Debye’s theory

The considerations on p. 530 require revision in the light of Debye s theory. 

Debye s theory, considered in Chapter 2, applies only to dense media, whereas spectroscopic investigations of orientational relaxation are possible for both gas and liquid. These data provide a clear presentation of the transformation of spectra during condensation of the medium (see Fig. 0.1 and Fig. 0.2). In order to describe this phenomenon, at least qualitatively, one should employ impact theory. The first reason for this is that it is able to describe correctly the shape of static spectra, corresponding to free rotation, and their impact broadening at low pressures. The second (and main) reason is that impact theory can reproduce spectral collapse and subsequent pressure narrowing while proceeding to the Debye limit. 

It is predicted that the dielectric constants of solid HC1, HBr, and HI at temperatures just below the melting points will be very high and dependent on the temperature, the values being given by Debye s theory of the orientation of electric dipole molecules while the low-temperature forms will have low dielectric constants nearly independent of the temperature. 

The SI unit for heat capacity is J-K k Molar heat capacities (Cm) are expressed as the ratio of heat supplied per unit amount of substance resulting in a change in temperature and have SI units of J-K -moC (at either constant volume or pressure). Specific heat capacities (Cy or Cp) are expressed as the ratio of heat supplied per unit mass resulting in a change in temperature (at constant volume or pressure, respectively) and have SI units of J-K -kg . Debye s theory of specific heat capacities applies quantum theory in the evaluation of certain heat capacities. 

As for graphite, its zero-point energy, ZPE = R6 + jR0 , is most conveniently deduced from Debye s theory [197,198] by separating the lattice vibrations into two approximately independent parts, with Debye temperatures (in plane) and 6j (perpendicular). A balanced evaluation gives ZPE 3.68 kcal/mol [199]. 

Perrin [223] extended Debye s theory of rotational relaxation to consider spheroids and ellipsoids. Using Edwards analysis [224] of the torque on such bodies, Perrin found two or three rotational relaxation times, respectively. However, except for bodies very far from spherical, these times are within a factor of two of the Debye rotational times [eqn. (108)] for stick boundary conditions. 

In polymer theory, the LDT result corresponds to the theory of large chain extensions P. J. Flory, Statistical Mechanics of Chain Molecules, Interscience, New York, 1969. Another mapping exists onto Debye s theory for dielectric properties of molecules with permanent dipoles P. Debye, Polar Molecules, reprinted, Dover, New York, 1958. 

Diffusion-Controlled Reactions. The specific rates of many of the reactions of elq exceed 10 Af-1 sec.-1, and it has been shown that many of these rates are diffusion controlled (92, 113). The parameters used in these calculations, which were carried out according to Debye s theory (41), were a diffusion coefficient of 10-4 sec.-1 (78, 113) and an effective radius of 2.5-3.0 A. (77). The energies of activation observed in e aq reactions are also of the order encountered in diffusion-controlled processes (121). A very recent experimental determination of the diffusion coefficient of e aq by electrical conductivity yielded the value 4.7 0.7 X 10 -5 cm.2 sec.-1 (65). This new value would imply a larger effective cross-section for e aq and would increase the number of diffusion-controlled reactions. A quantitative examination of the rate data for diffusion-controlled processes (47) compared with that of eaq reactions reveals however that most of the latter reactions with specific rates of < 1010 Af-1 sec.-1 are not diffusion controlled. 

The specific heat of Si3N4 ceramics is in the temperature range 293 up to 1200 K [Cp (293 K) = 0.67 KJ (K kg)-1] nearly independent of the composition of the additives. The isobaric specific heat values agree well with the isochoric specific heat calculated by Debye s theory. Also the Dulong Petit s rule can applied as an approximation of the Cv values [25 J(K mol)-1] at temperatures >1100 K [371]. From the Cp values at around 100 K the amount of the amorphous grain boundary phase can be calculated [371]. 

Debye s Theory and the Parameter y.—Debye s theory furnishes us with an approximation to the frequency spectrum of a solid, and we can use this approximation to find how the frequencies change with... 

This is a heat capacity proportional to the temperature, and in Sec. 5, Chap. V, we computed it for a particular case, showing that it amounted to only about 1 per cent of the corresponding specific heat of free electrons on the Boltzmann statistics, at room temperature. In Table XXIX-2 we show the value of the electronic specific heat at 300° abs., computed from the values of Wi which we have already found, in calories per mole. We verify the fact that this specific heat is small, and for ordinary purposes it can be neglected, so that the specific heat of a metal can be found from the Debye theory, considering only the atomic vibrations. At low temperatures, however, Eq. (2.4) gives a specific heat varying as the first power of the temperature, while Debye s theory, as given in Eq. (3.8),... 

Yet another cause for deviation from Eotvos s law may be found in Bom and Courant s theory1 of the motions of the molecules of liquids, which follows the lines of Debye s theory of the specific heat of solids. Assuming three degrees of freedom for the motions of the molecules, they obtained good agreement with experiment for several liquids for which the constant is about 2-1. If the number of degrees of freedom is n, the constant is altered on their theory in the ratio (n/3). ... 

The actual dependence of pATsp on the temperature is rather complicated because of the dependence of the specific heat Cp on T, which is given by Debye s theory of specific heat for the reacting oxides and corresponding lattice dynamical model for crystalline solids. Simple assumptions regarding the net change in specific heats of the components involved in the dissolution reactions, however, allow one to avoid these complications [3]. 

For a non-ideal (pseudo) two-phase system having interface layer, the overall scattering will show negative deviation from Porod s law (see the curve 11 in Fig.l) and Debye s theory (see the curve II in Fig.2), thence the Porod equation (1) becomes [6,9]... 

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