= Dulong-Petit law
{c}
{wiki=Dulong–Petit_law}
In the classical high-<temperature> limit, each independent <harmonic oscillator> contributes $k_B$ to the <heat capacity at constant volume>, where $k_B$ is the <Boltzmann constant>. A three-dimensional crystalline <solid> with $N$ <chemical atoms> has $3N$ vibrational <normal modes> in the bulk harmonic approximation. Its vibrational <internal energy> and <heat capacity at constant volume> therefore obey
$$
U_{\rm vib}=U_0+3Nk_BT,\qquad
C_V=\left(\frac{\partial U_{\rm vib}}{\partial T}\right)_{V,N}=3Nk_B.
$$
Here $U_0$ is independent of <temperature> at fixed <volume (thermodynamics)>. Thus the <Dulong-Petit law> predicts $C_V\simeq3Nk_B$ for a real <solid> in this regime. Per <mole> of <chemical atoms>, rather than per <mole> of a compound's formula units,
$$
C_{V,\rm mol}\longrightarrow3R\simeq24.9434\,\mathrm{J\,mol^{-1}\,K^{-1}},\qquad R=N_Ak_B,
$$
with <molar gas constant> $R$ and <Avogadro constant> $N_A$. A formula unit containing $r$ <chemical atoms> consequently has a classical vibrational prediction of $3rR$ per <mole> of formula units.
For example, in this ideal classical model, one <mole> of <chemical atoms> heated at fixed <volume (thermodynamics)> from $300$ to $600\,\mathrm K$ gains vibrational <internal energy>
$$
\Delta U_{\rm vib}=3R(600-300)\,\mathrm K\simeq7483.02\,\mathrm J.
$$
The <heat capacity at constant volume> stays the same even though the thermal part of the <internal energy> doubles. These <temperatures> illustrate the stipulated classical model; they do not assert that every real <solid> is classical at $300\,\mathrm K$.
\Image[/dulong-petit-classical-energy.png]
{title=Classical <equipartition theorem>: one <harmonic oscillator> stores $k_BT$, one <chemical atom> stores $3k_BT$}
{description=Mean <kinetic energy> and <potential energy> at $300$ and $600\,\mathrm K$, in units of $k_B(300\,\mathrm K)$. Each independent quadratic term contributes $k_BT/2$. The three-direction model has three kinetic and three potential terms per <chemical atom>; doubling <temperature> doubles both contributions.}
The proof outline is the <equipartition theorem>: one classical <harmonic oscillator> has a quadratic <kinetic energy> and a quadratic <potential energy>, each with mean $k_BT/2$. Their sum is $k_BT$, whose <derivative> with respect to <temperature> is $k_B$. Summing over $3N$ <normal modes> gives $3Nk_B$. The classical result does not require equal <angular frequencies>: each harmonic <normal mode> contributes the same amount once it is classical. Overall rigid motions give only a finite-size correction to the bulk count. This independence from the individual <angular frequencies> makes the <Dulong-Petit law> useful as a quick estimate of a <solid>'s heating requirement.
Experiments on freely expanding <solids> often measure the <heat capacity at constant pressure> $C_P$, whereas the harmonic prediction is for the <heat capacity at constant volume> $C_V$. In a smooth single <thermodynamic phase> of a simple compressible material, the <difference between constant-pressure and constant-volume heat capacities> is
$$
C_P-C_V=\frac{TV\alpha^2}{\kappa_T},
$$
where $\alpha=V^{-1}(\partial V/\partial T)_P$ is the volumetric <coefficient of thermal expansion> and $\kappa_T=-V^{-1}(\partial V/\partial P)_T$ is the <isothermal compressibility>. Using measured $C_P$ as an approximation to $C_V$ requires this correction to be small compared with $C_V$. https://davidtong.org/teaching/statistical-physics/statmechhtml/S4#S4.SS4.SSS2[David Tong, Statistical Physics, section 4.4.2: relation between the two heat capacities]{ref}
High <temperature> here means $k_BT\gg\hbar\omega$ for the relevant vibrational <normal modes>, with reduced <Planck constant> $\hbar$ and mode <angular frequency> $\omega$. At low <temperature>, excitation across a quantum <energy> spacing is suppressed: modes with $\hbar\omega\gg k_BT$ freeze out thermally, and the classical <equipartition theorem> fails. The <Einstein model of a solid> makes this loss of vibrational <heat capacity> explicit. The <Dulong-Petit law> is therefore an ideal high-<temperature> vibrational limit, not a statement that all <solids> have the same <heat capacity> at every <temperature>. Other contributions and departures from the harmonic approximation can matter even at high <temperature>. https://davidtong.org/teaching/statistical-physics/statmechhtml/S3#S3.SS3[David Tong, Statistical Physics, section 3.3: solid vibrations, the classical limit and quantum freeze-out]{ref}
Historically, Petit and Dulong's 1819 experiments compared the <specific heat capacities> of several elemental <solids> and found an approximately constant product with their relative atomic weights. Their original empirical regularity came before the classical harmonic explanation; it suggested that counting <chemical atoms> reveals a simpler pattern than comparing equal <masses>. https://webserver.lemoyne.edu/giunta/PETIT.html[Petit and Dulong, Recherches sur quelques points importants de la Théorie de la Chaleur, Annales de Chimie et de Physique 10, 395–413 (1819), contemporary English translation hosted by Carmen Giunta]{ref}
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