Debye theory

Inspection of Fig. 3.9 suggests that for polyisobutylene at 25°C, Ti is about lO hr. Use Eq. (3.101) to estimate the viscosity of this polymer, remembering that M = 1.56 X 10. As a check on the value obtained, use the Debye viscosity equation, as modified here, to evaluate M., the threshold for entanglements, if it is known that f = 4.47 X 10 kg sec at this temperature. Both the Debye theory and the Rouse theory assume the absence of entanglements. As a semi-empirical correction, multiply f by (M/M. ) to account for entanglements. Since the Debye equation predicts a first-power dependence of r) on M, inclusion of this factor brings the total dependence of 77 on M to the 3.4 power as observed. 

In applying the Debye theory to concentrated solutions, we must extrapolate the results measured at different concentrations to C2 = 0 to eliminate the effects of solute-solute interactions. 

Using the original Hc2/r values, recalculate M using the various refractive index gradients. On the basis of self-consistency, estimate the molecular weight of this polymer and select the best value of dn/dc2 in each solvent. Criticize or defend the following proposition Since the extension of the Debye theory to large particles requires that the difference between n for solute and solvent be small, this difference should routinely be minimized for best results. 

Chapter 10, the last chapter in this volume, presents the principles and applications of statistical thermodynamics. This chapter, which relates the macroscopic thermodynamic variables to molecular properties, serves as a capstone to the discussion of thermodynamics presented in this volume. It is a most satisfying exercise to calculate the thermodynamic properties of relatively simple gaseous systems where the calculation is often more accurate than the experimental measurement. Useful results can also be obtained for simple atomic solids from the Debye theory. While computer calculations are rapidly approaching the level of sophistication necessary to perform computations of... 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

The results of the Debye theory reproduced in the lowest order of perturbation theory are universal. Only higher order corrections are peculiar to the specific models of molecular motion. We have shown in conclusion how to discriminate the models by comparing deviations from Debye theory with available experimental data. 

The central Lorentzian part of the IR spectrum (2.55) has the same shape as in the classical Debye theory and may be of various origins. The impact mechanism of reorientation can be confirmed judging by the shape of the wings only. The inertial effects show themselves in the asymptotic relation... 

The logarithm of the rate ratios is plotted versus p1/2/(l + ji1/2) in Figure 6.1. In terms of the Bronsted model of ionic reactions and application to the Debye theory of ionic solutions, we may write ... 

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. 

Molar polarizability, P, can be evaluated from measurements of D by Eq. (18), and a critical test of the Debye theory (Debye, 1954) is provided by a plot of P against 1/T, which gives a straight line of slope... 

F = E + 47T//3). In fact, no Curie point can be observed with polar liquids as a function of l/T and the Debye theory is therefore not valid for our strongly polar mixtures. A much more refined theory than that of Debye, which is due to Kirkwood (1939), suffices to account for the general character of the dielectric constant of strongly polar liquids and its changes as a function of temperature. It is not the purpose of this article to give a full account of this theory. 

When Tc is of the order of or less than spectral density of quanta is less. Thus, for example in a liquid, it is found that Ti decreases with increasing viscosity (increasing tc) to a certain point (tc Po) and then increases with increasing viscosity. A simple proportionality between Tc and the viscosity is given by the Debye theory of liquids. This dependence of Ti on has also been verified for several solids such as the ammonium halides (62), solid benzene, and partially deuterated benzenes (63). 

Theoretical investigations trying to explain the k deviation from i.e., of the Debye theory, are summarized as follows ... 

Figure 7 shows an aberration-free intensity distribution at the focus of a typical objective lens similar to that used for DLW lithography. Calculations were carried out using a vectorial Debye theory, which accounts for the polarization effects. For the linearly polarized wave it can be seen that the spot is elongated along the polarization vector. To reduce this asymmetry, a X/4-plate can be used to convert the polarization of the incident beam to circular, which can be interpreted as a combination of two mutually perpendicular linearly polarized components. Thus, width of the photomodified line becomes independent of the beam scanning direction in the sample. 

Fig. 7 Plane wave focusing by a NA = 1.35 objective lens, calculated using vectorial Debye theory, a The normalized 3D intensity distribution with the cutoff threshold at 1% intensity. The lateral cross-sections are plotted on a log-scale at the axial positions z = 0 (b) and z = 7-/2 (c), respectively. Contour lines are plotted at 0.5 (inner) and 1/e (outer) levels, respectively. Polarization of the plane wave was horizontal (along x)...

The point spread function (PSF) resulting imder these circiunstances was derived in using scalar Debye theory [43] ... 

There is no oscillation the polarization merely relaxes toward zero with a time constant t. In the following paragraphs, we shall use (9.35), the basic assumption of the Debye theory, to derive an expression for the dielectric function of a collection of permanent dipoles. 

The Debye equations (9.42) are particularly important in interpreting the large dielectric functions of polar liquids one example is water, the most common liquid on our planet. In Fig. 9.15 measured values of the dielectric functions of water at microwave frequencies are compared with the Debye theory. The parameters tod, e0v, and r were chosen to give the best fit to the experimental data r = 0.8 X 10 -11 sec follows immediately from the frequency at which e" is a maximum e0d — e0v is 2e"ax. 

Section 5.5 moves on to an extension of the Rayleigh theory essential for colloid science, namely, the Debye theory for particles of the order of the wavelength of the radiation source. The important concept of interference effects, the form factor, the Zimm plot, and... 

Equation (67) shows clearly that i should be measured as a function of both concentration and angle of observation in order to take full advantage of the Debye theory. The light scattering photometer described in Section 5.4 is designed with this capability, so this requirement introduces no new experimental difficulties. The data collected then consist of an array of i/I0 values (i needs no subscript since it now applies to small and large particles) measured... 

The particles and the fluid are effectively transparent to x-rays and neutrons i.e., their effective refractive indices are nearly the same. Therefore, the criterion we specified in Equation (61) is easily satisfied, and we can avoid the need for the more complicated Mie theory (see Section 5.7b) and use the Rayleigh-Debye theories. 

What is the Debye theory of light scattering What are its assumptions and limitations ... 

The Debye theory [220] in which a sphere of volume V and radius a rotates in a liquid of coefficient of viscosity t has already been mentioned. There is angular momentum transfer across the sphere—liquid interface that is, the liquid sticks to the sphere so that the velocity of the sphere and liquid are identical at the sphere s surface. Solution of the rotational diffusion equation... 

Fig. 6.48. The role of non-equilibrium charge screening in eliminating the Coulomb catastrophe the dimensionless reaction rate vs time. Dotted curve - the Debye theory (no screening and similar particle correlation) broken curves - the solution of kinetic equations incorporating these correlations but neglecting screening full curves, screening is taken into account. Parameters L = 5, Da = Db. Curves 1 to 3 correspond to dimensionless concentrations ...

DEBYE THEORY OF SPECIFIC HEAT. The specific heal of solids is attributed to the excitation of thermal vibrations of the lattice, whose spectrum is taken to be similar to that of an elastic continuum, except that it is cut off at a maximum frequency in such a way that the total number of vibrational modes is equal to the total number of degrees of freedom of the lattice. 

The Einstein equation was the first approximation to a quantum theoretical explanation of the variation of specilic heat with temperature. It was later replaced by the Debye theory of specific heat and its modifications. 

Landau treats liquid helium hy an approach similar in lhat ol the Debye theory of solids. The longitudinal and transverse sound waves, which are the elementary cxcitalions of that theory of solids. corresponU in the case... 

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