Partition Function for Positronium

1. Nov 25, 2014

wilsonaj4

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)

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. Nov 25, 2014

Staff: Mentor

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 ;).

3. Nov 25, 2014

wilsonaj4

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!

4. Nov 25, 2014

wilsonaj4

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
5. Nov 25, 2014

Staff: Mentor

I don't see where you used the degeneracy now.

6. Nov 25, 2014

wilsonaj4

Sorry, I should've multiplied the partition function by 4 since it's four-fold degenerate. Is that correct?

7. Nov 25, 2014

Staff: Mentor

4? One specific part of it, yes.