Chapter 3.2, Problem 14P

Experimental measurements of the heat capacity of aluminum at low temperatures (below about 50 K) can be fit to the formula $C_V=aT+b{T}^3$, where $C_V$ is the heat capacity of one mole of aluminum, and the constants a and b are approximately $\text{a}=0.00\text{135 J}/{\text{K}}^{\text{2}}$ and $\text{b}=\text{2}.\text{48}\times \text{1}{0}^{-\text{5}}\text{J}/{\text{K}}^{\text{4}}$. From this data, find a formula for the entropy of a mole of aluminum as a function of temperature. Evaluate your formula at $\text{T}=\text{1 K}$ and at $\text{T}=\text{1}0\text{ K}$, expressing your answers both in conventional units (J/K) and as unitless numbers (dividing by Boltzmann’s constant). [Comment: In Chapter 7 I’ll explain why the heat capacity of a metal has this form. The linear term comes from energy stored in the conduction electrons, while the cubic term comes from lattice vibrations of the crystal.]