Free ion activity coefficients

Here m>u and m>UCd++ are molal concentrations of the unoccupied and occupied sites, respectively, and aCd++ is the activity of the free ion. Activity coefficients for the surface sites are not carried in the equation they are assumed to cancel. Equilibrium constants reported in the literature are in many cases tabulated in terms of the concentrations of free species, rather than their activities, as assumed here, and hence may require adjustment. 

The effects of this specific reaction would not he represented in MSM calculations. In fact, it is assumed that no ion pairs are formed at 1= 0.7 in the experiments with individual electrolytes used to calculate activity coefficients in the MSM. This is true mainly for salts of potassium and chlorine, which are the ones primarily used in the MSM calculations. Because any ions that join to form ion pairs in a mixed electrolyte solution are effectively removed from solution with respect to reaction potential, an accurate determination of the extent of ion pairing in addition to the free ion activity coefficient must he made to determine the total activity coefficient in seawater. 

The percentage of the total concentration of a specific ion in seawater that is involved in ion pairs can he determined from thermodynamic equilibrium information and will be demonstrated later in the chapter. For now, suffice it to say that the individual ion concentration is determined by multipl5dng the total ion concentration by a fraction between 0 and 1.0 representing the total ion that is free of ion pairs. The product, (%freej/100) Mj, can then be multiplied by the free ion activity coefficient to determine the activity. In general form, the total correction equation is... 

Compute the total or stoichiometric and effective ionic strengths, and corresponding total- and free-ion activity coefficients of Ca and SO in pure water saturated with gypsum at 25"C. It is given that the solubility product of gypsum, and that at gypsum saturation... 

Taking the above considerations into account, it turns out that reasonable agreement between theory and results is obtained if we express the free ion activity coefficient of an ion of valence z as... 

Note Because of the fairly high ionic strength, Eq. (2.30) is not accurate anymore, and the free ion activity coefficients are likely to be somewhat higher than calculated. On the other hand, other associations undoubtedly occur, for instance of citrate with K+, which is abundant in blood, thereby further decreasing the free ion concentrations. 

The choice of R is arbitrary within reasonable limits and then divides up the thermodynamic excess fimction, = vRT In y., into contributions from the so-defined ion pair (degree of dissociation, a) and from the free ions (activity coefficient of the free ions, y. ), cf. Eq. (7). Onsager characterized the situation as follows The distinction between free ions and associated pairs depends on an arbitrary convention. Bjerrum s choice is good, but we could vary it within reason. In a complete theory this would not matter what we remove from one page of the ledger would be entered elsewhere with the same effect. ... 

Major ions Concentration m Amount occurring as free ions % Activity coefficient Th Activity a... 

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). 

Both associated and nonassociated electrolytes exist in sea water, the latter (typified by the alkali metal ions U+, Na-, K+, Rb+, and Cs-) predominantly as solvated free cations. The major anions. Cl and Br, exist as free anions, whereas as much as 20% of the F in sea water may be associated as the ion-pair MgF+. and 103 may be a more important species of I than I-. Based on dissociation constants and individual ion activity coefficients the distribution of the major cations in sea water as sulfate, bicarbonate, or carbonate ion-pairs has been evaluated at specified conditions by Garrels and Thompson (19621. 

There is one rather nasty twist in the ion pair evaluation where a negative value for free carbonate ion concentration results, because the initial value which must be used for the carbonate ion activity coefficient is much larger than it is under the new conditions. This demands some maneuvers with both the calcite solubility constant and the calcium carbonate ion pair association constant. These will not be gone into here. 

Table 3.5. I Concentrations, single-ion activity coefficients, yi, percent of the ion that is free of ion pairing, and the total ion activity coefficient.

The increase in ion activity coefficients at elevated ionic strengths results from several effects, including the fact that an increasing fraction of water molecules are involved in hydration spheres around ions. This causes a proportionate decrease in the concentration of free water molecules in the solution. For example, assuming there are six water molecules associated with each pair of dissolved Na+ and Cl ions in a 0.01 molar NaCl solution (see Bockris and Reddy 1973), the moles of free water molecules in a liter of solution are 55.5 - 6 x (0.01) = 55.44. On the other hand, if the solution... 

Given this analysis and that the dissociation constants for the ion pairs at 9.8°C are A jj,soc(CaHCOJ) = Iq-0.97 and /fdissoc(MgHCOj+) = 10" , hand calculate the molal concentrations of free Ca, Mg-", and HCOf and of the bicarbonate ion pairs and compute their percent values. Use the ion activity coefficients computed in part (b) in this calculation, assuming a constant ionic strength, and that the activity coefficients of both bicarbonate ion pairs equal ynco - Also compute the apparent CO2 pressure and the saturation indices of calcite and dolomite. 

The ion-interaction model is a theoretically based approach that uses empirical data to account for complexing and ion pair formation by describing this change in free ion activity with a series of experimentally defined virial coefficients. Several philosophical difficulties have resulted from the introduction of this approach the lack of extensive experimental database for trace constituents or redox couples, incompatibility with the classical ion pairing model, the constant effort required to retrofit solubility data as the number of components in the model expand using the same historical fitting procedures, and the incompatibility of comparing thermodynamic solubility products obtained from model fits as opposed to solubility products obtained by other methods. 

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. 

Table I. Salts used to fit ion-activity-coefficient parameters and parameters for Equation 3 for each free ion...

The set of complexes the individual-ion, activity-coefficient parameters and the stability constants are listed in Table II, for each of the three models (1) The WATEQ model, (2) the amended WATEQ model, and (3) the fit model. The WATEQ and amended WATEQ models had the same set of complexes and stability constants. The two models differed only in the individual-ion activity co( fficients of the free ions. The fit model contained different complexes or stability constants or both compared to the WATEQ and amended WATEQ models (except for the OH complexes that were obtained from the WATEQ inodel). The activity coefficients for the free ions were the same in the fit and amended WATEQ models, but the activity coefficients of the complexes differed. 

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. 

Most stability constants from the fit model for two-ion complexes agreed with values used in the WATEQ model within 0.3 log unit There was considerably less agreement between the fit model and the WATEQ model for the stability constants of complexes with three or more ions. The calculations indicated that stability constants are too large for the Cl complexes of Cu", Mn, Ni 2, and Zn" " in the WATEQ model, regardless of the choice of individual-ion activity coefficients of the free ions. 

The ligand solutions were prepared by titration of the free acid with tetrame-thylammonium hydroxide (to a pH value of approximately 10). Aliquots containing the ligand, metal (as Ni(N03)2), and tetramethylammonium chloride were titrated with 0.0685 M HCl(aq), with the primary measurements being carried out at pH values between 4.5 and 6.5. The electrodes were probably calibrated against pH buffers rather than concentration standards, and the hydrogen ion activity coefficient, was assigned the value of 0.796 the SIT procedure described in Appendix B would have led to essentially the same value, y. = 0.80. 

Here a is the fraction of the ions that are free with activity coefficients and 1 - a is... 

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