the Dulong-Petit rule

Here is a surprising regularity: weigh out one mole of almost any solid element — iron, copper, silver, lead — and near room temperature each needs about the same heat, roughly 25 joules, to warm by one degree. Two French scientists, Pierre Dulong and Alexis Petit, spotted this in 1819. It is nature quietly telling us that heat in a solid is stored atom by atom, and every atom stores about the same amount.

The rule states that the molar heat capacity Cv is approximately 3R, where R is the gas constant 8.314 J/mol-K, so 3R is about 24.9 J/mol-K. The factor of 3 comes from each atom being able to vibrate in three independent directions (x, y, z); classical physics gives each vibrational direction an average energy worth of storage (kinetic plus potential), so each atom contributes 3 units and a mole contributes 3R. This also explains why heavy atoms have small specific heat per kilogram: a mole of lead weighs far more than a mole of aluminum, so the same 25 J/mol spread over more mass gives fewer J/kg.

Be honest about the limits: the rule is only the high-temperature ceiling. Cool a solid down and its heat capacity falls, dropping toward zero as the temperature approaches absolute zero — something classical physics simply cannot explain. Light, stiffly bonded solids like diamond fall short of 3R even at room temperature. Explaining the drop needs quantum theory (first Einstein, then Debye) and the idea of the Debye temperature.