# Partition Function for a helium atom

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1. Feb 9, 2016

### 1v1Dota2RightMeow

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. Feb 10, 2016

### 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.

3. Feb 10, 2016

### 1v1Dota2RightMeow

DrClaude, thank you for responding. What exactly is the ''population'' of a ground state of an atom?

4. Feb 10, 2016

### Staff: Mentor

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.

5. Feb 10, 2016

### 1v1Dota2RightMeow

Wouldn't the relative population then be $$P2/P1 = 90/10 = 9$$

6. Feb 11, 2016

### 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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