Debye-Hiickel reduced

The first assumption of the Debye-Hiickel theory is that is spherically symmetric. With the elimination of any angular dependence, the Poisson equation (expressed in spherical-polar coordinates) reduces to... 

Thus we have found that the screening should be more efficient than in the Debye-Hiickel theory. The Debye length l//c is shorter by the factor 1 — jl due to the hard sphere holes cut in the Coulomb integrals which reduce the repulsion associated with counterion accumulation. A comparison with Monte Carlo simulation results [20] bears out this view of the ion size effect [19]. 

As a check on the consistency of our mathematics, it is profitable to verify that Equation (63) reduces to Equation (37) in the limit of low potentials. Expanding the exponentials in T and truncating the series so that only one term survives in both the numerator and denominator results in the Debye-Hiickel expression, Equation (37). 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

Again we wish to emphasize that it is the modified moments (aWtU ) and (V5(D) Moi) arising from the use of Onsager s theory of the moments in liquids that are important. With reasonable values of internal refractive index, the moments in vacuum can be reduced. Furthermore, our theory does account for values of j greater than those predicted by the Debye-Hiickel limiting law if the ions can be assumed to have dipole characteristics. 

If the Debye-Hiickel approximation, zetf/JkT < 1, is made, equation (8.7) reduces to equation (8.5). 

Fig. 13 Measured shear modulus G as a function of the reduced intermicellar distance, at r/2Rm

For the spherical Debye-Hiickel approximation, the Poission-Boltz-mann Equation reduces to... 

Hence, the complicated problem of the time-averaged distribution of ions inside an electrolytic solution reduces, in the Debye-Hiickel model, to the mathematically simpler problem of finding out how the excess charge density q varies with distance r from the central ion. 

Eq. (178) only represents a limiting law owing to the reduced validity of the prerequisites on which the Debye-Hiickel theory is based. Thus, Eq. (178) is applicable to monovalent 1-1 electrolytes in aqueous solution to, at most, an ionic strength of 10-2 mol/L. 

At ionic strengths between 0,001 and 0.0001 mol/kg the separate curves for individual ions of the same valence converge on a single curve. At these low ionic strengths, the term B afs/T approaches zero, and the Debye-Hiickel equation is reduced to... 

If the treatment is restricted to very low concentrations these higher terms drop out and the equation reduces to the simple Debye-Hiickel-Onsager equation ... 

Hydration theory was first applied to aqueous electrolytes by Stokes and Robinson (1) in 1948. The approach is to correct for the fact that the actual solute species do not exist as bare ions as they are normally written, but as hydrated aqueous complexes. In the case of a solution of an aqueous nonelectrolyte, this reduces to the assumption that the solution is ideal, or nearly so, if one writes the formula of the solute to include the waters of hydration and computes its concentration and that of the solvent on this basis. In the case of an aqueous electrolyte, it reduces to the assumption that the Debye-Hiickel model becomes adequate (or at least more adequate) to describe the activity coefficients if this treatment is applied. 

A simple Debye-Hiickel model (15) may be employed to provide a first approximation to the dependence of E° on ionic strength. From Eq. (6), the measured potential E is related to the midpoint potential and the activities of the oxidized and reduced species ... 

FIG. 5.5 Convergence of the iteration to very small residuals in the reduced basis method, for systems contrived to have large positive and negative residuals at the start of the iteration. The tests assume Debye-Hiickel activity coefficients ( ) and an ideal solution in which the activity coefficients are unity (O). 

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