How do I solve this?

The Big Bang lled the universe with "cosmic background\" radiation that acts as a thermal bath Today the temperature of the bath is 2.7K. Imagine that an atom exists today (To : 2.7K) with 3 energy levels like this: HN The energy spacing betwcn the lowest two levels is (5. The energy spacing between the lst and 3rd levels is A : 10006. Assume that the ground state energy E1 : 0 and that the atom is always in thermal equilibrium with the universe and by chance the temperature of the universe today is almost. exactly the energy difference 6 (i.e. To 2 6/165). (a) (3 points) Write an expression for the probability of nding the atom in the highest level. (13) (3 points) W hat is the average energy of the atom in terms of 6? (e) (2 points) What value of 112 / n1 do we predict? [Asidez Note that astronomers can measure the relative number of atoms in a gas cloud that are in level 1 (m) and level 2 (mg) by observing the atom with a background source of light (a distant quasar7 perhaps), since atoms making transitions from 1 > 3 and from 2 > 3 absorb photons at slightly different energies] (d) (2 points) Because the Universe is expanding, at earlier times the temperature of the Coernic Background was much higher. At what temperature T' (in terms of 5) was the probability for nding an atom in level 3 equal to 1/6