Dulong-Petit limit

According to the classical statistical mechanics, at sufficiently high temperature, specific heat of a solid approaches a constant value C = 3R, also known as the Dulong-Petit limit. It is also well known experimentally that specific heat decreases with decreasing temperature. Generally, the specific heat of a solid is extremely small at low temperature, and it can be represented as C = yT + jST, where, y and P are constants. The physical meaning of these constants is clarified as follows. 

Dulong-Petit limit can be reproduced. In the Debye model, lattice vibrations are approximated as a continuous elastic body considering only acoustic modes, and therefore, phonon spectrum is treated linearly. In... 

Metals provided additional problems for the classical theory of heat capacity. Metals are generally much better conductors of heat than nonmetals because most of the heat is carried by the free electrons. According to the classical theory, this electron gas should contribute an additional (3/2)R to the heat capacity. But the measured heat capacity of metals approached nearly the same 3R Dulong-Petit limit as the nonmetals. How can the electrons be a major contributor to the thermal conductivity and not provide significant additional heat capacity ... 

Classical thermod)mamics using the equipartition of energy principle predicts that the lattice molar heat capacity will be given by 1/2R for each of the six degrees of freedom in a solid (three kinetic and three potential energy) for a total of 3R. As the temperature is increased, the observed heat capacity of materials approaches this value, which is known as the Dulong-Petit limit. However, at low temperatures, the observed heat capacity approaches zero as 7. ... 

Using the Planck distribution, Debye constructed a model in which the lattice heat capacity goes to zero as 7. The Debye model assumes a linear dispersion relation characteristic of a continuous medium rather than a chain of discrete atoms and cuts off the distribution at a frequency such that the number of normal modes is equal to 3 x the number of atoms. This frequency, knovm as the Debye frequency, is given by w-d = Vo 67t N/V), where Vq is the velocity of sound in the medium and N/V is the atoms per unit volume. A Debye temperature d is defined in terms of the Debye frequency d = hco-o/k. For T D, the Debye model approaches the classical Dulong-Petit limit. 

The incorporation of a defect also alters the vibrations locally. The influence of these energetic changes on the energy of formation of defects is relatively small and mostly negligible (cf. Dulong-Petit limit for a conserved number of vibrators, Chapter 3). However, the effect on entropy and, hence, on the temperature-dependence of the thermodynamic formation balance, as well as on the free enthalpy directly, is important. Even at small values of S the TS term can be of significance at high temperatures. 

Figure 33 Interpolation properties of Maier-Kelley functions, exemplified by compound clinoenstatite. Experimental data from Robie et al. (1978). B-B = Berman-Brown polynomial F-S = Fei-Saxena polynomial. Dashed line Dulong and Petit limit. Reprinted from Y. Fei and S. K. Saxena, Geochimica et Cosmochimica Acta, 51, 251-254, copyright 1987, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

Fisher polynomials can be used only within the T range for which they were created. Extrapolation beyond the T limits of validity normally implies substantial error progression in high-F entropy and enthalpy calculations. For instance, figure 3.4 compares Maier-Kelley, Haas-Fisher, and Berman-Brown polynomials for low albite. As can be seen, the first two interpolants, if extended to high T, definitely exceed the Dulong and Petit limit. The Berman-Brown interpolant also passes this limit, but the bias is less dramatic. 

Phonon velocity is constant and is the speed of sound for acoustic phonons. The only temperature dependence comes from the heat capacity. Since at low temperature, photons and phonons behave very similarly, the energy density of phonons follows the Stefan-Boltzmann relation oT lvs, where o is the Stefan-Boltzmann constant for phonons. Hence, the heat capacity follows as C T3 since it is the temperature derivative of the energy density. However, this T3 behavior prevails only below the Debye temperature which is defined as 0B = h( DlkB. The Debye temperature is a fictitious temperature which is characteristic of the material since it involves the upper cutoff frequency ooD which is related to the chemical bond strength and the mass of the atoms. The temperature range below the Debye temperature can be thought as the quantum requirement for phonons, whereas above the Debye temperature the heat capacity follows the classical Dulong-Petit law, C = 3t)/cb [2,4] where T is the number density of atoms. The thermal conductivity well below the Debye temperature shows the T3 behavior and is often called the Casimir limit. 

The discussion in this section has only been concerned with the enthalpy term. In order to determine the free energy, which is necessary for a calculation of the equilibrium defect concentration, the standard entropy change for the formation of a mole of defects may be estimated as follows. In the simplest case of the Einstein approximation for the limiting case of Dulong-Petit behaviour, the crystal with Nq lattice atoms is considered to be a system of... 

According to the Dulong-Petit law, for solids is 3R. This is however a high-temperature limit value, approached by different elements at different temperatures. is a function of temperature T and vibrational frequency v of the solid. 

By the late 1800s, study of heat capacities had led to the perplexing observation that the heat capacities of solids at quite low temperatures were very much below those expected from the Dulong-Petit rule. Some measurements gave heat capacities only 1% of the predicted value. About 1907, Einstein tied quantum behavior to heat capacities. He showed that if the vibrational energies of atoms in a solid were quantized, heat capacity would diminish sharply at low temperature. The high-temperature limit of Einstein s theory was the result achieved earlier by Boltzmann. Einstein s theory proved not as quantitatively accurate in its application to the heat capacity curves of solids as it was to those of diatomic gases. 

At high temperatures , T > 0d, the Dulong-Petit law applies once again. For T Partial integration allows successive reduction of the x term. At the end the exponential function remains. It disappears at the upper limit d/T 1 and becomes 1 at the lower limit. Hence, Cvib oc T results as a low temperature approximation in agreement with experiment -. Both theories presented have Cvib as a universal function of the reduced temperature... 

At high F, when the spacing of vibrational energy levels is low with respect to thermal energy, crystalline solids begin to show the classical behavior predicted by kinetic theory, and the heat capacity of the substance at constant volume (Cy) approaches the theoretical limit imposed by free motion of all atoms along three directions, in a compound with n moles of atoms per formula unit limit of Dulong and Petit) ... 

With increasing T, Cp, like Cy, approaches a limit imposed by the Dulong and Petit rule ... 

At the opposite extreme, if k T A> hvmax (where vmax is the highest vibrational frequency of the material) then there is a nearly continuous distribution of available energies. For example, the state with E = k T has about 37% as much population as the state with E = 0 the state with E = 2 k T has about 14% of the population of E = 0. If you sum over all of the possible states, it can be shown that the classical E = k T per vibration is recovered (Problem 5-5). Hence the rule of Dulong and Petit must be the high-temperature limit for all substances. 

This relation, commonly known as Einstein s (1905) law of heat capacities, correctly corresponds to the limiting cases Cy = Oat T = 0 and to Cy = 3R at infinite temperature—the law of Dulong and Petit. It immediately rationalizes the deviations of the observed Cy values from this law at lower temperatures that had mystified scientists before the advent of quantum mechanics. However, the above equation generally deviates to some extent from experiment in particular, at low T, the above relation shows that Cy, while experiments conform to aCy T variation. 

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