Generalized Debye-Hiickel theory, ionic

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). 

Activity coefficients for each species can be calculated from the Debye-Hiickel theory. The actual ionic strength must be calculated from the experimental conditions, and this generally involves successive approximations (see Worked Problem 8.22). ... 

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. 

The MSA is fundamentally connected to the Debye-Hiickel (DH) theory [7, 8], in which the linearized Poisson-Boltzmann equation is solved for a central ion surrounded by a neutralizing ionic cloud. In the DH framework, the main simplifying assumption is that the ions in the cloud are point ions. These ions are supposed to be able to approach the central ion to some minimum distance, the distance of closest approach. The MSA is the solution of the same linearized Poisson-Boltzmann equation but with finite size for all ions. The mathematical solution of the proper boundary conditions of this problem is more complex than for the DH theory. However, it is tractable and the MSA leads to analytical expressions. The latter shares with the DH theory the remarkable simplicity of being a function of a single screening parameter, generally denoted by r. For an arbitrary (neutral) mixture of ions, this parameter satisfies a simple equation which can be easily solved numerically by iterations. Its expression is explicit in the case of equisized ions (restricted case) [12]. One has... 

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