According to the Dulong–Petit law, the molar heat capacity of many solid elements near room temperature is about 3R (R is the gas constant, 8.31 J/(mol·K)). Estimating the specific heat per unit mass of aluminum (molar mass 26.98 g/mol) with this law, which value is closest?

Answer: C. About 0.92 J/(g·K)

Atoms in a solid vibrate in three directions, and the vibration in each direction holds kT/2 each as kinetic and potential energy, for a total of 3kT per atom, so the molar heat capacity is 3R ≈ 24.9 J/(mol·K). Dividing by the molar mass gives 24.9/26.98 ≈ 0.92 J/(g·K), close to the measured value for aluminum of about 0.90 J/(g·K). About 0.31 J/(g·K) drops the factor of 3 and divides R alone, and about 0.46 J/(g·K) uses (3/2)R, the constant-volume molar heat capacity of a monatomic gas, which leaves out the potential energy of the vibrations. Because the molar heat capacities are nearly the same, lighter elements have larger specific heats per unit mass.