Debye expression

Equation (2.61) predicts a 3.5-power dependence of viscosity on molecular weight, amazingly close to the observed 3.4-power dependence. In this respect the model is a success. Unfortunately, there are other mechanical properties of highly entangled molecules in which the agreement between the Bueche theory and experiment are less satisfactory. Since we have not established the basis for these other criteria, we shall not go into specific details. It is informative to recognize that Eq. (2.61) contains many of the same factors as Eq. (2.56), the Debye expression for viscosity, which we symbolize t . If we factor the Bueche expression so as to separate the Debye terms, we obtain... 

The dielectric constant D is related to the dipole moment fi by the Debye expression... 

The formation of strong hydrogen bonds in glycerol leads to a very high viscosity and also a long relaxation time (see Table 1.2) however, a comparison with the data for alcohols with comparable volumes suggests that the relationship between viscosity and relaxation time is not the simple linear one implied by the Debye expression. 

Complex dielectric susceptibility data such as those in Figure 15.6 provide a detailed view of the dynamics of polar nanodomains in rls. They define relaxation frequencies, /, corresponding to the e (T) peak temperatures Tm, characteristic relaxation times, r = 1/tu (where uj = 2nf is the angular frequency), and a measure of the interaction among nanodomains as represented by the deviation of the relaxation process from a Debye relaxation. Analysis of data on pmn and other rls clearly shows that their dipolar relaxations cannot be described by a single relaxation time represented by the Debye expression... 

Two atoms, p and q, bonded together at a distance rpq give a contribution, ipq(s), to the diffraction curve, which can be calculated according to the Debye expression... 

We used the Debye expression for the temperature dependence of the specific heat... 

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. 

As mentioned in Section II.B, the dielectric response in the frequency domain for most complex systems cannot be described by a simple Debye expression (17) with a single dielectric relaxation time. In a most general way this dielectric behavior can be described by the phenomenological Havriliak-Negami (HN) formula (21). 

The three rate constants, kq, kr, kd, and one coefficient a, theoretically can be solved by the slopes and intercepts in eqs. 83 and 88. However, the intercept of Eq. 88 is a virtual intercept, which was physically meaningless since the y-intercept did not exist in the case of high humic concentration, (i.e. humic concentration cannot be zero in this model). Therefore, another equation is needed to solve the four constants. Practically, it is normally assumed that the rate for exothermic quenching is diffusion-controlled. The quenching rate, kq, can therefore be evaluated from the modified Debye expression (William M., 1976). 

It is often found that the Curie law Eq. (21) is followed by many magnetically dilute substances other than free atoms or ions. There is, in addition, a second-order contribution to the paramagnetic susceptibility, the so-called temperature-independent paramagnetism Not. (also abbreviated TIP, cf. section 1,1.3.6) which arises from states separated from the ground state by an energy k T It follows that the molar susceptibility corrected for diamagnetism may be frequently represented by the Langevin-Debye expression... 

Similar property distributions occur throughout the frequency spectrum. The classical example for dielectric liquids at high frequencies is the bulk relaxation of dipoles present in a pseudoviscous liquid. Such behavior was represented by Cole and Cole [1941] by a modification of the Debye expression for the complex dielectric constant and was the first distribution involving the important constant phase element, the CPE, defined in Section 2.1.2.3. In normalized form the complex dielectric constant for the Cole-Cole distribution may be written... 

For random, isotropic and inhomogeneous materials, such as swollen networks, one has the Debye expression (19) for the Rayleigh ratio rco... 

Additional thermodynamic and conformational information is available in the functions P(q, c) and H(q, c). The well-known Debye expression for P(q, 0) is found to be a good representation for flexible chain polymers over a wide range inu = (qRo)"[6,45-47] ... 

Figure 28.9 shows the heat capacities of several elements, along with curves representing the Debye function for the Debye temperatures given. At high temperatures, the Debye expression conforms to the law of Dulong and Petit. 

The qualitative basis of Eq. (8) is the assumption that ev i layer and chain structures behave like an isotropic elastic continuum for long wavelength vibrations. For shorter wavelength, vibrations in different layers or chains are assumed to be completely decoupled. The Debye expression derives from Eq, (8) if we assume ... 

The volume and compressibility are both approximately temperature independent, and, since both vary in the same direction with temperature, their ratio remains essentially constant. Hence if y were temperature independent, one could surmise that a should vary in proportion to the specific heat, and therefore should have a temperature variation similar to the C (Debye) curve shown in Fig. 3.14. This is indeed the case, and provides a useful correlation for thermal expansion data. In fact, at low temperatures (T<0 )/12), the coefficient of thermal expansion is proportional to in accord with the Debye expression. 

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