Dulong Petit Law

The Dulong-Petit law (according to Pierre Louis Dulong and Alexis Thérèse Petit ) states that the molar heat capacity of from individual atoms composite solid body have a universal and constant value, namely three times the universal gas constant  R :

${\displaystyle {\begin{aligned}C_{\mathrm {m} }&=3\cdot R\\&\approx 3\cdot 8{,}3\,{\frac {\mathrm {J} }{\mathrm {mol} \cdot \mathrm {K} }}\\&=24{,}9\,{\frac {\mathrm {J} }{\mathrm {mol} \cdot \mathrm {K} }}\end{aligned}}}$

The molar heat capacities of most elements - here shown as a function of the order number - are between 2.8 R and 3.4 R .

The two namesake found experimentally that many of the substances they examined have practically the same molar heat capacity, and in 1819 published the assumption that this was a general law. Classical statistical thermodynamics (which was not yet familiar with quantum effects ) later actually found the constant value for the molar heat capacity of monoatomic (that is, made up of similar atoms) solids . Measurements extended to larger temperature ranges and theoretical investigations taking into account quantum mechanical principles show, however, that this law is only approximately valid. ${\displaystyle 3R}$

Derivation

The particles in a solid are bound to their places in the crystal lattice and vibrate around these central positions . As a first approximation, the oscillation of each particle can be described as a harmonic oscillator . According to the uniform distribution theorem of classical statistical thermodynamics, each of the three lattice vibration degrees of freedom of each particle (one each in -, - and - direction) carries the average kinetic energy at the temperature ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle T}$

${\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}\cdot k_{\mathrm {B} }\cdot T}$

with the Boltzmann constant . ${\displaystyle k_{\mathrm {B} }}$

The potential energy of the harmonic oscillator is a homogeneous function of the 2nd degree in the deflection . So according to the virial theorem it follows that the mean potential energy is equal to the mean kinetic energy:

${\displaystyle E_{\mathrm {kin} }=E_{\mathrm {pot} }}$

Therefore, on average, the energy is allotted to one degree of freedom and the energy to a particle with three degrees of freedom for the lattice vibration . One mole of such particles carries the energy ${\displaystyle \textstyle 2\cdot {\frac {1}{2}}\cdot k_{\mathrm {B} }\,T}$ ${\displaystyle \textstyle 3\,k_{\mathrm {B} }\,T}$

${\displaystyle E=3\cdot N_{\mathrm {A} }\cdot k_{\mathrm {B} }\cdot T=3\cdot R\cdot T}$

With

• ${\displaystyle N_{\mathrm {A} }}$the Avogadro constant
• ${\displaystyle R}$the universal gas constant

and is the molar heat capacity

${\displaystyle C_{\mathrm {m} ,V}={\frac {1}{n}}\left({\frac {\partial U_{\mathrm {m} }}{\partial T}}\right)_{V}\approx C_{\mathrm {m} ,p}={\frac {1}{n}}\left({\frac {\partial H_{\mathrm {m} }}{\partial T}}\right)_{p}\approx 3\cdot R=3\cdot N_{\mathrm {A} }\cdot k_{\mathrm {B} }}$

With

• ${\displaystyle n}$the amount of substance
• ${\displaystyle C_{\mathrm {m} ,p}}$the molar heat capacity at constant pressure ${\displaystyle p}$
• ${\displaystyle C_{\mathrm {m} ,V}}$the molar heat capacity at constant volume ${\displaystyle V}$
• ${\displaystyle U_{\mathrm {m} }}$the molar internal energy
• ${\displaystyle H_{\mathrm {m} }}$the molar enthalpy .

Limits

Deviations from Dulong and Petit's Law

Despite its simplicity, the Dulong-Petit law makes relatively good predictions for the specific heat capacity of solids with a simple crystal structure at sufficiently high temperatures (e.g. at room temperature ).

In areas of low temperatures, it increasingly deviates from the experimental findings. Since the lattice vibrations are quantized , they can only absorb energy quanta of the same size per degree of freedom ( Planck's quantum of action , vibration frequency ). In particular, at least the energy per degree of freedom is necessary to stimulate the oscillation at all. If the available thermal energy is too low, some degrees of freedom are not stimulated at all and cannot contribute to the heat capacity by absorbing energy. The heat capacity of solids therefore decreases noticeably at very low temperatures and tends towards zero (third law of thermodynamics ). The Debye model provides better predictions here. ${\displaystyle h\nu }$ ${\displaystyle h}$ ${\displaystyle \nu }$ ${\displaystyle 1\cdot h\nu }$ ${\displaystyle kT}$ ${\displaystyle T\to 0}$

If a solid is not made up of individual atoms, but of more complex molecules (e.g. CaSO ₄ ), then there are additional degrees of freedom of the molecular vibration for each particle in addition to the 3 degrees of freedom of the lattice vibration (the particles of the molecule vibrate against each other). The molar heat capacity of such a solid can be significantly higher than predicted by the Dulong-Petit law.

Molar heat capacity at 25 ° C as a function of the atomic number of the elements: The values ​​for bromine and iodine apply to the gas state. Carbon (C), boron (B), beryllium (Be) and silicon (Si) have a small heat capacity and are highlighted in light blue. Near room temperature, gadolinium has a higher heat capacity due to the phase transition from ferromagnetic to paramagnetic state.

Metals as monoatomic solids are mostly in good agreement with the Dulong-Petit law. Because of the metal bond , one would initially expect otherwise, since the atoms release electrons from their outer electron shell when they are bonded , which can move freely through the crystal ( electron gas ). Each electron would have to contribute 3  degrees of translational freedom, so that if each atom releases an electron, the molar heat capacity would have to be. However, since all the states below the Fermi distribution are already occupied in the electron gas, most electrons cannot transition into a state of higher energy and therefore cannot contribute to the heat capacity. ${\displaystyle 3R+3\cdot R/2=9/2R}$

literature

• Alexis Petit , Pierre Louis Dulong : Recherches sur quelques points importants de la théorie de la chaleur . Présentées à l'Académie des sciences le 12 avril 1819. In: Joseph Louis Gay-Lussac, François Arago (ed.): Annales de chimie et de physique . tape 10 . Crochard, Paris 1819, p. 395–413 (French, online at Gallica - Bibliothèque Nationale de France [accessed December 4, 2016] English translation online at Le Moyne College): “Reconnaitre immédiatement l'existence d'une loi physique susceptible d'être généralise et étendue à toutes substances élementaires [...] Les atomes de tous le corps simples ont exactement la même capacité pour la chaleur. »