Mean activity coefficient free ions

The mean activity coefficient of the free ions is calculated using the Debye-Hiickel Equation with a distance parameter a = 4.0 A. If we ignore AGt° (mic), the above relation can be rearranged as ... 

Activity coefficients in most concentrated solutions reflect deviation from ideal behavior because of (1) the general electric field of the ions, (2) solute-water interactions, and (3) specific ionic interactions (association by ion pair and complex formation). None of the major cations of seawater appears to interact significantly with chloride to form ion pairs hence activity coefficients in these solutions appear to depend primarily on the ionic strength modified by the extensive hydration of ions. Thus synthetic solutions of these chlorides provide reference solutions in obtaining activity coefficients of the cations. The single activity coefficients for free ions in seawater can be obtained from mean activity coefficient data in chloride solutions at the corresponding ionic strength by various ways. 

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. 

In this report, calculations made using ion-association aqueous models were compared to experimental mean activity coefficients for various salts to determine the range of applicability and the sources of errors in the models. An ion-association aqueous model must reproduce the mean activity coefficients for various salts accurately or it does not describe the thermodynamics of aqueous solutions correctly. Calculations were made using three aqueous models (1) The aqueous model obtained from WATEQ (3), WATEQF (4), and WATEQ2 (6), referred to as the WATEQ model (2) the WATEQ model with modifications to the individual-ion, activity-coefficient equations for the free ions, referred to as the amended WATEQ model and (3) an aqueous model derived from least-squares fitting of mean activity-coefficient data, referred to as the fit model. 

The individual-ion activity coefficients for the free ions were based on the Macinnis (18) convention, which defines the activity of Cl to be equal to the mean activity coefficient of KCl in a KCl solution of equivalent ionic strength. From this starting point, individual-ion activity coefficients for the free ions of other elements were derived from single-salt solutions. The method of Millero and Schreiber (14) was used to calculate the individual-ion, activity-coefficient parameters (Equation 5) from the parameters given by Pitzer (19). However, several different sets of salts could be used to derive the individual-ion activity coefficient for a free ion. For example, the individual-ion activity coefficient for OH could be calculated using mean activity-coefficient data for KOH and KCl, or from CsOH, CsCl, and KCl, and so forth. 

All possible sets of salts that could be used to calculate the individual-ion, activity-coefficient parameters (Equation 5) were considered for each ion. The parameters that produced the largest individual-ion activity coefficients for an ion were used in the amended WATEQ and fit models. This choice of individual-ion activity coefficients insured that complexing could account, at least in part, for the differences between the calculated and experimental values of the mean activity coefficients, because the effect of adding a complex to the aqueous model is to decrease the calculated mean activity coefficient. The salts used to calculate the individual-ion, activity-coefficient parameters of the free ions are listed in Table I. 

Calculations for MnCl2 solutions are shown in Figure 1. The calculated mean activity coefficients were much smaller than the experimental values for the WATEQ and amended WATEQ models. These models used different individual-ion activity coefficients for the free ions yet the calculated results were similar, which indicates that the discrepancy was caused by the stability constants of the Cl complexes. The calculations indicate that the WATEQ and amended WATEQ models do not reproduce the mean activity coefficients for CuCl2, MnCl2, NiCl2, and ZnCl2 at concentrations greater than 0.1 molal. 

The calculations of mean activity coefficients for various salts using the WATEQ model indicate that if Equation 2 is used for individual-ion activity coefficients of free ions, the results are reliable only for concentrations of 0.1 molal or less. If equations 3 or 5 are used (the amended WATEQ model), the calculated mean activity coefficients are accurate for the salts used to derive the individual-ion activity coefficients of the free ions, but are not accurate for other salts unless additional complexes are included. 

An ion-association aqueous model was derived by (1) Selecting the set of salts used to calculate the individual-ion activity coefficients of the free ions (2) hypothesizing an appropriate set of complexes and (3) fitting the stability constants and individual-ion, activity-coefficient parameters for the complexes using the mean- activity-coefficient data. 

At low electrolyte concentrations the activity coefficient of ion pairs may be set equal to unity and Eq. (77) is linked to the ion-pair association constant and the mean activity coefficient of the free ions by the relationships I — O 1... 

One way of probing the interactions between solvated ions is to study aqueous bulk electrolyte activity coefficients, y. For the most common choice of reference state, the mean activity coefficient is a measure of the excess free energy, fM = ksT In y, of transferring a solvated salt pair from an infinite dilution to a solution -with a finite salt concentration. In the present context, we are interested in the ion specijkity and therefore — for a fixed salt concentration — define a free energy of exchanging one counter ion -with another. For example,... 

In the virial methods, therefore, the activity coefficients account implicitly for the reduction in the free ion s activity due to the formation of whatever ion pairs and complex species are not included in the formulation. As such, they describe not only the factors traditionally accounted for by activity coefficient models, such as the effects of electrostatic interaction and ion hydration, but also the distribution of species in solution. There is no provision in the method for separating the traditional part of the coefficients from the portion attributable to speciation. For this reason, the coefficients differ (even in the absence of error) in meaning and value from activity coefficients given by other methods. It might be more accurate and less confusing to refer to the virial methods as activity models rather than as activity coefficient models. 

Once the composition of the aqueous solution phase has been determined, the activity of an electrolyte having the same chemical formula as the assumed precipitate can be calculated (11,12). This calculation may utilize either mean ionic activity coefficients and total concentrations of the ions in the electrolyte, or single-ion activity coefficients and free-species concentrations of the ions in the electrolyte (11). If the latter approach is used, the computed electrolyte activity is termed an ion-activity product (12). Regardless of which approach is adopted, the calculated electrolyte activity is compared to the solubility product constant of the assumed precipitate as a test for the existence of the solid phase. If the calculated ion-activity product is smaller than the candidate solubility product constant, the corresponding solid phase is concluded not to have formed in the time period of the solubility measurements. Ihis judgment must be tempered, of course, in light of the precision with which both electrolyte activities and solubility product constants can be determined (12). 

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