Dulong–Petit law

The empirical rule that many crystalline solids approach a molar heat capacity of three times the gas constant at sufficiently high temperatures. It follows classically from equipartition of energy among atomic vibrations.

Equipartition theorem: A classical statistical-mechanics result assigning equal average energy to each quadratic degree of freedom at thermal equilibrium. Each atom’s three-dimensional vibrations contribute six quadratic terms, yielding an average energy of 3RT per mole.
Pierre Louis Dulong: A French chemist and physicist who studied the thermal properties of matter. He formulated the heat-capacity rule with Alexis Petit.
Einstein model of a solid: A quantum model that treats atoms in a solid as identical oscillators with one characteristic frequency. Einstein’s model explains the low-temperature decline that the classical law cannot.
Atomic mass determination: The experimental determination of an element’s atomic mass relative to a standard. Before accurate mass-spectrometric methods, the law helped distinguish plausible atomic weights.
Lattice vibration: The collective oscillatory motion of atoms in a crystal around their equilibrium positions. The law treats these motions as classical harmonic oscillators.
Alexis Thérèse Petit: A French physicist who investigated heat and the thermal properties of solids. His collaboration with Dulong produced the empirical law in 1819.
Debye model: A quantum theory of solid heat capacity that approximates lattice vibrations by a spectrum of acoustic modes. Debye’s model improves on Einstein’s and predicts the observed low-temperature cubic dependence.
Molar heat capacity: The heat required to raise the temperature of one mole of a substance by one degree. The law’s characteristic value is approximately 3R per mole per kelvin.
Harmonic oscillator: A system whose restoring force is proportional to displacement from equilibrium. A crystal’s small-amplitude atomic vibrations are approximated as harmonic oscillators.
Dulong–Petit law (1819 paper): The 1819 publication in which Dulong and Petit reported their rule for the specific heats of solid elements. The paper established the experimental pattern before its classical theoretical explanation.

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