Activity coefficient individual ion

In contrast with the individual ion activity coefficients fit the mean activity coefficient ft can be measured, calculation of which can be achieved through eqn. 2.46 as follows ... 

These individual-ion activity coefficients have the desired property of approaching 1 at infinite dilution, because each ratio a,/(m,/m°) approaches 1. However, individual-ion activity coefficients, like individual-ion activities, cannot be determined experimentally. Therefore, it is customary to deal with the mean activity coefficient 7+ and the mean activity a which for a uni-univalent electrolyte can be related to measurable quantities as follows ... 

According to the Debye-Htickel theory, in the limit of the infinitely dilute solution, individual-ion activity coefficients are given by the equation... 

Various empirical relations are available for calculating individual ion activity coefficients [discussed by Stumm and Morgan (1996) for natural waters and Sposito (1984a, b), for soil solutions]. In the calculations in this book I used the Davies equation ... 

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. 

In saline soils and soils contaminated with geothermal brines, the ionic strengths of the soil solution may exceed 0.5 M. This fact poses the necessity of using equations which have been developed to describe the activity coefficients of ions in concentrated, multicomponent electrolyte solutions. As part of a study on the chemistry of ore-forming fluids, Helgeson (50) has proposed that the true individual ion activity coefficients for ions present in small concentrations in multicomponent electrolyte solution having sodium chloride as the dominant component be approximated by a modified form of the Stokes-Robinson equation. The equation proposed is ... 

According to Helgeson (50), equation 8 can be used to estimate the individual ion activity coefficient for ions present in small concentrations in sodium chloride solutions of true ionic strength up to 3.0 M. Since saline soils and geothermal brines are often dominated by sodium chloride, it will be appropriate to use the equation proposed by Helgeson (50). Therefore, in GEOCHEM, ionic activity coefficient calculations for such systems are performed by equation 8. 

Unfortunately, since neither liquid-junction potentials nor individual ion activity coefficients can be evaluated rigorously, the pan of this definition cannot be accurately related to experimental quantities. Several attempts have been made to determine pH scales that are not subject to the theoretical limitations of psH or pa and yet are capable of experimental measurement these have been described elsewhere. ... 

It would be difficult to find more comprehensive or more detailed studies on the physical chemistry of seawater than those done at the University of Miami (Millero, 2001). Several programs were developed for calculation of activity coefficients and speciation of both major ions and trace elements in seawater. The activity coefficient models have been influenced strongly by the Pitzer method but are best described as hybrid because of the need to use ion-pair formation constants (Millero and Schreiber, 1982). The current model is based on Quick Basic computes activity coefficients for 12 major cations and anions, 7 neutral solutes, and more than 36 minor or trace ions. At 25 °C the ionic strength range is 0-6 m. For major components, the temperature range has been extended to 0-50 °C, and in many cases the temperature dependence is reasonably estimated to 75 °C. Details of the model and the parameters and their sources can be found in Millero and Roy (1997) and Millero and Pierrot (1998). Comparison of some individual-ion activity coefficients and some speciation for seawater computed with the Miami model is shown in Section 5.02.8.6 on model reliability. 

Plummer L. N. and Sundquist E. T. (1982) Total individual ion activity coefficients of calcium and carbonate in seawater at 25°C and 35%o salinity, and implications to the agreement between apparent and thermodynamic constants of calcite and aragonite. Geochim. Cosmochim. Acta 46,... 

Aqueous electrolyte chemical potentials are described on the moial scale. To nlustrate the additional issues that enter into the thermodynamic interpretation of individual ion activity coefficients and chemical potentials and the relation of these to actual electrolyte experimental measurement, we briefly review the properties of the system NaCl(aq), that is, NaCl dissolved in water. [For a detailed discussion, see. e.g., Denbigh, 1971 Hamed and Owen, 1959 Klotz, 1964 Robinson and Stokes, 1959). 

In dilute solutions (/ < 10 M), that is, in fresh waters, our calculations are usually based on the infinite dilution activity convention and thermodynamic constants. In these dilute electrolyte mixtures, deviations from ideal behavior are primarily caused by long-range electrostatic interactions. The Debye-Huckel equation or one of its extended forms (see Table 3.3) is assumed to give an adequate description of these interactions and to define the properties of the ions. Correspondingly, individual ion activities are estimated by means of individual ion activity coefficients calculated with the help of the Guntelberg or Davies (equations 3 and 4 of Table 3.3) or it is often more convenient to calculate, with these activity coefficients, a concentration equilibrium constant valid at a given /,... 

In the mean-salt method, the behavior of KCl in solution is the standard basis for obtaining individual ion activity coefficients. Various lines of evidence indicate that /k+ and fc - have similar values that is, as an approximation,... 

We cannot measure the individual ion activity coefficients here, only their total effect on /f,p. It is convenient to lump this total effect in the geometric mean of the product of the individual activity coefficients and to call this the mean ion activity coefficient of the salt. Thus for a K2SO4 solution, by definition... 

In order for us to have flexibility in our modeling of natural water chemistry we need a way to obtain individual ion activity coefficients from mean values. To do so requires that we make an assumption, called the Macinnes convention (Macinnes 1919), which states = 7c - The convention is based on the observation that and Cr ions are of the same charge and nearly the same size, have similar electron structures (inert gas), and similar ionic mobilities. In support of this assumption, tracer diffusion coefficients, D°, of K+ and Cl" at infinite dilution are nearly equal at 19.6 and 20.3 X 10" cmVs (Lerman 1979). Also, limiting equivalent conductances, A°, of and Cl" are comparable at 73.50 and 76.35 cmV(ohm) (equiv.) at 25°C (Robinson and Stokes 1970),... 

The Macinnes convention leads to = Tci = 7 kci, We can now compute individual ion activity coefficients from their mean values measured in solutions of strong electrolytes using y Kci values as our starting point. (In the ideal strong electrolyte, cations and anions are unassociated with each other and thus do not form complexes [see Chap. 3].) It is important to remember that all such calculations must be done with y values for KCl and other salts measured at the same ionic strength, which is not the same molality except for monovalent-monovalent salts. 

Following are three example calculations of individual ion activity coefficients from mean salt data. Such calculations must always be performed using y values measured at the same ionic... 

Individual ion activity coefficients for Na+, HCO3, Ca2+, SO ", and computed from mean salt data and, assuming the Macinnes convention, are plotted in Fig. 4.2. As before, the mean salt data for KCl are from Hamer and Wu (1972). The y data for HCO3 and La + are from Roy et al. (1983) and Robinson and Stokes (1970), respectively. Sources for the other ions are as given above. 

Figure 4.2 Some individual ion activity coefficients computed from mean salt data assuming the Macinnes convention.

TABLE 4.2. Individual ion activity coefficients at 25°C for different ion sizes (a ) in angstroms (1 A = 10 cm) as a function of ionic strength, computed using the extended Debye-Huckel equation with A = 0.5091 and B = 0,3286... 

Some individual ion activity coefficients calculated using the extended Debye-Htickel... 

TABLE 4.3 Ion size a,) and b values for the Truesdetl-Jones equation for individual ion activity coefficients... 

The individual ion activity coefficients cannot be measured, but an analogous equation to Equation (8.70) can be written involving the mean activity coefficient y for the electrolyte as a whole. This is a quantity which can be measured (see emf measurements. Section 9.20.5, Worked Problems 9.28 and 9.29 and Sections 10.6.15, 10.7, 10.10 and 10.11). 

Calculations using the aqueous model from WATEQ and an aqueous model modified from WATEQ 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 was derived by least-squares fitting of ion-association stability constants and individual-ion, activity-coefficient parameters to experimental mean activity coefficients for various salts at 25°C. 

Cd+2, Co+2, Cs+, Cu+2, Fe+2, H+. K+, Li+, Mg+2 Mn+2, Na+, Ni+2 pb+2 Sr+2 Zn" 2, Cl, CIO4 , F , OH , and S04 2, The stability constants of the the derived model and the WATEQ model were in agreement for most two-ion complexes but were not in agreement for most complexes containing three or more ions. The largest discrepancies in stability constants were for complexes of Ni 2, and Zn with Cl ". The derived-model calculations matched the experimental data for all salts to a concentration of about 2 molal, but the parameters of the model could not be defined uniquely by the fitting process. Alternative choices for the complexes included in the model and for the individual-ion, activity-coefficient parameters could fit the experimental data equally well. 

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 results of calculations made using ion-association aqueous models are the molality, activity, and individual-ion activity coefficient for each species in the model. The mean activity coefficient, y , for a salt can be calculated as follows ... 

The mean activity coefficient for a salt can be calculated from experimental data, but the individual-ion activity coefficients used in ion-association aqueous models cannot be determined experimentally. Several formulas were used for individual-ion activity coefficients in the calculations presented in this report. Three formulas were used in the WATEQ model the extended Debye-Huckel formula including an ion-size parameter, a[ (Equation 2) modified extended Debye-Huckel formula with two fitted parameters, a[ and 6i (Equation 3) and the Davies equation (Equation 4). 

An individual-ion, activity-coefficient formula (Equation 5) derived by Millero and Schreiber (14) from the work of Pitzer (15) was used in the amended WATEQ and fit models ... 

The aqueous species stability constants and individual-ion, activity-coefficient parameters from WATEQ and WATEQ2 were used as published for one set of calculations. Perchlorate and Co" were not included in the WATEQ aqueous models, but were added for these calculations. The individual-ion activity coefficient for C104 and Co were calculated using Equation 2 and ion-size parameters (Uf) of 3.5 and 6.0, respectively (16). All calculations were made using a version of PHREEQE (10) modified to calculate mean activity coefficients for salts. 

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 are not completely dissociated in solution, then the individual-ion activity coefficients used in this report are too small. Further, the stability constants for the other salts also are too small. If the salts that are assumed to be associated in solution in this report are, in fact, completely dissociated in solution, then the ion-association aqueous model is inadequate. In this case, the ion-association aqueous model can not reproduce accurately the mean activity coefficients of all the salts and simultaneously maintain a correct physical description of the solutions.

For the fit model, aqueous complexes were added as needed in order to resolve the discrepancies between the calculated and experimental mean activity coefficients. The stability constants for the complexes and the individual-ion, activity-coefficient parameters for the complexes were estimated through least-squares fitting. The program that estimated the parameters used the modified version of PHREEQE (10) as a function subroutine that calculated the mean activity coefficient The stability constants and individual-ion, activity-coefficient parameters of a specified set of complexes were adjusted until a least-squares fit was obtained between the calculated and experimental mean activity coefficients for a series of salt-solution compositions. 

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