Eigenvalues and Boltzmann statistics

Problem:

Due to the presence everywhere of the cosmic background radiation the minimum possible temperature of a gas in interstellar space is 2.7 K.  This implies that a significant fraction of molecules in space may be in low-level excited states.  Consider a hypothetical molecule with one possible excited state.  What would the excitation energy have to be for 20% of the molecules to be in the excited state?

Solution:

Problem:

Consider a particle in a simple two level system with eigenstates ψ1 and ψ2 such that
1 = E1ψ1 and Hψ2 = E2ψ2.  Derive an expression for the specific heat per particle cV = ∂<E>/∂T of the system as a function of temperature.

Solution: