Debye corrections

The Debye correction becomes significant at temperatrrres of thermal plasma about... 

Debye Correetion of Thermodynamie Functions in Plasma. Estimate the Debye correction for Gibbs potential per unit volume of thermal Ar plasma at atmospheric pressure and temperature T = 20,000 K. The ionization degree can be calculated using the Saha equation (see Problem 3-1). 

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. 

Equation (10.82) is a correct but unwieldy form of the Debye scattering theory. The result benefits considerably from some additional manipulation which converts it into a useful form. Toward this end we assume that the quantity srj, is not too large, in which case sin (srj, ) can be expanded as a power series. Retaining only the first two terms of the series, we obtain... 

Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hiickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hiickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hiickel theory is to... 

Intermediate values for C m can be obtained from a numerical integration of equation (10.158). When all are put together the complete heat capacity curve with the correct limiting values is obtained. As an example, Figure 10.13 compares the experimental Cy, m for diamond with the Debye prediction. Also shown is the prediction from the Einstein equation (shown in Figure 10.12), demonstrating the improved fit of the Debye equation, especially at low temperatures. 

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. 

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. 

For materials which are available not in the form of substantial individual crystals but as powders, the technique pioneered by Debye and Scherrer is employed (Moore, 1972). The powder is placed into a thin-walled glass capillary or deposited as a thin film, and the sample is placed in the X-ray beam. Within the powder there are a very large number of small crystals of the substance under examination, and therefore all possible crystal orientations occur at random. Hence for each value of d some of the crystallites are correctly oriented to fulfil the Bragg condition. The reflections are recorded as lines by means of a film or detector from their positions, the d values are obtained (Mackay Mackay, 1972). 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

Abbas Z, Gunnarsson M, Ahlberg E, Nordholm S. 2002. Corrected Debye-Hiickel (CDH) analysis of surface complexation. J Phys Chem B 106 1403-1420. 

Gunnarsson M, Abbas Z, Ahlberg E, Nordhohn S. 2004. Corrected Debye-Hiickel analysis of surface complexation HI. Spherical particle charging including ion condensation. J Coll Interf Sci 274 563-578. 

Debye-Walter factors of the source and the absorber, respectively 5r, experimental correction factor, which is constant as function of the mercury concentration), experimental line width F/2 and isomer shift 5 as a function of the Hg content of the PtHg alloy (taken from [482])... 

This is the electrostatic energy arising from ions approaching within a of each other. When subtracted from the free energy functional above the corrected Debye Huckel equation becomes... 

The hole correction of the electrostatic energy is a nonlocal mechanism just like the excluded volume effect in the GvdW theory of simple fluids. We shall now consider the charge density around a hard sphere ion in an electrolyte solution still represented in the symmetrical primitive model. In order to account for this fact in the simplest way we shall assume that the charge density p,(r) around an ion of type i maintains its simple exponential form as obtained in the usual Debye-Hiickel theory, i.e.,... 

FIG. 4 The calculated internal energy of a 1-1 salt (line) is compared with the corresponding simulation results (open circles) obtained by Van Megen and Snook (Ref. 20). The Debye-Hiickel (DH, dashed line) and corrected Debye-Hiickel (CDH, full line) theory were used together with a GvdW(I) treatment of the uncharged hard-sphere mixture. The ion diameter was 4.25 A, the temperature was 298 K and the dielectric constant e was 78.36. 

Thus we can see that a combination of van der Waals treatment of hard sphere excluded volume and Debye-Huckel treatment of screening with a minor generalization to account for hole correction of electrostatic interactions yields quite accurate bulk thermodynamic data for symmetrical salt solutions. 

Hole Correction of the Debye-Huckel Coion Density Profile The DHH Theory... 

So far in our revision of the Debye-Hiickel theory we have focused our attention on the truncation of Coulomb integrals due to hard sphere holes formed around the ions. The corresponding corrections have redefined the inverse Debye length k but not altered the exponential form of the charge density. Now we shall take note of the fact that the exponential form of the charge density cannot be maintained at high /c-values, since this would imply a negative coion density for small separations. Recall that in the linear theory for symmetrical primitive electrolyte models we have... 

FIG. 10 The calculated internal energy of a one-component plasma as a function of coupling strength is compared with corresponding simulation results (open circles) by Brush, Sahlin, and Teller (J. Chem. Phys. 45 2102 (1966). The Debye-Huckel (DH) and hole-corrected Debye-Huckel (DHH) theories were used with results as shown (indicated lines). 

First-order estimates of entropy are often based on the observation that heat capacities and thereby entropies of complex compounds often are well represented by summing in stoichiometric proportions the heat capacities or entropies of simpler chemical entities. Latimer [12] used entropies of elements and molecular groups to estimate the entropy of more complex compounds see Spencer for revised tabulated values [13]. Fyfe et al. [14] pointed out a correlation between entropy and molar volume and introduced a simple volume correction factor in their scheme for estimation of the entropy of complex oxides based on the entropy of binary oxides. The latter approach was further developed by Holland [15], who looked into the effect of volume on the vibrational entropy derived from the Einstein and Debye models. 

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