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Energy levels of electrons

  1. Dec 4, 2004 #1
    How do I calculate the number of electrons with a given energy in
    a sample of hydrogen atoms at a certain temperature?
    For example how many electrons would be in the n=2 level
    at 3 Kelvin? And does it matter whether or not hydrogen is gas or liquid or solid i.e is a different calculation needed for a gas in place of a liquid.
  2. jcsd
  3. Dec 4, 2004 #2
    This is a classical problem; you weigh each possible state by the Boltzmann factor e^(-E/kT), write the partition function and calculate the probability of a given energy level. At 3K, the probability of electrons being in n=2 is vanishingly small.

    This only applies to a gas; in a liquid the interactions between atoms cannot be ignored and make for a messier problem.
  4. Dec 4, 2004 #3
    When you say vanishingly small, have you got an approximate number
    of decimal places for that small probability?
  5. Dec 5, 2004 #4
    Easy to get. The partition function is dominated by the ground state so we can approximate:

    Z = exp (-E1/kT).

    P(n=2) = g * exp (-E2/kT) / Z, where g is the degeneracy of the 2nd level (ie how many distinct states there are with that energy level). Ignoring fine structure, g=4.

    So P(n=2) = 4 exp (-(E2 - E1) / kT) = 4 e -39 455.3202 = 4 * 10 -17135

    Hmm... that's even smaller than I expected. Did I do something wrong?
  6. Dec 5, 2004 #5
    E1 = 10^-18 Joules.
    P(n=2) = 4 exp (-(E2 - E1) / kT) = 4 exp [(4 x 10^-18 - 1 x 10^-18)/10^-23]

    works out at about 10^10^-18

    There must be something wrong with the equation.
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