Heat capacity Einstein theory

The decrease in the heat capacity at low temperatures was not explained until 1907, when Einstein demonstrated that the temperature dependence of the heat capacity arose from quantum mechanical effects [1], Einstein also assumed that all atoms in a solid vibrate independently of each other and that they behave like harmonic oscillators. The motion of a single atom is again seen as the sum of three linear oscillators along three perpendicular axes, and one mole of atoms is treated by looking at 3L identical linear harmonic oscillators. Whereas the harmonic oscillator can take any energy in the classical limit, quantum theory allows the energy of the harmonic oscillator (en) to have only certain discrete values ( ) ... 

Both the Einstein and Debye theories show a clear relationship between apparently unrelated properties heat capacity and elastic properties. The Einstein temperature for copper is 244 K and corresponds to a vibrational frequency of 32 THz. Assuming that the elastic properties are due to the sum of the forces acting between two atoms this frequency can be calculated from the Young s modulus of copper, E = 13 x 1010 N m-2. The force constant K is obtained by dividing E by the number of atoms in a plane per m2 and by the distance between two neighbouring planes of atoms. K thus obtained is 14.4 N m-1 and the Einstein frequency, obtained using the mass of a copper atom into account, 18 THz, is in reasonable agreement with that deduced from the calorimetric Einstein temperature. 

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

Einstein showed that when a reversible reaction is present sound dispersion occurs at low frequency the equilibrium is shifted within the time of oscillation, the effective specific heat is at a maximum, and the speed of sound c0 is at a minimum. At high frequency the oscillations occur so rapidly that the equilibrium has no time to shift (it is frozen ). The corresponding Hugoniot adiabate (FHA) is shown in the figure. Here the effective heat capacity is minimal, the speed of sound c is maximal cx > c0. From consideration of the final state and the theory of shock waves it follows that C>c0. 

Einstein Theory of Low-Temperature Heat Capacity of Solids [2], When we consider the heat capacity of solids, we realize that they consist of vibrating atoms or molecules. Their vibrations are quantized, of course, and have the nice name of phonons. Einstein considered a single vibration of an oscillator, along with its partition function ... 

Further work on similar types of cells has been carried out, in which not only is use made of the Nernst Theorem but likewise of the Einstein theory of atomic heat of solids (as modified by Nernst and Lmdemann) This will be taken up after we have discussed Planck s Quantum Theory of radiation and Einstein s application of it to the heat capacity of solids (Vol. Ill)... 

The development of the quantum theory was at first slow. It was not until 1905 that Einstein2 suggested that the quantity of radiant energy hv was sent out in the process of emission of light not in all directions but instead unidirectionally, like a particle. The name light quantum or photon is applied to such a portion of radiant energy. Einstein also discussed the photoelectric effect, the fundamental processes of photochemistry, and the heat capacities of solid bodies in terms of the quantum theory. When light falls on a metal plate, electrons arc emitted from it. The maximum speed of these photoelectrons, however,... 

Einstein temperature (0 ) - In the Einstein theory of the heat capacity of a crystalline solid, 0 = hvik, where h is Planck s constant, k is the Boltzmann constant, and v is the vibrational frequency of the crystal. 

The Debye theory assumes that there is a continuous distribution of frequencies from V = 0 to a certain maximum value v = Vj). The final expression obtained for the heat capacity is complicated, but succeeds in interpreting the heat capacity of many solids over the entire temperature range rather more accurately than the Einstein expression. At low temperatures, the Debye theory yields the simple result... 

This difficulty does not arise in solid, where we have an energy spectrum corresponding to its 3N - 6 modes of vibration. If this spectrum is discrete, then the addition of the weight functions for each of these 3N - 6 modes of vibration gives the entropy of a solid (Einstein s theory of heat capacity). 

Just like gases, solids can lose their ability to absorb energy at low temperatures. According to equipartition Equation (11.54), each vibration contributes kT to the energy. If there are N atoms in a solid, and each atom has three vibrational modes (in the x-, y-, and z-directions), the heat capacity wall be C = 3Nk, independently of temperature. This is called the Law of Dulong and Petit, named after the experimentalists who first observ ed this behavior around 1819. But more recent experimental data, such as that shown in Figures 11.14 and 11.15, indicate that this law does not hold at low temperatures. As the temperature approaches zero, Cv 0. The Einstein model, developed in 1907, shows why. This work was among the first evidence for the quantum theory of matter. 

How do we determine the Einstein temperature Be without knowing the characteristic vibrational frequency of the atoms in the crystal Typically, experimental data is fitted to the mathematical expression in equation 18.64 and a value of the Einstein temperature is used to allow for the best possible fit to experimental results. For example, a plot of experimental measurements ofthe heat capacity versus Tdividedby E (which isproportional to T, whereas 0g/r is inversely proportional to T and less easy to graph as T —> 0 K) is shown in Figure 18.4. Notice that there is reasonable agreement between experiment and theory, suggesting that Einstein s statistical thermodynamic basis of the heat capacity of crystals has merit. Table 18.6 lists a few experimentally determined Einstein temperatures for crystals. 

FIGURE18.4 The Einstein theory of heat capacity of crystals agrees reasonably well with experimental measurements. 

The critical enhancement term AXc (p, T) was based on the theory of dynamic critical phenomena, which predicts that the critical part of the thermal diffusivity kj pep, where c JO is the heat capacity at constant pressure, close to the critical point satisfies a Stokes-Einstein diffusion law of the form (Chapter 6)... 

Using the Einstein theory, calculate the molar heat capacity of a diamond crystal at 298.15 K, at 500.0 K, and at 1320 K. Compare each value with 3R. The experimental value at 298.15 K is 6.113 JK-lmol-1. 

In this chapter, we have discussed the structure of solids and liquids. Many solids are crystalline, with molecular units arranged in a regular three-dimensional lattice. There are two principal theories for the vibrations of lattices of atoms, the Einstein and the Debye theories. In the Einstein theory the normal vibrational modes of the lattice are assumed to vibrate with the same frequency. In the Debye model, the normal modes of the lattice are assumed to vibrate with the same distribution of frequencies as would a structureless solid. In each theory, the formula for the heat capacity of the solid lattice conforms to the law of Dulong and Petit at high enough temperature. 

Although much work has been done on the three reference substances described here, there are still needs for refinement of heat capacity measurements and frequency spectra fit. Most emjdiasis has been placed on obtaining a picture of the vibrations in the ideal crystal lattice. The questions about vibrations in small and defect crystals has at pre nt only been opened. Historically, the universal Dulong Petit-Rule of heat capacity which fits only well at elevated temperatures was replaced between 1900 and 1920 mainly throi h the work of Einstein, Debye. Nemst, and Lindemaim by a theory with one characteristic constant for each substance, the 0-temperature. The Tarasov treatment of the 1950 s shows that in case of strong anisotropy of forces, as in one or two dimensionally strongly bonded crystals, a second constant... 

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. 

The physicochemical features, however, and in particular the nanoparticle s heat capacity in the imphcit description can depend on their forms, for instance, by the way it fastens to the substrate and the nature of interaction with it, i.e., from border conditions. A detailed consideration of all these factors forms the subject for specialized future studies. Therefore, the further evaluations are based on the assumption of the independence of the nanoparlicle s thermal characteristics from the particular type of border conditions (similar to that of the classical theories of Einstein and Debye). 

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