Consider a system of hydrogen atoms in thermal equilibrium. A hydrogen atom has degenerate energy levels En = -13.6 eV/n^2 . In fact there are 2 states with energy E1 and 8 states with energy E2.

(a) At room temperature, what fraction of atoms are in a particular excited state ψ2 with n = 2 compared to atoms in a particular ground state with ψ1 with n = 1?

(b) At what temperature will the average number of atoms with energy E2 be 1% of the number of atoms with energy E1? Remember about the degeneracy; an atom with energy E2 or E1 could be in one of several states].

(c) Is it possible (at any finite temperature) to have the average number of atoms in a particular state ψ2 with energy E2 equal to the average number of atoms in a particular state ψ1 with energy E1? If so, determine that temperature. If not, explain why not.

(d) Is it possible (at any finite temperature) to have the average number of atoms in all states with E2 equal to the average number of atoms in all states with n = 1? If so, determine that temperature. If not, explain why not.