A simple fundamental equation, which exhibits some of the thermodynamic properties of solids is:

U(s, v) = Ae^{b(v-v_0)^2}S^4/3e^S/3R

U, s and v are the specific energy, entropy, and volume per mole. A, v₀ and b are constants characteristic of solid. R is the idea gas constant.

a) Show that Cv, the specific heat at constant volume is proportional to T³ at low temperatures. This is the Debye law.

b) Show that Cv goes to 3R in the high temperature limit (the Dulong Petit Law).

c) Show that the thermal expansion coefficient of the solid vanishes at 0 pressure.