Einstein heat capacity

These heat capacity approximations take no account of the quantal nature of atomic vibrations as discussed by Einstein and Debye. The Debye equation proposed a relationship for the heat capacity, the temperature dependence of which is related to a characteristic temperature, Oy, by a universal expression by making a simplified approximation to the vibrational spectimii of die... 

Figure 10.12 Comparison for diamond of the experimental Cr.m (circles) and the prediction of the Einstein heat capacity equation with = 1400 K (solid line). The experimental results below T = 300 K are closely spaced in temperature, and not all are shown in the figure.

Intermediate values for C m can be obtained from a numerical integration of equation (10.158). When all are put together the complete heat capacity curve with the correct limiting values is obtained. As an example, Figure 10.13 compares the experimental Cy, m for diamond with the Debye prediction. Also shown is the prediction from the Einstein equation (shown in Figure 10.12), demonstrating the improved fit of the Debye equation, especially at low temperatures. 

Einstein heat capacity equation 569-72 Schottky effect 580—5 solid + solid phase transitions 399-404 first-order 402-4 solutes 6... 

Debye heat capacity equation 572-80 Einstein heat capacity equation 569-72 heat capacity from low-lying electronic levels 580-5 Schottky effect 580-5 statistical weight factors in energy levels of ideal gas molecule 513 Stirling s approximation 514, 615-16 Streett, W. B. 412... 

The data shown in the table indicate that the law of Dulong and Petit holds surprisingly well for metals. For 1 mole of NaCl, there are 2 moles of particles, so the heat capacity is approximately 12 cal/mol deg or 50 J/mol K. However, the heat capacity of a solid is not a constant, but rather it decreases rapidly at lower temperatures as shown in Figure 7.19 for copper. A more complete explanation of the heat capacity of a solid as outlined next was developed by Einstein. 

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 ( ) ... 

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. 

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. 

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. 

This is, of course, the reciprocal of the time taken for an atom to move a lattice site distance into a vacancy. um can be estimated from the heat capacity of the crystalline material using the Einstein or Debye models6 of atoms as harmonic oscillators in a lattice. Combining Equations (2.30) and (2.31) gives the number of atoms moving per second as... 

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... 

Here V is the crystal volume, k-p and ks are the isothermal and adiabatic compressibility (i.e., the contraction under pressure), P is the expansivity (expansion/contraction with temperature), Cp and Cv are heat capacities, and 0e,d are the Einstein or Debye Temperatures. Because P is only weakly temperature dependent,... 

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. 

HEAT CAPACITY EQUATION (Einstein). A quantum relationship for the heat capacity at constant volume of an element of the form ... 

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

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. 

In the case of polyethylene it is interesting to attempt to calculate the heat capacity per mole of chain atoms using the Einstein function... 

Fig. 3, Comparison of heat capacities of crystalline Q (C ), open circles, and of a low density polyethylene Cv), solid circles, with values calculated from molecular frequencies and the Einstein function...

Another paper of Einstein s showed that quantization of energy also predicted that Dulong and Petit s heat capacity rule would only be valid at high temperatures. Assume that the only allowed energies are E = 0, hv, 2hv,... nhv, where n can be arbitrarily large. The average energy is... 

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 ... 

Our discussion of the specific heat capacity of polymers on the preceding pages has been quite empirical. There are, in fact, few fundamental rules that can be used for the prediction of specific heat capacity. At very low temperatures, the equations of Debye and Einstein may be used. 

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