The law that the molar heat capacity of many solid elements near room temperature is about 3R (about 25 J/(mol·K)). It corresponds to the classical limit of lattice vibrations at high temperature.
Japanese name: デュロン・プティの法則
Dulong–Petit law appears in 2 practice questions across 1 topic, including Thermal properties of materials.
The Dulong–Petit law is an empirical rule presented in 1819 by Dulong and Petit, who found that the product of specific heat and atomic weight is nearly constant for many solid elements. In today's terms, the molar heat capacity of solid elements is about 3R, or about 25 J/(mol·K). Atomic weights were not yet accurately established at the time, so the law was also used to estimate atomic weights from measured specific heats and to correct wrong values. For copper, for example, dividing 3R by the molar mass of 63.55 g/mol gives about 0.39 J/(g·K), in good agreement with the measured value of about 0.385 J/(g·K).
By the equipartition theorem of classical statistical mechanics, each atom in a solid can be regarded as a harmonic oscillator in three directions and has an energy of 3kT, kinetic and potential energy combined. Differentiating this with respect to temperature gives a heat capacity of 3N_A k = 3R per mole. In this explanation, however, the heat capacity should not depend on temperature, and its decrease at low temperature cannot be explained. Einstein and Debye explained the low-temperature behavior from the quantization of vibrational energy, and the Dulong–Petit law came to be understood as the high-temperature limit in which all vibrations are fully excited.
Elements whose room-temperature values deviate strongly from 3R are those in which light atoms are strongly bonded and the Debye temperature is far above room temperature. The molar heat capacity of diamond at room temperature is about 6.1 J/(mol·K), only about a quarter of 3R, and beryllium (about 16 J/(mol·K)) and silicon (about 20 J/(mol·K)) are also below 3R. In these, the high-energy vibrations are not fully excited even at room temperature. Well above room temperature, on the other hand, the constant-pressure heat capacity of many elements slightly exceeds 3R, because the effects of thermal expansion, vibrational anharmonicity and, in metals, conduction electrons are added.
According to the Neumann–Kopp rule, the molar heat capacity of a compound is nearly equal to the sum of the molar heat capacities of its elements, and combined with the Dulong–Petit law this gives about 3R per atom in the formula unit. NaCl, for example, has two atoms and is estimated at about 6R ≈ 50 J/(mol·K), close to the measured room-temperature value of about 50.5 J/(mol·K). Even for materials without heat capacity data, a rough value can be estimated this way if the Debye temperature is not much above room temperature. Expressed per unit mass, materials made of elements with smaller atomic weights have larger specific heats.
Answer: C. About 0.92 J/(g·K)
Atoms in a solid vibrate in three directions, and the vibration in each direction holds kT/2 each as kinetic and potential energy, for a total of 3kT per atom, so the molar heat capacity is 3R ≈ 24.9 J/(mol·K). Dividing by the molar mass gives 24.9/26.98 ≈ 0.92 J/(g·K), close to the measured value for aluminum of about 0.90 J/(g·K). About 0.31 J/(g·K) drops the factor of 3 and divides R alone, and about 0.46 J/(g·K) uses (3/2)R, the constant-volume molar heat capacity of a monatomic gas, which leaves out the potential energy of the vibrations. Because the molar heat capacities are nearly the same, lighter elements have larger specific heats per unit mass.
The theoretical 3R is the value at constant volume. In solids, the difference between constant-pressure and constant-volume heat capacities is small at room temperature, a few percent, so comparisons are often made with the more easily measured constant-pressure value. At high temperature the difference grows and the constant-pressure heat capacity exceeds 3R.
Because carbon atoms are light and very strongly bonded, the lattice vibration frequencies are high and the Debye temperature, about 1900–2200 K, is far above room temperature. At room temperature many vibrations are not excited and few vibrations can take up energy, so the molar heat capacity stays at about 6.1 J/(mol·K).
Divide 3R ≈ 24.9 J/(mol·K) by the molar mass. For iron (55.85 g/mol), for example, this gives about 0.45 J/(g·K), in good agreement with the measured value of about 0.45 J/(g·K). The larger the atomic weight of an element, the smaller its specific heat per unit mass.
A nearly constant 3R is the classical limit at high temperatures comparable to or above the Debye temperature, corresponding to the Dulong–Petit law.
At temperatures well below the Debye temperature, how does …Debye temperature Phonon Coefficient of linear thermal expansion Invar Thermal diffusivity Umklapp process Wiedemann–Franz law