Using the notion of entropy, show that two systems that can freely exchange energy reach the same temperature. Show that the energy of a system increases with temperature.

A system consists of $N$ distinguishable, non-interacting spin $\frac{1}{2}$ atoms in a magnetic field, where $N$ is large. The energy of an atom is $\varepsilon>0$ if the spin is up and $-\varepsilon$ if the spin is down. Find the entropy and energy if a fraction $\alpha$ of the atoms have spin up. Determine $\alpha$ as a function of temperature, and deduce the allowed range of $\alpha$. Verify that the energy of the system increases with temperature in this range.