Debye-Huckel approach

The first term on the right-hand side of this equation is zero, since it is simply the sum of the electrical charge in solution, which must be zero for a neutral electrolyte solution. The third term is also zero for electrolytes with equal numbers of positive and negative ions, such as NaCl and MgSC>4. It would not be zero for asymmetric electrolytes such as CaCE. However, in the Debye-Huckel approach, all terms except the second are ignored for all ionic solutions. Substitution of the resulting expression into equation (7.20) gives the linear second-order differential equation... 

A number of other attempts have been made to account for the properties of concentrated aqueous solutions of ionic compounds by procedures that represent further improvements on the simple Debye-Huckel approach. However, they lie outside the scope of the present chapter. The important point to emphasize is that the concentrated aqueous solutions that are generally employed in the preparation of AB cements tend to exhibit significant ion-ion interactions such interactions lead to significant deviations from ideality which may be accounted for by substantial extension of the ideas of simple dilute solution theory. 

In what way does the Gouy-Chapman theory extend the Debye-Huckel approach ... 

Michaeli and Overbeek used the Debye-Huckel approach to estimate the enthalpy of the PE-PE and PE-solvent interaction [23]. In this model, the Flory entropy is used as the entropic term of the PE chain, and the electrostatic free enthalpy is calculated from the electrostatic energy of a strong PE [23]. By thermodynamic considerations, Michaeli and Overbeek showed that for small univalent counterions no coacervation (complex-formation between PE) occurs... 

When A > A the ions are free and the Debye-Huckel theory applies. When A < A the two ions tend to approach each other and form an ion-pair, and there is no contribution to the electrostatic energy from the interaction between an ion and its atmosphere. 

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... 

Changes in activity coefficients (and hence the relationship between concentration and chemical activity) due to the increased electrostatic interaction between ions in solution can be nicely modeled with well-known theoretical approaches such as the Debye-Huckel equation ... 

The derivation of the equations of the Debye-Huckel theory did not require differentiation between a solution of a single electrolyte and an electrolyte mixture provided that the limiting law approximation Eq. (1.3.24), was used, which does not contain any specific ionic parameter. If, however, approximation (1.3.29) is to be used, containing the effective ionic diameter ay it must be recalled that this quantity was introduced as the minimal mean distance of approach of both positive and negative ions to the central ion. Thus, this quantity a is in a certain sense an average of effects of all the ions but, at the same time, a characteristic value for the given central... 

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 parameter67 and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation.68 The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Huckel theory because the model includes the finite size of the solute molecules. 

Siveral approaches are available in the case of mixed electrolyte solutions. The Guntelberg equation can be used at very high dilutions to avoid the ambiguity in the meaning of aD, the distance of closest approach, when several electrolytes are present. This equation is empirical and has fewer terms than the Debye-Huckel extended equation. I found it to yield poor agreement with experimental results even at m = 0.01 for NaCl at 25°C (y+ caic = 0.8985 and y+ exp = 0.9024). For the Davies equation for m = 0.20 one obtains y+ calc = 0.752 andy+exo = 0.735 also for NaCl at 25°C. 

The statistical thermodynamic approach of Pitzer (14), involving specific interaction terms on the basis of the kinetic core effect, has provided coefficients which are a function of the ionic strength. The coefficients, as the stoichiometric association constants in our ion-pairing model, are obtained empirically in simple solutions and are then used to predict the activity coefficients in complex solutions. The Pitzer approach uses, however, a first term akin to the Debye-Huckel one to represent nonspecific effects at all concentrations. This weakens somewhat its theoretical foundation. 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

A polyelectrolyte solution contains the salt of a polyion, a polymer comprised of repeating ionized units. In dilute solutions, a substantial fraction of sodium ions are bound to polyacrylate at concentrations where sodium acetate exhibits only dissoci-atedions. Thus counterion binding plays a central role in polyelectrolyte solutions [1], Close approach of counterions to polyions results in mutual perturbation of the hydration layers and the description of the electrical potential around polyions is different to both the Debye-Huckel treatment for soluble ions and the Gouy-Chapman model for a surface charge distribution, with Manning condensation of ions around the polyelectrolyte. 

The discussion above is a description of problem that requires answers to the following (1) the determination of the distribution of ions around a reference ion, and (2) the determination of the thickness (radius) of the ionic atmosphere. Obviously this is a complex problem. To solve this problem Debye and Huckel used a rather general approach they suggested an oversimplified model in order to obtain an approximate solutions. The Debye-Huckel model has two basic assumptions. The first is continuous dielectric assumption. In this assumption water (or the solvent) is a continuous dielectric and is not considered to be composed of molecular species. The second, is a continuous charge distribution in the ionic atmosphere. Put differently, charges of the ions in the ionic surrounding atmosphere are smoothened out (continuously distributed). 

The primary particle involved in the screening process is the mobile electron. One has then the problem of a self-consistent calculation of the charge distribution in the neighborhood of a test charge. The Thomas-Fermi approach to this problem is the analog of the Debye-Huckel calculation wherein allowance has been made for the Pauli exclusion principle. From any standard text one can obtain the Poisson equation (19)... 

In the Debye-Huckel theory, an ion in solution is treated as a conducting sphere. The distance of closest approach of two ions is a.4 The solution beyond a... 

The left-hand side of this equation can be calculated from measurements of cell voltage as a function of concentration. The second term on the right-hand side becomes zero at infinite dilution. However, because no meaningful measurements can be made at zero concentration of reactants, we must extrapolate the equation to infinite dilution using the known concentration behavior of activity coefficients. In approaching infinite dilution, it is sufficient to use the Debye-Huckel... 

Abstract, The solution of the Debye-Huckel equation for a system of spheres with arbitrary radii and surface charge in electrolyte solutions is described. The general theoretical approach to describe such systems is elaborated. The practically important case of two spheres is considered in detail. Finite closed formulae to calculate the interaction energy of two spherical particles with constant surface charges are obtained from general expressions in zero approximation. Known relationships follow from our formulae in limiting cases. 

As the dependency does not include any specific property of the ion (in particular its chemical identity) but only its charge the explanation of this dependency invokes properties of the ionic cloud around the ion. In a similar approach the Debye-Huckel-Onsager theory attempts to explain the observed relationship of the conductivity on c1/2. It takes into account the - electrophoretic effect (interactions between ionic clouds of the oppositely moving ions) and the relaxation effect (the displacement of the central ion with respect to the center of the ionic cloud because of the slightly faster field-induced movement of the central ion, - Debye-Falkenhagen effect). The obtained equation gives the Kohlrausch constant ... 

The classical approach to correct charge carrier interactions in liquid systems is the Debye-Huckel theory which is extensively discussed in textbooks.79 The decisive parameter is the screening length... 

The basis for this is the theory of specific ion interaction, proposed by J. N. Bronsted, J. Am. Chem. Soc., 44, 877 (1922), to the effect that ions will be uniformly influenced by ions of the same sign but specifically influenced by ions of opposite sign. This is quite reasonable in terms of the above equation, in which the uniform influence is included in the Debye-Huckel terms. The implication is that only ions of opposite sign will approach each other closely enough to produce specific interactions. What the nature of the interactions is, however, is not specified. 

---