Dulong–Petit law
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The Dulong–Petit law, a chemical law proposed in 1819 by French physicists and chemists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the specific heat capacity of a crystal due to its lattice vibrations.
The result states that regardless of the nature of the crystal, the specific heat capacity (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). In other words, the dimensionless heat capacity is equal to 3.
Despite its simplicity, Dulong–Petit law offers fairly good prediction for the specific heat capacity of solids with relatively simple crystal structure at high temperatures. It fails, however, in the low temperature region, where the quantum mechanical nature of the solid manifests itself. There, the Debye model works well.
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[edit] 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[1]
Where the index α sums over all the degrees of freedom. Suppose we consider the limit 
. Then, 
and we have
Define geometric mean frequency by
, where M measures the total number of degrees of freedom of the system.
Thus we have 
Using energy 
, we have 
This gives specific heat 
, which is independent of the temperature
[edit] See also
[edit] References
- ^ Landau, L. D.; Lifshitz, E. M. (1980). Statistical Physics Pt. 1. Course in Theoretical Physics. 5 (3rd ed.). Oxford: Pergammmon Press. p. 193,196. ISBN 0750633727.
[edit] External links
- Petit A.-T., Dulong P.-L.: Recherches sur quelques points importants de la Théorie de la Chaleur. In: Annales de Chimie et de Physique 10, 395-413 (1819) (Translation)
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