Heat Capacity and Debye Temperature

A Debye temperature of 0d = 176 16 K is derived from X-ray diffraction intensities at room temperature by Subhadra, Sirdeshmukh [13]. The following 0q values are estimated by Berton 

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U. Piesbergen, Heat Capacity and Debye Temperatures G.Giesecke, Lattice Constants J.R. Drabble, Elastic Properties... 

Table 4.1-8 Heat capacities and Debye temperatures of Group IV semiconductors and IV-IV compounds...

Fig. ft.1-52 BP. Temperature dependence of specific heat capacity and Debye temperature [1.51]... 

Table 4.1-122 Heat capacity and Debye temperature of oxides of Ca, Sr, and Ba...

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)... 

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. 

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,... 

Tn the critical region of mixtures of two or more components some physical properties such as light scattering, ultrasonic absorption, heat capacity, and viscosity show anomalous behavior. At the critical concentration of a binary system the sound absorption (13, 26), dissymmetry ratio of scattered light (2, 4-7, II, 12, 23), temperature coefficient of the viscosity (8,14,15,18), and the heat capacity (15) show a maximum at the critical temperature, whereas the diffusion coefficient (27, 28) tends to a minimum. Starting from the fluctuation theory and the basic considerations of Omstein and Zemike (25), Debye (3) made the assumption that near the critical point, the work which is necessary to establish a composition fluctuation depends not only on the average square of the amplitude but also on the average square of the local... 

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. 

It is necessary to specify zero ionic strength here because Debye-HUckel adjustments for ionic strength depend on the temperature. Heat capacities and transformed heat capacities are discussed in an Appendix to this chapter. However, since there is not very much information in the literature on heat capacities of species or transformed heat capacities of reactants, the treatments described here are based on the assumption that heat capacities of species are equal to zero. When molar heat capacities of species can be taken as zero, both standard enthalpies of formation and standard entropies of formation of species are independent of temperature. When Af H° and Af 5° are independent of temperature, standard Gibbs energies of formation of species at zero ionic strength can be calculated using... 

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 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. 

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).

Below the Debye temperature, only the acoustic modes contribute to heat capacity. It turns out that within a plane there is a quadratic correlation to the temperature, whereas linear behavior is observed for a perpendicular orientation. These assumptions hold for graphite, which indeed exhibits two acoustic modes within its layers and one at right angles to them. In carbon nanotubes, on the other hand, there are four acoustic modes, and they consequently differ from graphite in their thermal properties. StiU at room temperature enough phonon levels are occupied for the specific heat capacity to resemble that of graphite. Only at very low temperatures the quantized phonon structure makes itself felt and a linear correlation of the specific heat capacity to the temperature is observed. This is true up to about 8 K, but above this value, the heat capacity exhibits a faster-than-Unear increase as the first quantized subbands make their contribution in addition to the acoustic modes. 

The last term is evaluated from experimental measurements of heat capacities and heats of transition (Sec. 6-1). The extrapolated term is usually evaluated with the aid of the Debye equation for the heat capacity of crystals. The Debye equation for the heat capacity of crystals at low temperature is... 

Numerous physical properties of nanoparticles are not only unique to the nanosized domain, but also vary with the particle size within this domain - they include melting and Debye temperatures, heat capacities, magnetic properties, and trends in optical and infrared absorption spectra [23-30]. 

Gaur U, Pultz G, Wiedemeier H, Wunderlich B (1981) Analysis of the Heat Capacities of Group IV Chalcogenides using Debye Temperatures. J Thermal Anal 21 309-326. Baur H, Wunderlich B (1998) About Complex Heat Capacities and Temperature-modulated Calorimetry. J Thermal Anal and Calorimetry 54 437 65. 

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