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# Dulong–Petit law

The Dulong–Petit law, a chemical law proposed in 1819 by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the molar specific heat capacity of a crystal. Experimentally the two scientists had found that the heat capacity per weight (the mass-specific heat capacity) for a number of substances became close to a constant value, after it had been multiplied by number-ratio representing the presumed relative atomic weight of the substance. These atomic weights had shortly before been suggested by Dalton.

In modern terms, Dulong and Petit found that the heat capacity of a mole of many solid substances is about 3R, where R is the modern constant called the universal gas constant. Dulong and Petit were unaware of the relationship with R, since this constant had not yet been defined from the later kinetic theory of gasses. The value of 3R is about 25 joules per Kelvin.

The modern theory of the heat capacity of solids states that it is due to lattice vibrations in the solid, and was first derived in crude form from this assumption by Albert Einstein, in 1907. The Einstein solid model thus gave for the first time a reason why the Dulong–Petit law should be stated in terms of the classical heat capacities for gases.

## Equivalent forms of statement of the law

An equivalent statement of the Dulong–Petit law in modern terms is that, regardless of the nature of the substance or crystal, the specific heat capacity c of a solid substance (measured in joule per kelvin per kilogram) is equal to 3R/M, where R is the gas constant (measured in joule per kelvin per mole) and M is the molar mass (measured in kilogram per mole). Thus, the heat capacity per mole of many solids is 3R.

Thus, in modern form the Dulong–Petit law is:

c M = 3R or c = 3R/M

Dulong and Petit did not state their law in terms of the gas constant R (which was not then known). Instead, they measured the values of heat capacities (per weight) of substances and found them smaller for substances of greater "atomic weight" as inferred by Dalton. Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity (which would now be the heat capacity per mole in modern terms) was nearly constant, and equal to a value which was later recognized to be 3 R.

In other modern terminology, the dimensionless heat capacity is equal to 3.

## Application limits

Despite its simplicity, Dulong–Petit law offers fairly good prediction for the specific heat capacity of many solids with relatively simple crystal structure at high temperatures. This is because in the classical theory of heat capacity the heat capacity of solids approaches a maximum of 3R per mole of atoms, due to the fact that full vibrational-mode degrees of freedom amount to 3 degrees of freedom per atom each corresponding to a quadratic kinetic energy term and a quadratic potential energy term. By the equipartition theorem, the average of each quadratic term is 1/2 kT, or 1/2 RT per mole (see derivation below). Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity.

The Dulong-Petit law fails at room temperatures for light atoms bonded strongly to each other, such as in metallic beryllium, and in carbon as diamond. Here, it predicts higher heat capacities than are actually found, with the difference due to higher-energy vibrational modes not being populated at room temperatures in these substances.

In the very low (cryogenic) temperature region, where the quantum mechanical nature of energy storage in all solids manifests itself with larger and larger effect, the law fails for all substances. For crystals under such conditions, the Debye model, an extension of the Einstein theory that accounts for statistical distributions in atomic vibration when there are lower amounts of energy to distribute, works well.

## Derivation

A system of vibrations in a crystalline solid lattice can be modelled by considering harmonic oscillator potentials along each degree of freedom. Then, the free energy of the system can be written as $F=N\varepsilon_0+k_BT\sum_\alpha \log\left(1-e^{-\hbar\omega_{\alpha}/k_BT}\right)$

where the index α sums over all the degrees of freedom. In the 1907 Einstein model (as opposed to the later Debye model) we consider only the high-energy limit: $k_BT\gg\hbar\omega_\alpha. \,$

Then $1-e^{-\hbar\omega_\alpha/k_BT} \approx \hbar\omega_\alpha/k_BT. \,$

and we have $F=N\varepsilon_0+k_BT\sum_{\alpha}\log\left(\frac{\hbar\omega_{\alpha}}{k_BT}\right)$

Define geometric mean frequency by $\log\bar{\omega}=\frac{1}{M}\sum_\alpha \log\omega_\alpha,$

where M measures the total number of degrees of freedom of the system.

Thus we have $F=N\varepsilon_0-Mk_BT\log k_BT+Mk_BT\log\hbar\bar{\omega} \,$

Using energy $E=F-T\left(\frac{\partial F}{\partial T}\right)_V,$

we have $E=N\varepsilon_0+Mk_BT. \,$

This gives specific heat $C_V=\left(\frac{\partial E}{\partial T}\right)_V=Mk_B,$

which is independent of the temperature.

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