Mean activity coefficients, their theory

The beginning of the twentieth century also marked a continuation of studies of the structure and properties of electrolyte solution and of the electrode-electrolyte interface. In 1907, Gilbert Newton Lewis (1875-1946) introduced the notion of thermodynamic activity, which proved to be extremally valuable for the description of properties of solutions of strong electrolytes. In 1923, Peter Debye (1884-1966 Nobel prize, 1936) and Erich Hiickel (1896-1981) developed their theory of strong electrolyte solutions, which for the first time allowed calculation of a hitherto purely empiric parameter—the mean activity coefficients of ions in solutions. 

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. 

Mean activity coefficients can, in theory, be determined without ambiguity. For this reason, considerable attention has been directed to the use of solution theory as a guide to separating mean activities into their cationic and anionic components. The Macinnes as-... 

An obvious improvement of the theory consisted in removing the assumption of point-charge ions and taking into account their finite size. With the use of an ion size parameter a, the expression for the mean ionic-activity coefficient became... 

The values of y are nearly independent of the kind of ions in the compound so long as the compounds are of the same valence type. For example, KCl and NaBr have nearly the same activity coefficients at the same concentration, as do K2SO4 and Ca(N03)2. In Section 16.7 we shall see that the theory of Deby e and Huckel predicts that in a sufficiently dilute solution the mean ionic activity coefficient should depend only on the charges on the ions and their concentration, but not on any other individual characteristics of the ions. 

What does this mean First, observe that this simple theory of solubility uses no information about the solvent. The above theoretical calculation is for any solvent. If the solute (NaCl in this case) does not interact with the solvent, then this may be a fair estimate. The reported solubility of NaCl in ethanol at 25°C is 0.00025 mol fraction, 1.7 times the value calculated above. If we wanted to know the solubility of NaCl in gasoline or diesel fuel, with which it would not be expected to interact much, the 0.00014 mol fraction computed in Example 11.13 would be a fair estimate. However, we know that water and NaCl interact strongly. The salt ionizes, the ions solvate with the water molecules. We may think of their interaction as a strong example of type HI (Section 8.4.3) with a calculated activity coefficient of 0.000,14. All of the salts that dissolve to high concentrations in water are similar to NaCl in this behavior. 

In addition to neglecting ion correlation, using the mean electrostatic potential has the undesirable consequence that the (nonlinear) PB equation no longer satisfies a reciprocity condition that use of the potential of mean force would obey. Linearization of the equation by Debye and Hiickel regained this condition. These considerations led Outhwaite and others to propose modifications of the PB equation to treat these problems. Within this modified Poisson-Boltzmaim (MPB) theory, the effect of ion correlation is expressed in terms of a fluctuation potential for which a first-order (local) expression, written as an activity coefficient, can be derived. Their result for bulk hard-sphere electrolyte ions of valence z, and common radius a gives the formula ... 

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