Debye heat capacity function

The measured heat capacities of organic compounds can usually be extrapolated to T = 0 by fitting of Debye heat capacity functions (or a combination of Debye and Einstein functions) to the measured values below 20 K. Smoothed heat capacities from the experimental results and from the Debye functions are used to obtain the thermodjmamic properties S (X), H%T) - and [G%T) - H%0)yT of the crystals and... 

From Eq. (3) the frequency distribution can be calculated following the Debye treatment by making use of the fact that an actual atomic system must have a limited number of frequencies, limited by the number of degrees of freedom N. The distribution p(v) is thus simply given by Eq. (4). This frequency distribution is drawn in the sketch on the right-hand side in Fig. 2.36. The heat capacity is calculated by using a properly scaled Einstein term for each frequency. The heat capacity function for one mole of vibrators depends only on Vj, the maximum frequency of the distribution, which can be converted again into a theta-temperature, j. Equation (5) shows that at temperature T is equal to R multiplied by the one-dimensional Debye... 

Figure 3.6 The heat capacity of a solid as a function of the temperature divided by the Debye temperature...

The discovery of a transition which we identify with this has been reported by Simon, Mendelssohn, and Ruhemann,16 who measured the heat capacity of hydrogen with nA = 1/2 down to 3°K. They found that the heat capacity, after following the Debye curve down to about 11°K, rose at lower temperatures, having the value 0.4 cal/deg., 25 times that of the Debye function, at 3°K. The observed entropy of transition down to 3°K, at which the transition is not completed, was found to be about 0.5 E.U. That predicted by Eq. (15) for the transition is 2.47 E.U. 

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

Boyer et al. [20] have measured the heat capacity of crystalline adenine, a compound of biologic importance, with high precision, from about 7 K to over 300 K, and calculated the standard entropy of adenine. Table 11.8 contains a sampling of their data over the range from 7.404 K to 298.15 K. Use those data to calculate the standard entropy of adenine at 298.15 K, which assume the Debye relationship for Cp. The value for 298.15 K is calculated by the authors from a function fitted to the original data. 

Figure 4.24 Molar heat capacity as a function of temperature, based on the Debye model. From K. M. Ralls, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission John Wiley Sons, Inc.

The enthalpy and the entropy at 5 K were estimated from the extrapolated heat capacity at this temperature according to the Debye T3 law. The thermodynamic functions at higher temperatures were calculated from the obtained Cp(7) dependence. 

We must also consider the conditions that are implied in the extrapolation from the lowest experimental temperature to 0 K. The Debye theory of the heat capacity of solids is concerned only with the linear vibrations of molecules about the crystal lattice sites. The integration from the lowest experimental temperature to 0 K then determines the decrease in the value of the entropy function resulting from the decrease in the distribution of the molecules among the quantum states associated solely with these vibrations. Therefore, if all of the molecules are not in the same quantum state at the lowest experimental temperature, excluding the lattice vibrations, the state of the system, figuratively obtained on extrapolating to 0 K, will not be one for which the value of the entropy function is zero. 

Figure 3.20 Heat capacity of copper (0 = 343 K [22]), magnesia (0 = 946 K [23]), and diamond (0 = 2230 K [23]) as a function of temperature, as predicted by Debye theory ...

Heat capacity is (9.1 + 2.9 x 10 3 T(K)) cal mol 1 K 1 between 298 and 1273 K, while entropy is 10.4 cal mol 1 K 1 at 298.15 K [30], More recent heat capacity data between 153 and 293 K are quoted in TABLE 3, Datareview A1.4 of this volume. There is a 7.5% discrepancy between the two sets of data at room temperature. The inferred Debye temperature is 660 K [29], Enthalpies, entropies and Gibbs functions of fusion and formation are given in TABLE 4. Thermochemical data should also be regarded with circumspection in particular the early value of AH°f = -5 kcal mol 1 [1] is often quoted but should now be discarded in favour of more recent results [24,31] found to lie close to those for the other nitrides. 

The low temperature heat capacities in the temperature range from 52-298 K are obtained from Todd s measurements (7). Two peaks at 193.5 K and 230.9 K were found in his heat capacity data. Below 50 K, the heat capacities were extrapolated using a combination of 1 Debye (0 = 139) and 2 Einstein (0 = 260) functions as suggested by Todd. This extrapolation yields the entropy from lattice contribution as 3.278 cal k" mol at 50 K. By neutron diffraction. Smith et al. ( ) found an antiferromagnetic transition at 7 K which indicates the existence of an unpaired electron in KOgCcr). We tentatively adopt S (50 K) = 4.656 1 cal K" mol" which includes both lattice (3.278 cal k" mol" ) and unpaired electron (Rtn2) contributions. Heat capacities above 298 K are estimated graphically. 

King ( ) measured the heat capacity of the high temperature o-phase NbgOg from 53.24 to 296.64 K and fitted the data (29 data points) with a combination of Debye and Einstein functions. These functions fit the data over the entire measured temperature range with a maximum deviation of 0.6% (14) and are used to calculate S (50 K) 2.42 cal K" mol". ... 

Anderson (1 ) measured the heat capacity of V20g(cr) in the range 57-287 K. The data indicated an anomaly in the region 165-182 K. These heat capacity data are Joined smoothly at 298 K with the high temperature heat capacity values as derived from the enthalpy measurements of 0>ok (JJ.). The adopted C values are based on these two studies (1, 13 ). Using the combination of Debye and Einstein functions as suggested by Anderson (12), we calculate S (50 K) 0.783 cal K mol and H (50 0) 0.0284 kcal mol. There is considerable scatter in the data of Cook (13) the deviations from the adopted values range from -0.8 to 0.6% except for the data point at 369.1 K which is -1.4% low ( 25 cal K mol ),... 

Low temperature heat capacities of ZnS0 (cr, o) have been measured by Weller ( ) from 51.7 - 296.5 K. A small heat capacity maximum was observed at 124.37 K. Our adopted value of S°(298.15 K) = 26.42+0.3 cal K mol obtained from C is based on S (51 K) = 2.27 cal K mol obtained by Weller (1 ) by extrapolation of the measured heat capacity with a combination of Debye and Einstein functions. We have smoothed the data of Weller (H)) by fitting the data with orthogonal polynomials over selected overlapping temperature intervals. 

Low temperature heat capacities have been measured by Cristescu and Simon 76) from 13 to 210 K., and by Weertman, Burk, and Goldman (545) from 50 to 200 K. Since the latter workers have not substantiated the anomaly reported by the former workers, we have adopted the values of the latter group and have extrapolated them to absolute zero with a Debye function. From this information, we calculate the entropy at 298 K. to be 10.91 e. u. and the enthalpy at 298 K. to be 1448 cal./gram atom. We have estimated the heat capacity of the solid above 298 K. and of the liquid. A transition point has been reported by Duwez 91) and by Fast 110). The melting point has been reported by Adenstedt (5), Litton (575), and Zwikker 352). Considerable disagreement is evidenced by these values. There is probably a transition in the vicinit> of the melting point, but in view of the uncertainty existing, we have elected to minimize the necessary... 

Brown, Zemansky, and Boorse (4S) have measured the heat capacity up to 12° K. and also in the range from 65° to 75° K. We have used these meager data with a Debye function to calculate the entropy and enthalpy at 298° K. as 8.73 e.u. and 1264 cal./gram atom, respectively. Kelley (18S) lists the heat capacity for the solid above 298° K. Reimann and Grant (366) have determined the melting point as 2770° K. We estimate the heat of melting of 6400 cal./gram atom and the liquid heat capacity. 

The final subblock of the Heat Capacity Data Bank involves programs for needed calculations in the thermal analysis field. The simple stages involve data treatment for input and output, calculation of derived functions as given, for example, in egs. 1 to 3-Further stages include the data analysis in form of Debye and Tarasov 0-temperatures and group vibration frequencies, a stage already completed (VI). Self-... 

It will be seen from the Debye equation (17.2) that Cr is a function of 9/T only, and hence the plot of Cy against T/0 (or log T/9) should yield a curve that is the same for all solid elements.f The nature of the curve is shown in Fig. 9, and it is an experimental fact that the heat capacities of many elements, and even of a few ample compounds, e.g., ionic cr stals such... 

Table V-52. Heat capacity and third law entropy of a-CdSe at 298.15 K. Original values including an estimated entropy term for the temperature range 0 to 50 K are denoted by (a), values corrected using the experimental mean entropy at 50 K from [76PET/KOF] and [92SIR/GAV] are denoted by (b), and values derived from estimated Debye-Einstein functions are denoted by (c).

The entropy of Nd3Sc4(cr) at 298.15 K was evaluated in [2002BOL/KOP] by extrapolation of the measured heat capacity to 0 K using Debye functions. The value obtained is selected by the review, but the uncertainty limits have been increased in order to account for unknown systematic errors... 

To calculate the thermodynamic functions for pure metals, one needs the thermal heat capacity C at ambient pressure. Above the Debye temperature, Cp consists of three parts a) the Dulong-Petit value of 3/cB, b) an additional linear increase proportional to temperature, which can also be seen in the thermal expansion coefficient, and c) an additional amount close to the melting temperature Tm, which results from the formation of defects (mainly vacancies). The last part can be approximated for small concentration as... 

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