Calculating Electron Distribution in Hydrogen Atoms

[Rothiemurchus]

How do I calculate the number of electrons with a given energy in
a sample of hydrogen atoms at a certain temperature?
For example how many electrons would be in the n=2 level
at 3 Kelvin? And does it matter whether or not hydrogen is gas or liquid or solid i.e is a different calculation needed for a gas in place of a liquid.

 

[zefram_c]

This is a classical problem; you weigh each possible state by the Boltzmann factor e^(-E/kT), write the partition function and calculate the probability of a given energy level. At 3K, the probability of electrons being in n=2 is vanishingly small.

This only applies to a gas; in a liquid the interactions between atoms cannot be ignored and make for a messier problem.

 

[Rothiemurchus]

When you say vanishingly small, have you got an approximate number
of decimal places for that small probability?

 

[zefram_c]

Easy to get. The partition function is dominated by the ground state so we can approximate:

Z = exp (-E₁/kT).

P(n=2) = g * exp (-E₂/kT) / Z, where g is the degeneracy of the 2nd level (ie how many distinct states there are with that energy level). Ignoring fine structure, g=4.

So P(n=2) = 4 exp (-(E₂ - E₁) / kT) = 4 e ^{-39 455.3202} = 4 * 10 ^-17135

Hmm... that's even smaller than I expected. Did I do something wrong?

 

[Rothiemurchus]

E1 = 10^-18 Joules.
P(n=2) = 4 exp (-(E2 - E1) / kT) = 4 exp [(4 x 10^-18 - 1 x 10^-18)/10^-23]

works out at about 10^10^-18

There must be something wrong with the equation.