Dulong–Petit law

The law states that the heat capacity of a crystalline elemental solid approaches 3R per mole of atoms when its temperature is high enough. Here, R is the molar gas constant, whose exact SI value is 8.314462618 J mol−1 K−1, so 3R is approximately 24.94 J mol−1 K−1.² Pierre-Louis Dulong and Alexis-Thérèse Petit reported the relationship in 1819 after measuring the specific heats of solid elements and combining them with their estimated atomic weights.¹

The result was striking because it linked a bulk thermal property to atomic mass and helped chemists improve atomic-weight estimates. Its strongest form applies to crystalline elemental solids, and the relevant quantity is usually the heat capacity per mole of atoms rather than per mole of an arbitrarily chosen compound.

The classical explanation is that each atom in a crystal participates in three independent vibrational directions, producing three pairs of quadratic energy terms. The equipartition theorem assigns an average energy of kB T to each direction pair, giving 3kB per atom and therefore 3R per mole of atoms. This argument makes the law independent of atomic mass, even though mass affects the frequencies of the vibrations.

The law concerns a macroscopic heat capacity, not a universal exact constant. Measurements may refer to C_V, heat capacity at constant volume, or C_P, heat capacity at constant pressure. In solids they are often close, but thermal expansion and elastic properties make them distinguishable. The classical prediction became a benchmark for theories of lattice dynamics.

The law fails at low temperature because crystal vibrations are quantized and high-frequency modes are not thermally excited. Albert Einstein introduced a quantum model in 1907 in which all atoms vibrate at one characteristic frequency; it correctly produced a falling heat capacity but oversimplified the spectrum. Peter Debye’s 1912 model replaced that single frequency with a continuum of acoustic modes and predicted the characteristic low-temperature proportionality C_V ∝ T³ for many insulating crystals.

The crossover is described using a material’s Debye temperature. Above it, most vibrational modes are populated and the Dulong–Petit value is approached; below it, progressively fewer modes contribute. The approach can also be altered by electronic, magnetic, structural, or anharmonic contributions.

The law is most useful as a high-temperature limit and as a diagnostic, not as a precise room-temperature rule for every element. Diamond is a famous exception near ordinary temperatures because its stiff carbon lattice has very high vibrational frequencies; its heat capacity rises toward 3R only at much higher temperatures. Light elements and strongly bonded crystals can show similarly delayed convergence.

For a compound containing n atoms in each formula unit, the classical lattice contribution is approximately 3nR per mole of formula units, provided all relevant modes are excited. This generalization explains why “per mole” must be specified carefully. The law also helped expose the limits of classical physics: its low-temperature failure was among the thermal anomalies that motivated the quantum treatment of solids, later supported by spectroscopy and low-temperature calorimetry.

Molar heat capacity

The heat required to raise the temperature of one mole of a substance by one kelvin, under specified conditions.

Equipartition theorem

A classical statistical-mechanical result assigning an average energy of one-half kB T to each independent quadratic term in the energy.

Debye temperature

A characteristic temperature that represents the upper scale of a crystal’s acoustic-vibration spectrum and helps determine when quantum effects become important.

C_V and C_P

Heat capacities measured at constant volume and constant pressure, respectively.