Generalized Debye-Huckel Theory

The discussion of continuum electrostatics in Section 11.2.1 was limited to solution of Poisson s equation, which can be achieved exactly (for classical solutes) or to a good approximation (for QM solutes) using PCMs. In biomolecular applications, however, the objective is usually solution of the Poisson-Boltzmann equation [4, 33]. For low concentrations of dissolved ions, the latter is often replaced by the linearized Poisson-Boltzmann equation (LPBE), 

k = Qne X/ekgT is the inverse Debye length, for a solution whose ionic strength is X. The LPBE was derived by Debye and Huckel [28], and its anal5ftic solution for a spherical cavity forms the basis of the eponymous theory. In this section, we discuss how PCMs can be modified to solve the LPBE, but first we present an alternative derivation of GCOSMO that will be useful in this respect. 

The original derivation of COSMO was based on taking e oo, in which case Sq = -V is the exact solution to the molecular electrostatics problem, then rescaling the solution for finite e [37, 78]. Recently, we presented a much more satisfying derivation [43]. Our approach starts from an ansatz 

Noting that 9s Ao continuous across F, the reaction-field potential must be solely responsible for the jump in the electric field [16], This condition can be expressed as [43] 

The normal derivative of the ansatz in Eq. (11.27) lacks the second term in Eq. (11.28) hence, C-PCM/GCOSMO engenders errors of order as compared to an exact treatment of classical electrostatics. Such errors are negligible in water [44], as seen in Fig. 11.3. 

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The Poisson-Boltzmann equation is a modification of the Poisson equation. It has an additional term describing the solvent charge separation and can also be viewed mathematically as a generalization of Debye-Huckel theory. 

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. 

Then, about 1904, it was pointed out by A. A. Noyes in this country and Sutherland in England that many properties of solutions of salts and strong acids (such as their color) suggest that most salts and strong acids are completely ionized in dilute solution. This view has been generally accepted since 1923, when a quantitative theory of the interactions of ions in solution was developed by Debye and Hiickel. This theory is called the Debye-Huckel theory of electrolytes. 

Heat Capacities and the Debye-Hfickel Theory.—By combining the general equation (44.46) with the expression for L2 [equation (44.39)3 derived from the Debye-Huckel theory, it is found that for a solution containing a single strong electrolyte,... 

For an overall charged ion pair, a mean ionic activity coefficient must always be assigned to it, generally calculated from the Debye-Huckel theory on the basis of the overall charge. 

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. 

Comparison between thermodynamic values is generally made with standard state functions. To obtain the standard enthalpy of solution, A//g°in, it is necessary to extrapolate directly measured enthalpies of solution at finite concentrations to infinite dilution some form of the Debye-Huckel theory is generally used in this extrapolation (see sect. 2.5.2). 

Classically, one treats phases of two components as ideal, regular, or real solutions. Usually, however, one concentrates for the non-ideal case only on solutions of salts by discussing the Debye-Huckel theory. Polymer science, in turn, adds the effect of different molecular sizes with the Hory-Huggins equation as of basic importance (Chap. 7). Considerable differences in size may, however, also occur in small molecules and their effects are hidden falsely in the activity coefficients of the general description. 

There is large literature, and several computational methods, for this problem of a solute M immersed in a salt solution. In general, fine and detailed descriptions of the ions in the liquid are discarded, in favor of their approximation to point charges which are assumed to follow the Boltzmann distribution as in the Debye-Huckel theory of electrolytes. Since the Poisson-Boltzmann equation is not linear, an approximation based on its linearization is often used. By assuming e = constant, the equation takes the simple form ... 

Effects of the inert inorganic salts on the rate constants (k) for the reactions involving ionic reactants are generally explained in terms of the Debye-Huckel or extended Debye-Huckel theory. In actuality, the extended Debye-Huckel theory involves an empirical term, which makes the theory a semiempirical theory. However, there are many reports in which the effects of salts on k of such ionic reactions cannot be explained by the Debye-Huckel theory. For instance, pseudo-first-order rate constants (k bs) for the reaction of HO with acetyl salicylate ion (aspirin anion) show a fast increase at low salt concentration followed by a slow increase at high concentration of several salts. But the lowest salt concentration for each salt remains much higher than the limiting concentration (0.01 M for salts such as M+X ) above which the Debye-Huckel theory is no longer valid. These k bs values fit reasonably well to Equation 7.48... 

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 highly charged macromolecules Eq. (5.8.16) does not apply. It is then necessary to generalize these arguments to include hydration and deviations from spherical shape. However there are even more formidable complications to consider first. In aqueous solution, the macroion is surrounded by an ion atmosphere composed mainly of ions of opposite charge. This means that the local field —the field felt by the macroion— will be considerably different than the applied field. Corrections for this effect can be made if one uses the Debye-Huckel-Henry theory to calculate the properties of the ion atmosphere (see Chapters 9 and 13). An estimate (Tanford, 1961) gives... 

Clearly if Ya is unity then the solution is ideal. Otherwise the solution is nonideal and the extent to which ya deviates from unity is a measure of the solution s non-ideality. In any solution we usually know [A] but not either a a or Ya- However we shall see in this chapter that for the special case of dilute electrolytic solutions it is possible to calculate ya- This calculation involves the Debye-Hdckel theory to which we turn in Section 2.4. It provides a method by which activities may be quantified through a knowledge of the concentration combined with the Debye-Huckel calculation of ya- First, however, we consider some relevant results pertaining to ideal solutions and, second in Section 2.3, a general interpretation of Ya-... 

In the specific examples of the theory given in this section the Debye-Huckel contribution is small or negligible. This circumstance is not general, as will be seen. 

Going beyond solutions of electrolytes in water, several other possibilities need consideration electrolytes in nonaqueous solvents, nonelectrolytic behavior in solutions, and nonelectrolytes in nonaqueous solvents. None of the theories proposed for the quantitative prediction of solution behavior has been as successful as that of Debye and Huckel for dilute ionic aqueous solutions. Nevertheless, general trends can be predicted. 

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