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Partition Function for Positronium

  1. Nov 25, 2014 #1
    First off, I'm glad I'm finally a member on this board. It has helped me TREMENDOUSLY over the past few years with various problems I've had. You guys/gals are awesome and hopefully I can make some contributions to this site.

    1. The problem statement, all variables and given/known data

    A. Write down the partition function for positronium, assuming that only the levels illustrated in the diagram exist. Evaluate the partition function for T=20000K. Remember to include the degeneracies for each level. (I couldn't figure out how to copy and paste the diagram I was given, so I googled the one below. It's the same diagram, but the diagram I was given stops at the N=3 energy level)

    images?q=tbn:ANd9GcRsDOxvDPL91NPAZx92QcxENKIvdq_Q1dBLbCOYSGcAvyMY8PZLQw.png

    B. Write down an expression for the probability that the atom will exist in the state given by N=3, l=1, and determine that probability for T=20000K

    C. Find an expression for the mean energy and evaluate that expression for T=20000K

    2. Relevant equations

    1. En = 6.803eV (1 - 1/n2)
    2. Z= ∑e-E(s)/KT
    3. P(s)= 1/z * e-En/KT
    4. ∑ E(s)* P(s)

    3. The attempt at a solution
    A. So, we haven't done anything even close to this in class, so I'm a little coonfused, but to start, I substituted eq1 into the partition function in eq2 to get Z= ∑e-6.803eV (1 - 1/n2)/KT. After this, I'm completely stuck because we didn't really cover degeneracy in class very well. I know I'm supposed to multiply the expression by the degeneracy, but I'm not exactly sure how to do that.

    B. Once A is found, this should be straightforward
    C. Once A is found, this should be straightforward





    My work is due by 11/25 at 5pm (EST), any help would be greatly appreciated!
     
    Last edited: Nov 25, 2014
  2. jcsd
  3. Nov 25, 2014 #2

    mfb

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    You can treat the degenerate states like different states with a very similar energy (they just happen to have a difference of exactly zero here), I think. The real states are not degenerate either.

    5 pm where? In my time zone, you posted the thread after 5 pm ;).
     
  4. Nov 25, 2014 #3
    So for my n in my energy equation, I can just use the degeneracy?

    and i've edited my post to include time zone, thanks!
     
  5. Nov 25, 2014 #4
    So, I've rewritten the partition function using what I think is correct regarding the degeneracy.

    Z= e-6.803(1-1/9)/KT + e-6.803(1-1/4)/KT + e-6.803(1-1/1)/KT
    = e-6.047/KT + e-5.102/KT + e0
    = (4)(1+ e-6.047/KT + e-5.102/KT)

    I multiplied the partition function by 4 since it's four-fold degenerate.

    am I at least on the right track?
     
    Last edited: Nov 25, 2014
  6. Nov 25, 2014 #5

    mfb

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    I don't see where you used the degeneracy now.
     
  7. Nov 25, 2014 #6
    Sorry, I should've multiplied the partition function by 4 since it's four-fold degenerate. Is that correct?
     
  8. Nov 25, 2014 #7

    mfb

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    4? One specific part of it, yes.
     
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