Debye-Huckel theory extension

Marshall s extensive review (16) concentrates mainly on conductance and solubility studies of simple (non-transition metal) electrolytes and the application of extended Debye-Huckel equations in describing the ionic strength dependence of equilibrium constants. The conductance studies covered conditions to 4 kbar and 800 C while the solubility studies were mostly at SVP up to 350 C. In the latter studies above 300°C deviations from Debye-Huckel behaviour were found. This is not surprising since the Debye-Huckel theory treats the solvent as incompressible and, as seen in Fig. 3, water rapidly becomes more compressible above 300 C. Until a theory which accounts for electrostriction in a compressible fluid becomes available, extrapolation to infinite dilution at temperatures much above 300 C must be considered untrustworthy. Since water becomes infinitely compressible at the critical point, the standard entropy of an ion becomes infinitely negative, so that the concept of a standard ionic free energy becomes meaningless. 

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... 

Another policy in writing the book has been the attempt to base the deduction of all equations on first principles. What actually constitutes such principles is, to an extent, a matter of individual preference. Any attempt at definition would immediately lead one into the field of the professional philosopher. Such an intrusion the author is, above everything, anxious to avoid. Fie feels, however, that the attempt to build from the ground up has been accomplished in most of the subjects considered. Exceptions are, however, the extension of the Debye-Huckel theory, and the application of the interionic attraction theory to electrolytic conductance. In the latter case the fundamentals lie in the field of statistical mechanics, which cannot be adequately treated short of a book the size of this one, and which, in any case, would not be written by the author. 

Table II. Constants for the Third and Fifth Approximations of the Gronwall, LaMer and Sandved Extension of the Debye-Huckel Theory...

The components of an ion-association aqueous model are (1) The set of aqueous species (free ions and complexes), (2) stability constants for all complexes, and (3) individual-ion activity coefficients for each aqueous species. The Debye-Huckel theory or one of its extensions is used to estimate individual-ion activity coefficients. For most general-purpose ion-association models, the set of aqueous complexes and their stability constants are selected from diverse sources, including studies of specific aqueous reactions, other literature sources, or from published tabulations (for example, Smith and Martell, (13)). In most models, stability constants have been chosen independently from the individual-ion, activity-coefficient expressions and without consideration of other aqueous species in the model. Generally, no attempt has been made to insure that the choices of aqueous species, stability constants, and individual-ion activity coefficients are consistent with experimental data for mineral solubilities or mean-activity coefficients. 

C. W. Outhwalte, /. Chem. Phys., 50, 2277 (1969). Extension of the Debye-Huckel Theory of Electrolyte Solutions. 

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. 

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

As in the previous chapter on the Smoluchowski theory and its extensions, similar boundary and initial conditions may be used. The reaction of a species A with a vast excess of B (yet still sufficiently dilute to ensure that Debye- HUckel screening is unimportant) can be considered as one where the A species are statistically independent of each other and are surrounded by a sea of B species. An ionic reactant A has a rate of reaction with all the B reactants equal to the sum of the rates of reaction of individual A—B pairs. This rate for large initial separations of A and B is... 

The criterion used to choose the topics covered in this book was their usefulness in application to problems in chemistry and biochemistry. Thus cluster expansion methods for a real gas, although very useful for the development of the theory of real gases per se, was judged not useful except for the second virial coefficient. Similarly, the statistical mechanical extensions of the theory of ionic solutions beyond the Debye-Huckel limiting law were judged not useful in actual applications. Some important topics may have been missed either because of my lack of familiarity with them or because I failed to appreciate their potential usefulness. I would be grateful to receive comments or criticism from readers on this matter or on any other aspect of this book. 

The terms in the new series are ordered differently from those in the original expansion and Mayer showed that the Debye-Huckel limiting law follows as the leading correction to the ideal behavior for ionic solutions. In principle, the theory enables systematic corrections to the limiting law to be obtained as the concentration of the electrolyte increases for any Hamiltonian which defines the short-range potential u j (r), not just the one which corresponds to the RPM. A modified (or renormalized) second virial coefficient was tabulated by Porrier (1953), while Meeron (1957) and Abe (1959) derived an expression for this in closed form. Extensions of the theory to non-pairwise additive solute potentials have been discussed by Friedman (1962). 

Conductivity equations based on Debye-Huckel-Onsager theory, such as Equation 17.9, cannot predict the conductance maxima. They are valuable tools to study dilute solutions in a concentration range below the maximum where the solvent may be described as a homogeneous medium with permittivity and viscosity of the pure solvent compound. For extension of the transport equations in the theory of transport properties, especially the continuity equation approach, the reader is referred to Ref [183] and the references given there. 

In order to describe the effects of the double layer on the particle motion, the Poisson equation is used. The Poisson equation relates the electrostatic potential field to the charge density in the double layer, and this gives rise to the concepts of zeta-potential and surface of shear. Using extensions of the double-layer theory, Debye and Huckel, Smoluchowski,... 

Moreover in the spherical double layer with 1/x large compared with the radius of the particle, the distribution of the potential is governed more by the factor 1/r in eq. (79) which comes from the spherical extension of the lines of force, than from double layer effects and therefore a slight incorrectness in the treatment of the double layer does not influence the distribution of the potential very much Table 1 in which the approximate theory of Debye and HUckel and the exact solution of eq. (40) by MULLER are compared for a case where 1/x = 5 a, illustrate this clearly. 

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