Mean energy behavior as a function of T.

[quasar987]

Here's a problem the TA made but now that I look back at it, I wonder how he did it.

A system contains N weakly interacting particles, each of which can be in either one of two states of respective energies $\epsilon_1$ and $\epsilon_2$ with $\epsilon_1<\epsilon_2$.

a) With no explicit computation, draw a qualitative representation of the mean energy $\bar{E}$ of the system as a function of the temperature T. What happens to $\bar{E}$ in the limit of very small and very large temparatures? Approximately at which value of T does $\bar{E}$ changes from its low temperature limit to its high temperature limit?

He drew a curve that starts at T=0 with $\bar{E}(0)=N\epsilon_1$, rises up, appears to have an inflexion point at $(\epsilon_2-\epsilon_1)/k$, and approaches $N(\epsilon_1+\epsilon_2)/2$ as $T\rightarrow +\infty$.

I just really don't know how he knows all that. The only relation btw T and E I know is really not helpful:

$$\frac{1}{kT}=\frac{\partial \ln(\Omega)}{\partial E}$$

There's one in term of the partition function too but we're not allowed to calculate it.