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Partition Function for a helium atom

  1. Feb 9, 2016 #1
    1. The problem statement, all variables and given/known data
    The first excited state of the helium atom lies at an energy 19.82 eV above the ground state. If this excited state is three-fold degenerate while the ground state is non-degenerate, find the relative populations of the first excited and the ground states for helium gas in thermal equilibrium at 10,000K.

    2. Relevant equations
    $$Z=\sum_i e^{-\varepsilon /kT}$$

    3. The attempt at a solution
    $$Z=\sum_i e^{-\varepsilon /kT}$$
    $$Z=3e^{-(19.82eV) /(8.617e-5eV/K)(10000K)}$$
    $$Z=3.074e-10$$

    I found the partition constant for the first excited state, but then I'm not sure what to do. What do they mean by "the relative populations"?
     
  2. jcsd
  3. Feb 10, 2016 #2

    DrClaude

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    Staff: Mentor

    That statement doesn't make sense. The partition function is a sum over all states. You can't find the partition function for a single state.

    ##P_2/P_1##, where ##P_1## is the population of the ground state and ##P_2## the population of the excited state.
     
  4. Feb 10, 2016 #3
    DrClaude, thank you for responding. What exactly is the ''population'' of a ground state of an atom?
     
  5. Feb 10, 2016 #4

    DrClaude

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    It's the number of atoms in the ground state. The term comes from the idea that you have an ensemble of identically prepared systems: if you have 100 atoms with a probability of .9 of being in the ground state, then the population of the ground state is 90, and the relative population is .9.
     
  6. Feb 10, 2016 #5
    Wouldn't the relative population then be $$P2/P1 = 90/10 = 9$$
     
  7. Feb 11, 2016 #6

    DrClaude

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    Staff: Mentor

    It's a question of interpretation. Re-reading the problem as you stated it in the OP, I think indeed that you need to calculate the relative population of the excited state as ##P_2/P_1##, which in my example would give ##0.1/0.9 \approx 0.111##.
     
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