Physics · Ch 13 — Kinetic Theory

Specific Heat Capacity of Solids

13.6.4

Specific Heat Capacity of Solids

The kinetic theory of gases gives us a molecular picture of heat capacity, but the same ideas can be extended — with important modifications — to solids. In a solid, atoms are not free to wander; they are locked into a crystal lattice, each atom vibrating about a fixed equilibrium position. The energy of the solid is essentially the vibrational energy of these atoms.

Consider a solid made of $N$ atoms. Each atom can vibrate in three independent directions (along the $x$ , $y$ , and $z$ axes). According to the equipartition theorem, each vibrational degree of freedom contributes two quadratic terms to the energy: one for kinetic energy ( $\frac{1}{2}k_BT$ ) and one for potential energy ( $\frac{1}{2}k_BT$ ). So each vibrational mode contributes a total of $k_BT$ to the internal energy.

Since each atom has three vibrational degrees of freedom, the total internal energy $U$ of the solid is:

U = N \times 3 \times k_B T = 3N k_B T

For one mole of the solid, $N = N_A$ (Avogadro's number). Therefore, the molar internal energy is:

U = 3 N_A k_B T = 3 R T

where $R = N_A k_B$ is the universal gas constant.

The molar specific heat capacity at constant volume, $C_V$ , is defined as the rate of change of internal energy with temperature:

C_V = \left(\frac{dU}{dT}\right)_V

For the solid, since $U = 3RT$ , we get:

C_V = \frac{d}{dT}(3RT) = 3R

C_V = 3R \approx 24.94 \text{ J mol}^{-1} \text{K}^{-1}

This is the Dulong–Petit law, stated in 1819: the molar specific heat capacity of all solids at constant volume is approximately $3R$ , or about $25 \text{ J mol}^{-1} \text{K}^{-1}$ .

Note

The Dulong–Petit law is a high-temperature limit. At room temperature, many solids obey it quite well, but at very low temperatures, $C_V$ drops toward zero — a fact that classical physics could not explain. That required quantum theory (Einstein's and Debye's models).

Experimental Verification and Deviations

The table below shows the measured molar specific heat capacities of some common solids at room temperature (around 300 K). The values are remarkably close to $3R$ , confirming the classical prediction for most elements.

Element	Atomic Mass (u)	Specific Heat (J g⁻¹ K⁻¹)	Molar Specific Heat (J mol⁻¹ K⁻¹)
Aluminium	27.0	0.900	24.4
Carbon	12.0	0.507	6.1
Copper	63.5	0.386	24.5
Lead	207.2	0.128	26.5
Silver	107.9	0.236	25.5
Tungsten	184.0	0.134	24.9

Note

Carbon is an exception. Its molar specific heat capacity (6.1 J mol⁻¹ K⁻¹) is far below the Dulong–Petit prediction of $3R \approx 24.94$ J mol⁻¹ K⁻¹ — carbon is one of the clearest exceptions to the law at room temperature. This happens because carbon's atoms are bound very rigidly by strong covalent bonds, giving carbon an unusually high Debye temperature; at ordinary room temperature, carbon is still well below its own Debye temperature, so many of its vibrational modes remain quantum-mechanically "frozen out" and do not yet contribute their full classical share of energy. Only at much higher temperatures does carbon's molar specific heat rise toward the Dulong–Petit value of $3R$ .

Watch out

| The specific heat capacity given in tables is often the specific heat capacity (per gram), not the molar specific heat capacity. To compare with $3R$ , you must multiply the specific heat by the atomic mass to get the molar value. For example, for aluminium: $0.900 \text{ J g}^{-1} \text{K}^{-1} \times 27.0 \text{ g mol}^{-1} \approx 24.3\text{-}24.4 \text{ J mol}^{-1} \text{K}^{-1}$ (the small difference from the table's 24.4 is rounding in the atomic mass used), which is very close to $3R \approx 24.94 \text{ J mol}^{-1} \text{K}^{-1}$ .

Why the Dulong–Petit Law Works (and When It Fails) …

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