Serway & Jewett — The Kinetic Theory of Gases: Problem 28.

In a sample of a solid metal, each atom is free to vibrate about some equilibrium position. The atom's energy consists of kinetic energy for motion in the \(x\), \(y\), and \(z\) directions plus elastic potential energy associated with the Hooke's law forces exerted by neighboring atoms on it in the \(x\), \(y\), and \(z\) directions. According to the theorem of equipartition of energy, assume the average energy of each atom is \(\frac{1}{2}k_B T\) for each degree of freedom. (a) Prove that the molar specific heat of the solid is \(3R\). The Dulong–Petit law states that this result generally describes pure solids at sufficiently high temperatures. (You may ignore the difference between the specific heat at constant pressure and the specific heat at constant volume.) (b) Evaluate the specific heat \(c\) of iron. Explain how it compares with the value listed in Table 19.1. (c) Repeat the evaluation and comparison for gold.