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Specific heat of solids
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Dulong–Petit law and Debye model.
C = 3RClassically each atom has 6 quadratic terms, so C = 3R ≈ 25 J/mol·K (Dulong–Petit), valid at high T. At low temperatures C falls, which classical theory couldn't explain.
Copper 24.5 J/mol·K at room temperature.
Diamond is an exception at room T.
In more detail
At high temperatures, each vibrational mode of a solid contributes kT to its energy, giving a molar heat capacity of about 3R, the Dulong-Petit law. At low temperatures, quantum effects reduce it.
Real-world uses
- Estimating heat capacities of metals at room temperature.
- Engineers use heat capacities in thermal design of solids.
- Engineers estimated the heat capacity of solids using the Dulong–Petit law before quantum effects were understood.
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Frequently asked questions
What is Dulong-Petit?
The approximately constant heat capacity of solids at high temperature.
Why does it fail at low temperature?
Because quantum vibrational modes freeze out and cannot store energy.
What does 3R per mole mean?
Each atom in a solid vibrates in three directions, so it stores about three times the gas constant in heat capacity.
Related topics in Kinetic Theory & Statistical Physics
Bose–Einstein statistics · Bose–Einstein condensate · Fermi energy · Debye model · Photon gas · Planck distribution