Debye-Hiickel theory model

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.,... 

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... 

Raji Heyrovska [18] has developed a model based on incomplete dissociation, Bjermm s theory of ion-pair formation, and hydration numbers that she has found fits the data for NaCl solutions from infinite dilution to saturation, as well as several other strong electrolytes. She describes the use of activity coefficients and extensions of the Debye-Hiickel theory as best-fitting parameters rather than as explaining the significance of the observed results. ... 

Dubye-Hiickel Theory of Activity Coefficient Point-Charge Model. The Debye-Hiickel theory of ion-ion interactions (Chapter 2) gives the following theoretical... 

Debye-Hiickel Theory Finite-Ion-Size Model. If the approximation of the point... 

The above equation is known as the linearized Poisson-Boltzmann equation since the assumption of low potentials made in reaching this result from Equation (29) has allowed us make the right-hand side of the equation linear in p. This assumption is also made in the Debye-Hiickel theory and prompts us to call this model the Debye-Hiickel approximation. Equation (33) has an explicit solution. Since potential is the quantity of special interest in Equation (33), let us evaluate the potential at 25°C for a monovalent ion that satisfies the condition e p = kBT ... 

The contribution of electrostatic interactions to fast association was analyzed by applying the classical Debye-Hiickel theory of electrostatic interactions between ions to mutants of bamase and barstar whose ionic side chains had been altered by protein engineering (Chapter 14).16 The association fits a two-step model that is probably general (equation 4.84). 

Issue is taken here, not with the mathematical treatment of the Debye-Hiickel model but rather with the underlying assumptions on which it is based. Friedman (58) has been concerned with extending the primitive model of electrolytes, and recently Wu and Friedman (159) have shown that not only are there theoretical objections to the Debye-Hiickel theory, but present experimental evidence also points to shortcomings in the theory. Thus, Wu and Friedman emphasize that since the dielectric constant and relative temperature coefficient of the dielectric constant differ by only 0.4 and 0.8% respectively for D O and H20, the thermodynamic results based on the Debye-Hiickel theory should be similar for salt solutions in these two solvents. Experimentally, the excess entropies in D >0 are far greater than in ordinary water and indeed are approximately linearly proportional to the aquamolality of the salts. In this connection, see also Ref. 129. 

The Debye-Hiickel Theory of Activity Coefficient The Point-Charge Model. The... 

A theoretical approach for explaining the relationship between S and the characteristics of the electrolyte was provided by Onsager on the basis of the model of ions plus ionic cloud developed in the Debye-Hiickel theory, obtaining [4]... 

In a free solution, the electrophoretic mobility (i.e., peiec, the particle velocity per unit applied electric field) is a function of the net charge, the hydrodynamic drag on a molecule, and the properties of the solutions (viscosity present ions—their concentration and mobility). It can be expressed as the ratio of its electric charge Z (Z = q-e, with e the charge if an electron and q the valance) to its electrophoretic friction coefficient. Different predictive models have been demonstrated involving the size, flexibility, and permeability of the molecules or particles. Henry s theoretical model of pdcc for colloids (Henry, 1931) can be combined with the Debye-Hiickel theory predicting a linear relation between mobility and the charge Z ... 

The thermodynamic model of Krissmann [53] was used in the calculations of these experiments, though this was limited by the phase equilibrium (Eq. (3)) and the reaction equilibrium (Eq. (4)). Calculation of the activity coefficients of the H+ ions and HSOj was performed according to the extended Debye-Hiickel theory, using the approximation of Pitzer... 

Debye-Hiickel theory — The interactions between the ions inside an electrolyte solution result in a nonideal behavior as described with the concepts of mixed-phase thermodynamics. Assuming only electrostatic (i.e., coulombic) interactions - Debye and - Hiickel suggested a model describing these interactions resulting in - activity coefficients y suitable for further thermodynamic considerations. Their model is based on several simplifications ... 

It was necessary to measure the dielectric constant and density of each solvent mixture studied. Densities were determined in a constant-temperature bath maintained to within 0.02°C. Gay-Lussac pycnometers with a capacity of 25 mL were used for density measurements. Dielectric constants were determined with a Balsbaugh Model 2TN50 conductivity cell having a cell constant of 0.001. A Janz-Mclntyre a-c bridge (17) was used. The dielectric constants and densities of the solvents are listed in Table I, along with the constants A and B of the Debye-Hiickel theory. 

The simplest model capable of reproducing the ion-ion interactions at high dilutions is that given by the Debye-Hiickel Theory."" The model assumes that the solvent can be characterized as a structureless, isotropic medium of dielectric constant D which is identical with its macroscopic dielectric constant. The ions are regarded as point charges imbedded in hard spheres of fixed radius and of dielectric constant 1. 

A second approximation is to use the Debye-Hiickel theory to give the activity of the ions i and j, while a third approximation is to use the Kirkwood or Amis-Jaffe models for ion pairs to evaluate the activity coefficient of the ion pair. Under these conditions we can write ... 

The electrostatic methods just discussed suitable for nonelectrolytic solvent. However, both the GB and Poisson approaches may be extended to salt solutions, the former by introducing a Debye-Huckel parameter and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation. The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Hiickel theory because the model includes the finite size of the solute molecules. 

So here, the term theory will be used in a way that embraces the typical named theories of chemistry such things as molecular orbital theory, valence shell electron pair repulsion theory, transition state theory of reactions, and Debye Hiickel theory of electrolyte solutions. No decisive distinction will be made between theory, model, and other similar terms. But there is one distinction that we do make. The term theory is considered in an epistemological sense—as an expression of oin best knowledge and belief about the way chemical systems work. 

In chapter 3, it was shown that the Debye-Hiickel theory for ion-ion interactions is able to account for solution non-ideality in very dilute systems. The same model forms the basis for understanding the concentration dependence of the conductance observed for strong electrolytes. Thus, Onsager [9] showed in 1927 that the limiting conductance law for 1-1 electrolytes has the form... 

In summary, Onsager s extension of the Debye-Hiickel theory to the nonequilibrium properties of electrolyte solutions provides a valuable tool for deriving single ion properties in electrolyte solutions. Examination of the large body of experimental data for aqueous electrolyte solutions helped confirm the model for a strong electrolyte. In more recent years, these studies have been extended to non-aqueous solutions. Results in these systems are discussed in the following section. 

Numerous models predict the activity coefficient of individual ions in solution. The one by Debye and Hiickel [8] considers only electrostatic (columbic) interactions between cations and anions in a dilute solution of a single, completely dissociated salt. It is assumed that ion-ion interactions (as opposed to other phenomena such as ion-solvent interactions, ion solvation effects, and variations in the solvent dielectric constant with salt concentration) cause the ion activity coefficients to deviate from 1.0. From a practical point, only the Debye-Hiickel activity coefficient relationship is needed, along with some knowledge of the theory s shortcomings, which restrict its application. For a dilute electrolytic solution containing a binary salt (i.e., a salt with one type each of cation and anion species), the ion activity coefficient from Debye-Hiickel theory is given by... 

Equation (26.41) predicts to within approximately 10% mean molal activity coefficients for salt concentrations up to 0.1 molal. The more accurate form of the activity coefficient equation [Equation (26.40)] allows the model to be extended to salt concentrations up to 0.5 molal. To expand the applicability of the Debye-Hiickel theory to higher concentrations, additional terms are added to Equation (26.40), such as [4]... 

Debye-Hiickel theory that has not been criticized. From this perspective, electrolyte models are simply the best tools available to assess whether the dependence of electron transfer rate constants on ionic strength is sufficiently well behaved to justify use of the Marcus model. For this, and despite their shortcomings, they are indispensable. 

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