Hydrogen ground state/excited states due to temperature problem

[emtilt]

I apologize if this is the incorrect forum for this problem; I was unsure which would be most suitable because the problem is from a low level astrophysics class but is not precisely or exclusively astrophysics.

Homework Statement

For a gas of neutral hydrogen atoms, at what temperature will equal numbers of atoms have electrons in the:
A) ground state (n=1) and in the first excited state (n=2)?
B) ground state and second excited state (n=3)?

(I'm assuming that this problem is dealing with the Bohr atom, not something more complex.)

Homework Equations

Perhaps the Boltzmann distribution ($$\frac{n_j}{n_i}=\frac{g_j}{g_i}e^{\frac{-(E_j-E_i)}{kT}}$$) with the Balmer formula to get the enrgies? But then where do I get the statistical weights?


I really do not know how to do this problem, or what equations to use to relate the temperature to the electron states. Any help is appreciated.

 

[dynamicsolo]

Homework Statement

For a gas of neutral hydrogen atoms, at what temperature will equal numbers of atoms have electrons in the:
A) ground state (n=1) and in the first excited state (n=2)?
B) ground state and second excited state (n=3)?

(I'm assuming that this problem is dealing with the Bohr atom, not something more complex.)

IIRC, it will be a little more complex than the Bohr atom, but you may able to deal with this from something you learned in chemistry...

Homework Equations

Perhaps the Boltzmann distribution ($$\frac{n_j}{n_i}=\frac{g_j}{g_i}e^{\frac{-(E_j-E_i)}{kT}}$$) with the Balmer formula to get the enrgies? But then where do I get the statistical weights?

I believe this gets called the Saha equation in astrophysics, but it basically comes from the partition function concept in statistical mechanics.

I should check this to be sure, but we are dealing with hydrogen, so there is only one electron per atom. Recall what you learned of atomic orbitals in chemistry. In the "ground state" (n=1 -- the 1s-shell), how many "slots" are there for an electron? How many "slots" are there at n=2 altogether (what orbitals are available)? And what of n=3? Those will be your statistical weights g_sub-i at each energy level. (It's not quite so simple for other atoms...)

As for the energies E_sub-i, what are the energy levels of the orbitals according to the Bohr model? (Those work OK in solving this problem for hydrogen...)

 

[KaiserBrandon]

I'm doing the exact same question as you, except only the second part, we have the first part for n=1 and n=2 as an example. Basically, using the formula you gave, and also the formulas gn=2n^2, and En= -13.6/n^2, you can figure out the temperature since Nj=Ni. At least that's how we were taught to do it.