The Debye Heat Capacity Equation

Equation (10.148) does not correctly predict CV.m at low temperatures because it assumes all the atoms are vibrating with the same frequency. In the ideal gas, this is a good assumption, and in the previous section we used an equation similar to (10.148) to calculate the vibrational contribution to the heat capacity of an ideal gas. 

But in a solid, the motions of the atoms are coupled together. The vibrational motion involves large numbers of atoms, and the frequency for these vibrations is less than the value used in equation (10.148). Thus, vibrations occur over a range of frequencies from v = 0 to a maximum u = vm. When this occurs the energy can be calculated by summing over all the frequencies. The 

To integrate equation (10.149) we must know how dn is related to v. Debye assumed that a crystal is a continuous medium that supports standing (stationary) waves with frequencies varying continuously from v = 0 to v = t/m. The situation is similar to that for a black-body radiator, for which it can be shown that 

Integrating the right side of equation (10.152) and setting it equal to 3NA gives 

Differentiation of equation (10.154) with respect to T gives an equation for Cy. m. The result is 

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Figure 10.15 Comparison of the fit of the Debye heat capacity equation for several elements. Reproduced from K. S. Pitzer. Thermodynamics. McGraw-Hill, Inc., New York, 1995, p. 78. Reproduced with permission of the McGraw-Hill Companies.

In deriving the Debye heat capacity equation, one assumes that the atoms in an atomic solid are vibrating with a range or frequencies v varying from u = 0 to a maximum u — vm. The resulting equation for calculating Cv, m is... 

Show that if the Debye heat capacity equation were applicable, the entropy of a perfect solid at very low temperatures should be equal to JCp, where Cp is the heat capacity at the given temperature. What would be the value in terms of the Debye characteristic temperature ... 

For Cy/T to approach zero as T approaches zero, CV must go to zero at a rate at least proportional to T. Earlier, we summarized the temperature dependence of Cy on T for different substances and showed that this is true. For example, most solids follow the Debye low-temperature heat capacity equation of low T for which... 

Thus Cp m and CVm differ little from one another at low temperatures. The Debye low-temperature heat capacity equation (and other low-temperature relationships) we have summarized calculates Cp.m, as well as CV. m, without significant error. 

E4.1 Show that at very low temperatures where the Debye low temperature heat capacity equation applies that the entropy is one third of the heat capacity. 

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. 

Figure 10.14 Graph showing the limiting behavior at low temperatures of the heat capacity of (a), krypton, a nonconductor, and (b). copper, a conductor. The straight line in (a) follows the prediction of the Debye low-temperature heat capacity equation. In (b), the heat capacity of the conduction electrons displaces the Debye straight line so that it does not go to zero at 0 K.

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. 

The heat capacity and entropy of TiBr Ccr) have been measured over the temperature range 51 to 800 K by King et al. (2). Heat capacities above 800 K are estimated from graphical extrapolation. The value of S"(298.15 K) is derived from these data, based on S (51 K) - 8.60 cal K mol. The value of S (51 K) is estimated from a Debye-Einsteln extrapolation of the measured heat capacities, the equation being C - D(70.0/T) + E(120/T) + 2E(306/T). It is assumed that all electronic entropy is... 

The important conclusions, therefore, to be drawn from the Debye theory are that at low temperatures the atomic heat capacity of an element should be proportional to T, and that it should become zero at the absolute zero of temperature. In order for equation (17.4) to hold, it is necessary that the temperature should be less than about 9/10 this means that for most... 

The heat capacity equation (V.81) was used to derive an expression for the entropy change in the temperature range 53 to 1173 K. Below 50 K, Kelley [41KEL] extrapolated a value for the entropy change of 2.34 J-K -mor from the funetion sum (using Debye and Einstein terms) derived between 51 and 298 K the Einstein terms, however, only contribute a negligible amount to the value. The values obtained below and above 50 K were then combined to derive a value for the standard entropy at... 

At the time of the heat capacity measurements of Lounasmaa (1964c) (0.4. 0 K), the lattice and magnetic contributimis could not be separated but later Rosen (1968a) determined the limiting Debye temperature from elastic constants measurements to be 118 K, equivalent to a lattice contribution to the heat capacity of 1.18 mJ/(mol K ) so that the magnetic contribution could then be separately calculated from a reassessment of the measurements of Lounasmaa (1964c) over the temperature range of 0.50-1.25 K. When combined with the nuclear terms as determined in Part 11.11, results in a revised heat capacity equation valid to 1.25 K ... 

Additional contributions to the total heat capacity with the nuclear contribution omitted were fitted to the equation C°p T) = yT+BT, where y is the electronic coefficient and B is a combination of the lattice and magnetic contributions to the heat capacity. From elastic constant measurements on single crystals, Rosen et al. (1974) determined a limiting Debye temperature of 191.5 K, equivalent to a lattice contribution of 0.28 mJ/(mol K" ), while similar measurements by Palmer (1978) determined 186.8 K, equivalent to 0.30 mJ/(mol K" ). Averaging these values to 0.29 mJ/(mol K ), then the magnetic contribution to the heat capacity is given by the subtraction is U=B-0.29mJ/(molK ) (Table 143). 

Total thermal conductivity is a sum of the lattice and electronic parts, K = Ki + Ke- The lattice part of the thermal conductivity describes the scattering of phonons on the vibrations of atoms, whereas the electronic part describes thermal conductivity appearing due to conduction electrons and is related to the electrical conductivity Wiedemann-Franz equation, = a T Lo, where T is the absolute temperature and Lq is the ideal Lorenz number, 2.45 X 10 Wf2K [64]. The electronic part of the thermal conductivity is typically low for low-gap semiconductors. For the tin-based cationic clathrates it was calculated to contribute less than 1% to the total thermal conductivity. The lattice part of the thermal conductivity can be estimated based on the Debye equation /Cl = 1 /3(CvAvj), where C is the volumetric heat capacity, X is the mean free path of phonons and is the velocity of sound [64]. The latter is related to the Debye characteristic temperature 6 as Vs = [67t (7V/F)] . Extracting the... 

Calculate the standard entropy of silver at 25 C (298 K) from the third lav/ of thermodynamics, assuming that the molar heat capacity at temperatures below -258°C HS K) follows Debye s equation. 

Solution. Equation (3.17) may be used to determine the electron heat capacity, while Eq. (3.10) may be used to calculate the lattice specific heat provided T/6 is less than 1/12. Table 3.3 gives a Debye temperature of 310 K for copper, for copper from Table 3.5 is 0.011 J/kgK. The molecular weight of copper is 63.55. 

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... 

Equation (4.2) requires that the total area above 0 Kelvin be obtained, but heat capacity measurements cannot be made to the absolute zero of temperature. The lowest practical limit is usually in the range from 5 K to 10 K, and heat capacity below this temperature must be obtained by extrapolation. In the limit of low temperatures, Cp for most substances follows the Debye low-temperature heat capacity relationship11 given by equation (4.4)... 

Graph the above data in the form Cp,m/T against T2 to test the validity of the Debye low-temperature heat capacity relationship [equation (4.4)] and find a value for the constant in the equation, (b) The heat capacity study also revealed that quinoline undergoes equilibrium phase transitions, with enthalpies as follows ... 

The Debye temperature, can be calculated from the elastic properties of the solid. Required are the molecular weight, molar volume, compressibility, and Poisson s ratio.11 More commonly, do is obtained from a fit of experimental heat capacity results to the Debye equation as shown above. Representative values for 9o are as follows ... 

Since in our temperature range, the Debye temperature of Ge is 370K [47], the phonon contribution to the heat capacity can be neglected. Hence, the heat capacity of our samples is expected to follow the equation ... 

Although the Debye model reproduces the essential features of the low- and high-temperature behaviour of crystals, the model has its limitations. A temperature-dependent Debye temperature, d(F), can be calculated by reproducing the heat capacity at each single temperature using the equation... 

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... 

Data for a large number of organic compounds can be found in E. S. Domalski, W. H. Evans, and E. D. Hearing, Heat capacities and entropies in the condensed phase, J. Phys. Chem. Ref. Data, Supplement No. 1, 13 (1984). It is impossible to predict values of heat capacities for solids by purely thermodynamic reasoning. However, the problem of the solid state has received much consideration in statistical thermodynamics, and several important expressions for the heat capacity have been derived. For our purposes, it will be sufficient to consider only the Debye equation and, in particular, its limiting form at very low temperamres ... 

The symbol 9 is called the characteristic temperamre and can be calculated from an experimental determination of the heat capacity at a low temperature. This equation has been very useful in the extrapolation of measured heat capacities [16] down to OK, particularly in connection with calculations of entropies from the third law of thermodynamics (see Chapter 11). Strictly speaking, the Debye equation was derived only for an isotropic elementary substance nevertheless, it is applicable to most compounds, particularly in the region close to absolute zero [17]. 

Use of Debye Equation at Very Low Temperatures. Generally, it is assumed that the Debye equation expresses the behavior of the heat capacity adequately below about 20 K [9]. This relationship [Equation (4.68)],... 

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