Which constants to use in Saha Equation

[leroyjenkens]

I'll post a picture of the equation and a picture of the problem I'm being asked. My problem is I can't get the units to cancel. This is a ratio, so the final answer is supposed to be unitless. The exponential term obviously needs the units to cancel, so I chose the correct Boltzmann constant for the units to cancel in that one, so I think I'm good there. The first term with the P_e in the denominator always gives me meters cubed as the units. So if I can get inverse meters cubed from the second term, then I'd be ok and all the units would be cancelled. But that crazy 2nd term with the 3/2 exponent gives me a crazy result for units, no matter what I do. I've tried expressing electron mass in both kilograms and electronvolts, and I've tried expressing Planck's constant in both electronvolts and in Joules, and no matter what I do, I end up with units like this: K^3/2s⁶kg^-3/2m^-6
Stuff like that. That's no where near being able to cancel, no matter what I do.
So even ignoring the fact that the units don't cancel, I still don't get the result that's in the solution manual.
If anyone here has had experience with the Saha equation, you've probably done something like this and dealt with this problem before. It seems unavoidable.
Thanks.

 

[Chris B]

It's been a while since I had Stellar Astrophysics, but let's see:

mass of the electron should be in eV/c² (I like eV better than SI units)
k is eV/K
T is K
h is eV⋅s

so that's eV/c²⋅eV/K⋅K/(eV⋅s)² all to the 3/2

looks like everything cancels except c² and s², but c²=m²/s² so all you're left with is (m²)^3/2, which is just meters cubed. But, P_e is a density right? So, that is something per meters cubed which will cancel... I think. But, I don't think I made any mistakes in there.

 

[Chris B]

Sorry it's (1/(m²))^3/2

 

[leroyjenkens]

Sorry it's (1/(m²))^3/2

Thanks a lot. Turns out I wrote down the equation wrong, so every time I referenced the equation to see if I had it right, I kept looking at the wrong one that didn't include k in the numerator.
Thanks.