Temperature Calculation for a Distant Star with Thermal Excitation

[Vitani11]

Homework Statement

The spectrum of a distant star shows that one in 3x10⁶ of the atoms of a particular element in its first excited state 7.3 eV above the ground state. What is the temperature of the star? (You can ignore the other excited states and assume the ratio of statistical weights is 4)

Homework Equations

(N₂/N₁) = e^{-(E₂-E₁)/K_bT)}

The Attempt at a Solution

All variables are given to plug into the equations but it won't work.
Kb is Boltzmann constant. I used the difference in energies as 7.3 eV. I used N₂ / N₁ as 4. I know how to solve log equations. I don't understand what else needs to be done here though. I've also tried to divide the final answer by 3x10⁶ because I figured that would scale this down to one atom instead of many, but that is not correct.

 

[DrClaude]

What does "statistical weight" mean? Can N₂ be greater than N¹? What does 3×10⁶ represent?

 

[Vitani11]

Statistical weight is the relative probability of a particular feature/state. Statistical weight is the ratio of N₂ / N₁? I'm honestly not sure whether N₂ can be greater then one. If there are less excited states than ground states then I could see why this would be true. Why is this true? I think that N₂ / N₁ should be the ratio 1/3x10⁶ because this gives the number of excited atoms out of the total amount - but then there is a factor of 4. What am I supposed to do with that?

 

[DrClaude]

The statistical weight is the degeneracy factor. It tells you how many states correspond to N₂. It goes with the Boltzmann factor, see https://en.wikipedia.org/wiki/Maxwell–Boltzmann_statistics