^2[/tex]

    [tex]\frac{\sqrt{<(N-<N>)^2>}}{<N>} = \frac{\sqrt{<N^2> - <N>^2}}{<N>}[/tex]

    Now from here I can substitute into the regular [tex]<N>[/tex] terms but I don't know how I'm supposed to find [tex]<N^2>[/tex]?? Any help would be greatly appreciated.. thanks!
    Last edited by a moderator: May 2, 2017
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  3. Jan 26, 2006 #2


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    Homework Helper

    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.
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