Specific heat vibrational contribution

Table III presents integral excess entropies of formation for some solid and liquid solutions obtained by means of equilibrium techniques. Except for the alloys marked by a letter b, the excess entropy can be taken as a measure of the effect of the change of the vibrational spectrum in the formation of the solution. The entropy change associated with the electrons, although a real effect as shown by Rayne s54 measurements of the electronic specific heat of a-brasses, is too small to be of importance in these numbers. Attention is directed to the very appreciable magnitude of the vibrational entropy contribution in many of these alloys, and to the fact that whether the alloy is solid or liquid is not of primary importance. It is difficult to relate even the sign of the excess entropy to the properties of the individual constituents.

The high-temperature contribution of vibrational modes to the molar heat capacity of a solid at constant volume is R for each mode of vibrational motion. Hence, for an atomic solid, the molar heat capacity at constant volume is approximately 3/. (a) The specific heat capacity of a certain atomic solid is 0.392 J-K 1 -g. The chloride of this element (XC12) is 52.7% chlorine by mass. Identify the element, (b) This element crystallizes in a face-centered cubic unit cell and its atomic radius is 128 pm. What is the density of this atomic solid ... 

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. 

Differences in specific heats can be obtained in a similar fashion. Since translational and rotational contributions to Cp at elevated temperatures are minor, the differences to be accounted for are entirely due to vibrational effects. The most effective way to accomplish this is to identify the incremental contribution of each atom or group to Cp, and add or subtract this value from... 

The transport of heat in metallic materials depends on both electronic transport and lattice vibrations, phonon transport. A decrease in thermal conductivity at the transition temperature is identified with the reduced number of charge carriers as the superconducting electrons do not carry thermal energy. The specific heat and thermal conductivity data are important to determine the contribution of charge carriers to the superconductivity. The interpretation of the linear dependence of the specific heat data on temperature in terms of defects of the material suggests care in interpreting the thermal conductivity results to be described. 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

The possibility of deactivation of vibrationally excited molecules by spontaneous radiation is always present for infrared-active vibrational modes, but this is usually much slower than collisional deactivation and plays no significant role (this is obviously not the case for infrared gas lasers). CO is a particular exception in possessing an infrared-active vibration of high frequency (2144 cm-1). The probability of spontaneous emission depends on the cube of the frequency, so that the radiative life decreases as the third power of the frequency, and is, of course, independent of both pressure and temperature the collisional life, in contrast, increases exponentially with the frequency. Reference to the vibrational relaxation times given in Table 2, where CO has the highest vibrational frequency and shortest radiative lifetime of the polar molecules listed, shows that most vibrational relaxation times are much shorter than the 3 x 104 /isec radiative lifetime of CO. For CO itself radiative deactivation only becomes important at lower temperatures, where collisional deactivation is very slow indeed, and the specific heat contribution of vibrational energy is infinitesimal. Radiative processes do play an important role in reactions in the upper atmosphere, where collision rates are extremely slow. 

The specific heat of a semiconductor has contributions from lattice vibrations, free carriers and point and extended defects. For good quality semi-insulating crystals only the lattice contribution is of major significance. Defect-free crystals of group III nitrides are difficult to obtain, and thus the specific heat measurements are affected by the contributions from the free carriers and the defects. While the specific heat of AIN is affected by the contribution of oxygen impurities, the data for GaN and InN are affected by free electrons, especially at very low temperatures. 

Using the Debye expression for the specific heat to fit these data, the Debye temperature of InN was obtained 0d = 660 K [20], The resulting specific heat curve and the experimental data are plotted in FIGURE 1. Since the temperature range of these measurements is rather narrow, it is difficult to compare these results and the Debye curve. Good quality, pure InN crystals are extremely difficult to grow and the deviations from the Debye curve indicate that the InN samples have significant contributions from non-vibrational modes. 

Each of the vibrational degrees of freedom given by Eq. (6.1) would have a mean kinetic and potential energy of kT according to equipartition, and would contribute an amount R to the specific heat. As with diatomic molecules, however, the quantum theory tells us, and wc find... 

Once we have obtained the dispersion curves for the metal, wc may proceed to other properties just as we did with the covalent solids. In particular, we may quantize the vibrations as was done for the covalent solids and obtain the appropriate contribution to the specific heat. We shall not repeat that analysis now for the simple metals but shall wish to use the customary terminology by referring to the vibrations as phonons. 

A thermodynamic quantity not very often measured for organic superconductors is the specific heat, C. Usually the crystal sizes are rather small and consequently a high sensitivity of the apparatus is needed. In most experiments, therefore, an assembly of many pieces of material is necessary to gain better resolution. In addition, the jump of C at Tc is expected to be rather small especially for compounds with higher transition temperatures because of the comparatively large lattice contribution to C owing to the low electron density and the low vibrational frequencies. 

The heat capacity was estimated in the same manner as for ZrBr (cr) [see ZrBr (cr) table]. The values for 9p and 0 were taken to be the same as those estimated for ZrBr (cr). The internal contribution was obtained from the estimated ZrBr vibrational frequencies and the anharmonicity factor a" was taken to be 2.5 x 10". The specific heat above 300 K was obtained by graphical extrapolation. 

The contribution of each vibration to the specific heat increases with increasing temperature and may be written in the form RP where... 

At high temperatures (i.e. large Tj ) this function tends to unity, and the contribution to the specific heat from the vibration tends to R per mole. At low temperatures the function tends to zero. 

We may, therefore, write the vibrational contribution to the specific heat as... 

The solution arrived at in our linear elastic model may be contrasted with those determined earlier in the lattice treatment of the same problem. In fig. 5.13 the dispersion relation along an arbitrary direction in g-space is shown for our elastic model of vibrations. Note that as a result of the presumed isotropy of the medium, no g-directions are singled out and the dispersion relation is the same in every direction in g-space. Though our elastic model of the vibrations of solids is of more far reaching significance, at present our main interest in it is as the basis for a deeper analysis of the specific heats of solids. From the standpoint of the contribution of the thermal vibrations to the specific heat, we now need to determine the density of states associated with this dispersion relation. 

The specific heat (C) is the amount of energy required, per unit mass or per mole, to raise the temperature of a substance by one degree. This is the derivative of its internal energy dU/dT, and since magnetic levels make a contribution to this their separations can in principle be measured from C(T) measurements. However, the magnetic contribution to the specific heat must be disentangled from that of lattice vibrational modes. 

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