Show, for the two-level system of Problem 5.3-8, that as the temperature increases from zero to infinity the energy increases from N~εu to N~(εu+εd)/2. Thus, at zero temperature all atoms are in their ``ground state'' (with energy εu), and at infinite temperature the atoms are equally likely to be in either state. Energies higher than N(εu+εd)/2 are inaccessible in thermal equilibrium! (This upper bound on the energy is a consequence of the unphysical oversimplification of the model; it will be discussed again in Section 15.3.)
Show that the Helmholtz potential of a mixture of simple ideal gases is the sum of the Helmholtz potentials of each individual gas.