Debye's T³

Plate 60 speaks for itself: “Forget heat.”

The Physics

At room temperature a solid's heat capacity is about 3R per mole (Dulong–Petit). Far below the Debye temperature only long-wavelength vibrations are excited and the heat capacity falls as T³.

C_V = (12π⁴/5) N k_B (T/Θ_D)³

• C_V: heat capacity at constant volume, J K⁻¹
• N: number of atoms, dimensionless
• k_B: Boltzmann constant, J K⁻¹
• T: temperature, K
• Θ_D: Debye temperature (Cu ≈ 343 K), K
• R: gas constant 8.314 J mol⁻¹ K⁻¹

Room temperature: about 3R per mole. Far below Θ_D it falls as T³.

Honest caveat: In metals the electrons add a term linear in T that wins at the lowest temperatures.

Sources

• P. Debye, Ann. Phys. 344, 789 (1912).
• C. Kittel, Introduction to Solid State Physics, 8th ed., Ch. 5.

On the poster

Plate 060, “Debye's T³”: the pixel-art plate with its slogan “Forget heat.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.