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Partition function/probability

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Consider the lowest 3 energy levels of a hydrogen atom:

Ground state energy : -13.6 eV degeneracy = 0
First excited state energy : -3.4 eV degeneracy = 4
Second excited state energy: -1.5 eV degeneracy = 9

Ignore higher states.

a) Estimate the partition function Z for H-atom at 5800 K. Do not forget to take degeneracy into account.
b) Calculate the probability of finding an H-atom in the first excited state.
c) How does the result in part (b) compare with the same at 300 K?

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Let's first evaluate 1 eV/(kT) at T = 5800 K. I find that:

1 eV/(k* 5800 K) = 2.

So, the partition function is:

Z = Exp[13.6 eV/(kT)] + 4*Exp[3.4 eV/(kT)] + 9*Exp[1.5 eV/(kT)] =
Exp[2*13.6] + 4*Exp[2*3.4] + 9*Exp[2*1.5] = 6.5*10^11

The factors in front of ...

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