Heat capacity Einstein model

In Fig. 9.5, the Einstein model heat capacity is shown as a function of the characteristic temperature. Equation 9.69 correctly predicts the heat capacity at the limits of T 0 and T oo. 

The value of e that fits the Einstein crystal model heat-capacity formula to data for aluminum is 240 K. 

This model, the Einstein model for heat capacity, predicts that the heat capacity is reduced on cooling and that the heat capacity becomes zero at 0 K. At high temperatures the constant-volume heat capacity approaches the classical value 3R. The Einstein model represented a substantial improvement compared with the classical models. The experimental heat capacity of copper at constant pressure is compared in Figure 8.3 to Cy m calculated using the Einstein model with 0g = 244 K. The insert to the figure shows the Einstein frequency of Cu. All 3L vibrational modes have the same frequency, v = 32 THz. However, whereas Cy m is observed experimentally to vary proportionally with T3 at low temperatures, the Einstein heat capacity decreases more rapidly it is proportional to exp(0E IT) at low temperatures. In order to reproduce the observed low temperature behaviour qualitatively, one more essential factor must be taken into account the lattice vibrations of each individual atom are not independent of each other - collective lattice vibrations must be considered. 

Figure 8.3 Experimental heat capacity of Cu at constant pressure compared with Cv m calculated by the Einstein model using 0E = 244 K. The vibrational frequency used in the Einstein model is shown in the insert.

The experimental constant-pressure heat capacity of copper is given together with the Einstein and Debye constant volume heat capacities in Figure 8.12 (recall that the difference between the heat capacity at constant pressure and constant volume is small at low temperatures). The Einstein and Debye temperatures that give the best representation of the experimental heat capacity are e = 244 K and D = 315 K and schematic representations of the resulting density of vibrational modes in the Einstein and Debye approximations are given in the insert to Figure 8.12. The Debye model clearly represents the low-temperature behaviour better than the Einstein model. 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

The vibrational heat capacity is the largest contribution to the total heat capacity and determines to a large extent the entropy. Analytical expressions for the entropy of the models described in the previous section can be derived. The entropy corresponding to the Einstein heat capacity is... 

First-order estimates of entropy are often based on the observation that heat capacities and thereby entropies of complex compounds often are well represented by summing in stoichiometric proportions the heat capacities or entropies of simpler chemical entities. Latimer [12] used entropies of elements and molecular groups to estimate the entropy of more complex compounds see Spencer for revised tabulated values [13]. Fyfe et al. [14] pointed out a correlation between entropy and molar volume and introduced a simple volume correction factor in their scheme for estimation of the entropy of complex oxides based on the entropy of binary oxides. The latter approach was further developed by Holland [15], who looked into the effect of volume on the vibrational entropy derived from the Einstein and Debye models. 

Entropies and heat capacities can thus now be calculated using more elaborate models for the vibrational densities of states than the Einstein and Debye models discussed in Chapter 8. We emphasize that the results are only valid in the quasiharmonic approximation and can only be as good as the accuracy of the underlying force-field calculation of such properties can thus be a very sensitive test of interatomic potentials. 

Dahl, J. P. On the Einstein-Stern model of rotational heat capacities. J. Chem. Phys. 109, 10688-10691 (1998). 

Holland (1989) reconsidered the significance of constant K in fight of Einstein s model for the heat capacity of solids (see eq. 3.35 and 3.45) ... 

The important message from Einstein or Debye models is that vibrations of atoms in a crystal contribute to Entropy S and to Heat Capacity C therefore they affect the thermodynamic equilibrium of a crystal by modifying both the Eree energy F, which... 

The Einstein model for the molar heat capacity of a solid at constant volume, Cy, yields the formula ... 

In our present study To = 298.15 K thus AHo and ASo are respectively, the standard heat of formation and entropy. In our computational model we consider the heat capacity for the liquid phase to be temperamre independent and set it to 99.036 J/mole-K from literature data. For the solid phase, we employ a single Einstein oscillator to compute heat capacity ... 

There are no measurements of the heat capacity of ThD2(cr) or ThT2(cr), but Flotow et al. [1984FLO/HAS] have estimated the following values by modelling the heat capacity contributions from the lattice vibrations, represented by an Einstein function, conduction electrons and acoustic modes. This method had been shown to be valid for the deuterides of uranium, yttrium and zirconium. 

For the thermal properties of solids, Einstein developed an equation that could predict the heat capacity of solids in 1907. This model was then refined by Debye in 1912. Both models predict a temperature dependence of the heat capacity. At... 

As mentioned, many experimental results have shown that the specific heat for composites increases sHghtly with temperature before decomposition. In some previous models, the specific heat was described as a Hnear function. Theoretically, however, the specific heat capacity for materials wiU change as a function of temperature, as on the micro level, heat is the vibration of the atoms in the lattice. Einstein (1906) and Debye (1912) individually developed models for estimating the contribution of atom vibration to the specific heat capacity of a sohd. The dimensionless heat capacity is defined according to Eq. (4.32) and Eq. (4.33) and illustrated in Figure 4.12 [25] ... 

Most of the char material was composed of glass fiber and Cp was therefore considered as the specific heat capacity of the glass fibers. The results from Eq. 4.42 are compared with the results from the Einstein model (Eq. (4.33) in Figure 4.13 [12], as well as with the model used in previous studies [4, 5]. A linear function dependent on temperature for the specific heat capacity of fibers was used by Samanta et al. [4] and Looyeh et al. in 1997 [5], however, without direct experimental validation. As shown in Figure 4.13, the theoretical curve based on the Einstein model (Eq. (4.33) gives a reasonable estimation for the specific heat capacity of glass fibers. 

The true specific heat capacity of a composite material was obtained by the mle of mixture and the mass fraction of each phase was determined by the decomposition and mass transfer model. The true specific heat capacity of resin or fiber was derived based on the Einstein or Debye model. The effective specific heat capacity was obtained by assembhng the trae specific heat capacity with the decomposition heat that was also described by the decomposition model. The modeling approach for effective specific heat capacity is useful in capturing the endothermic decomposition of resin and was further verified by a comparison to DSC curves. 

The quite complicated temperature dependence of the macroscopic heat capacity in Fig. 2.46 must now be explained by a microscopic model of thermal motion, as developed in Sect. 2.3.4. Neither a single Einstein function nor any of the Debye functions have any resemblance to the experimental data for the solid state, while the heat capacity of the liquid seems to be a simple straight line, not only for polyethylene, but also for many other polymers (but not for all ). Based on the ATHAS Data Bank of experimental heat capacities [21], abbreviated as Appendix 1, the analysis system for solids and liquids was derived. 

There were many experimental proofs of the BCS model. In one proof, N. E. Phillips (1959) compared the heat capacity of aluminum in the superconducting and nonsuperconducting phase at low temperature. In the latter case, superconductivity was destroyed by a strong magnetic field. Phillips found the expected heat capacity behavior as a function of temperature for the nonsuperconducting phase. In the superconducting phase, the heat capacity increased very rapidly from zero and reached a value much higher than normal, as the temperature approached the critical temperature from below. This behavior is typical for the Bose-Einstein condensation and depends on the rapid increase of the entropy as T approaches Tc from below. 

The specific heat capacities are represented by functions derived from statistical models - e.g. for solids [SOU 15a], the Einstein function ... 

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. 

A major success of statistical mechanics is the ability to predict the thermodynamic properties of gases and simple solids from quantum mechanical energy levels. Monatomic gases have translational freedom, which we have treated by using the particle-in-a-box model. Diatomic gases also have vibrational freedom, which we have treated by using the harmonic oscillator model, and rotational freedom, for which we used the rigid-rotor model. The atoms in simple solids can be treated by the Einstein model. More complex systems can require more sophisticated treatments of coupled vibrations or internal rotations or electronic excitations. But these simple models provide a microscopic interpretation of temperature and heat capacity in Chapter 12, and they predict chemical reaction equilibria in Chapter 13, and kinetics in Chapter 19. 

This expression shows that the low-temperature heat capacity varies with the cube of the absolute temperature. This is what is seen experimentally (remember that a major failing of the Einstein treatment was that it didn t predict the proper low-temperature behavior of Cy), so the Debye treatment of the heat capacity of crystals is considered more successful. Once again, because absolute temperature and dy, always appear together as a ratio, Debye s model of crystals implies a law of corresponding states. A plot of the heat capacity versus TIdo should (and does) look virtually identical for all materials. 

Historical Note 1. Einstein s first work on heat capacity dates to 1907 [6]. In that article, Einstein limited his attention to solids formed by equivalent oscillators—the so-called Einstein model. In this case, the frequency distribution function is a Dirac 5 distribution centered on a... 

Chapter 1 discusses the modeling of pure solids. Oscillator models (Einstein s and Debye s) are used to calculate canonical partition functions for four types of solid atomic, ionic, molecular and metallic. These canonical partition functions can be employed, first to calculate the specific heat capacities at constant voliune, and second to determine the expansion coefficients with the Griineisen parameters. 

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