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Homework Help: Using Maxwell Boltzmann distribution to find number of atoms

  1. May 2, 2013 #1
    1. The problem statement, all variables and given/known data

    You will recall from our discussion of the Franck-Hertz experiment that the energy difference between the first excited state of mercury and the ground state is 4.86 eV. If a sample of mercury vaporized in a flame contains 1.06×1020 atoms in thermal equilibrium at 1563 K, calculate the number of atoms in the first excited state. Assume that the Maxwell-Boltzmann distribution applies and that the n = 1 (ground) and n = 2 (first excited) states have equal statistical weights.

    2. Relevant equations

    The Maxwell-Boltzmann distribution:

    [itex]f(E)=A e^{\frac{E}{k_{b}T}}[/itex]

    where A is a constant
    k_b is the boltzmann constant = 8.617*10^-5 eV/K
    and T is the temperature

    n(E)dE = g(E) * f(E) dE

    where n(E) = # per unit energy
    g(E) = # of states per energy E
    f(E) = MB distribution

    3. The attempt at a solution

    I'm still having trouble learning how to use distributions, so bear with me (or help me understand it :))

    I started off by normalizing the distribution by:

    [tex]\int\limits_0^\infty f(E)^2 dE [/tex]

    then I set it equal to 1 and solved for A. I found A=3.854.

    I think that was the right first step.

    and I know that g(E) for n=1 is 2 possible states
    and n=2 has 8.

    This is as far as I have gotten, as my professor hasn't really gone over how to use these distributions.

    My guess is that I set

    [itex] 1.06*10^{20} = n_{1}(E) dE + n_{2} (E) dE [itex]

    after this I know I have to integrate, but I'm not sure how to set up the limits, among other things.
  2. jcsd
  3. May 2, 2013 #2
    never mind I figured it out!
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