Post a New Question


posted by .

. Consider a large number, N(0)
, molecules contained in a volume V
. Assume that
there is no correlation between the locations of the molecules (ideal gas). Do not
use the partition function in this problem.
(a) Calculate the probability P (V; N) that an arbitrary region of volume V contains
exactly N molecules.
(b) Calculate the average value N¹
and the standard deviation of N.
(c) Show that if both V and V
(0) ¡ V are large, the function P (V; N) assumes a
Gaussian form for N close to N¹
(d) Show that if both V ¿ V
and N ¿ N(0)
, the function P (V; N) assumes a
Poisson form.

  • physics -

    Probability for one molecule to be in that region is

    q = V/V0


    P(V,N) =

    N0!/(N! (N0-N)!) q^N (1-q)^(N - N0)

    The average number N' is q N0 because the probability q for each molecule to be in the region is independent. You can also find this by direct computation:

    N' = Sum from N = 0 to N0 of N P(V,N)

    To perform the summation, consider the function

    Q(V,N) =

    N0!/(N! (N0-N)!) q^N r^(N - N0)

    where q and r are considered to be independent variables. Then we can compute the summation by differentiating w.r.t. q while keeping r constant and then we put r = 1-q

    So, we have:

    N' = f(q,1-q)


    f(q,r) = Sum from N = 0 to N0 of
    q d/dq Q(V,N) =

    q d/dq Sum from N = 0 to N0 of Q(V,N) =

    q d/dq (q+r)^N0 =

    N0 q (q+r)^(N0-1)

    We thus have:

    N' = f(q,1-q) = N0 q

    The standard deviation can be evaluated in a similar way.

    <N^2> = g(q,1-q)


    g(q,r) = q d/dq q d/dq (q+r)^N0 =

    N0 q d/dq q (q+r)^(N0-1) =

    N0 q (q+r)^(N0-1) +
    N0 q^2 (N0-1)(q+r)^(N0-2)


    <N^2> = N0 q + N0(N0-1)q^2

    <N^2> - <N>^2 =

    N0 q + N0(N0-1)q^2 - N0^2 q^2 =

    N0 q - N0 q^2 =

    N0 q (1-q)

    So, the standard deviation is:

    sqrt[No q (1-q)]

    To derive the Gaussian and Poisson approximation in the appropriate limits, you need to expand
    Log(P) using the Stirling approximation for log(N!) for large N. This only involves trivial manipulations.

Answer This Question

First Name
School Subject
Your Answer

Related Questions

More Related Questions

Post a New Question