Activity Pitzer model

Rard also employed Pitzer s electrolyte activity coefficient model to correlate the data. It was found that the quality of the fit depended on the range of molalities that were used. In particular, the fit was very good when the molalities were less than 3 mol/kg. 

Estimate Pitzer s electrolyte activity coefficient model by minimizing the objective function given by Equation 15.1 and using the following osmotic coefficient data from Rard (1992) given in Table 15.5. First, use the data for molalities less than 3 mol/kg and then all the data together. Compare your estimated values with those reported by Rard (1992). Use a constant value for in Equation 15.1. 

Two activity coefficient models have been developed for vapor-liquid equilibrium of electrolyte systems. The first model is an extension of the Pitzer equation and is applicable to aqueous electrolyte systems containing any number of molecular and ionic solutes. The validity of the model has been shown by data correlation studies on three aqueous electrolyte systems of industrial interest. The second model is based on the local composition concept and is designed to be applicable to all kinds of electrolyte systems. Preliminary data correlation results on many binary and ternary electrolyte systems suggest the validity of the local composition model. 

The species concentrations are formulated in activities using the Pitzer model (207) for the aqueous phase and the Hildebrand-Scott solubility parameter (208) for the organic phase. 

Reactive absorption processes occur mostly in aqueous systems, with both molecular and electrolyte species. These systems demonstrate substantially non-ideal behavior. The electrolyte components represent reaction products of absorbed gases or dissociation products of dissolved salts. There are two basic models applied for the description of electrolyte-containing mixtures, namely the Electrolyte NRTL model and the Pitzer model. The Electrolyte NRTL model [37-39] is able to estimate the activity coefficients for both ionic and molecular species in aqueous and mixed solvent electrolyte systems based on the binary pair parameters. The model reduces to the well-known NRTL model when electrolyte concentrations in the liquid phase approach zero [40]. 

The Pitzer model can be used to obtain activity coefficients for solutes in low (<0.1 mol L ), intermediate (0.1-3.5 mol L ) and high (>3.5 mol L ) ionic strength solutions. The Pitzer equations include terms for binary and ternary interactions between solute species as well as a modified DH expression. The general formula is... 

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. 

Weare (1987) (see also Millero 1983) suggested the following conceptual equation to describe the activity coefficient of an individual ion in the Pitzer model approach... 

Following is a relatively simple calculation using the Pitzer model to compute the activity coefficient of HCOj in seawater. The exercise is based largely on Millero (1983). (See also Harvie et al. 1984 Pitzer 1987). The activity coefficient of a trace cation in NaCl electrolyte solution can be written... 

DH-type, low ionic-strength term. Because the DH-type term lacks an ion size parameter, the Pitzer model is also less accurate than the extended DH equation in dilute solutions. However, a.ssuming the necessary interaction parameters (virial coefficients) have been measured in concentrated salt solutions, the model can accurately model ion activity coefficients and thus mineral solubilities in the most concentrated of brines. 

Models are often developed to explain certain kinds of data, ignoring other kinds that also might be pertinent. The initial development of Pitzer s equations (33.34) for activity coefficients in concentrated solutions was focused on explaining measurements of vapor pressure equilibrium and of electromotive force (emf). The data could be explained by assuming that the electrolytes examined were, at least in a formal sense, fully dissociated. Later work using these equations to explain solubility data required the formal adoption of a few ion pair species (30). Even so, no speciation/activity coefficient model based on Pitzer s equations is presently consistent with the picture of much more extensive ion-pairing based on other sources, such as Smith and Martell s (35) compilation of association constants. This compilation is a collective attempt to explain other kinds of data, such as electrical conductance, spectrophotometry, and acoustic absorption. 

As illustrated in Figure Vt-5, the different approaches and ion interaction coefficients can lead to appreciably discrepant activity coefficients for the Th" ioa For instance the values of in 0.1 or 0.5 m NaCl and hence the solubility constants calculated in [2000RAI/MOO] and [2003NEC/ALT] with the Pitzer model and SIT respectively for microcrystalline Th02(cr) at low pH, where hydrolysis is actually negligible, differ by about two orders of magnitude. 

As discussed earlier, the observed large difference in the standard state equilibrium constants based on the SIT and Pitzer models must result primarily from the differences in the standard state equilibrium constants for the major aqueous species used in these interpretations. Of course, the activity coefficients for the species involved, in particular that for the Th" ion, are also different in the two models cf. Section VI.3.2). 

To model the brine, we enter the chemical composition and set the activity model. The REACT commands debye-huckel, hmw, and pitzer, respectively, set the Debye-Hiickel (B-dot), Harvie-Mller-Weare, and Pitzer activity models. Here, the Pitzer model refers to the method of Pitzer (1979), as adapted at LLNL by Jackson and Wolery (1985). The HMW model does not account for bromine, so we must type remove Br- before invoking it. Similarly, the Pitzer model does not contain HCO3. 

Figure 7.8 shows the resulting saturation indices for halite and anhydrite, calculated for the first four samples in Table 7.8. The Debye-Hiickel (B-dot) method, which of course is not intended to be used to model saline fluids, predicts that the minerals are significantly undersaturated in the brine samples. The Harvie-M< >ller-Weare and Pitzer models, on the other hand, predict that halite and anhydrite are near equilibrium with the brine, as we would expect. As usual, we cannot determine whether the remaining discrepancies result from the analytical error, error in the activity model, or error from other sources. 

This Weiss equation is applicable to surface and shallow subsurface water within a limited temperature range. For more complex conditions (higher temperature and salinity) it is necessary to determine directly activities coefficient of nonpolar components using corresponding equations of state of solutions. For instance, according to Pitzer model... 

One of the most widely used activity coefficient models has been proposed by Pitzer in 1973 [6-10]. In principle, it is a series expansion of the Gibbs energy, analogous to the virial equation of state however, unlike to that, it is not directly justified by statistical mechanics. The expression is... 

In the following section we discuss the problems of activities of ionic species. Following that we discuss the conventions used to obtain numerical values for the state variables of individual ions, and we discuss the theory underlying the two major approaches to systematizing the data on electrolytes, the HKF and the Pitzer models. Because these are essentially equations of state, we introduced them in Chapter 13 ( 13.6.2 and 13.6.3). 

As mentioned earlier, most experimental data is measured at 25 C. This limits the applicability of the published parameters for the activity coefficient models discussed. The foDowing pages compare available experimental data and the activity coefficients calculated using Bromley, Meissner, Pitzer and Chen s models at temperatures other that 25"C. 

This review should cover more than one of these sources as they do not always present the same complexes. In those cases where there are differences, a choice of values will have to be made. This decision can be aided by a literature search which often will yield articles detailing work done to identify the ionic and molecular species in solution. Articles may be found describing modeling efforts similar to the one desired. Such articles are becoming increasingly common and they often use one of the activity coefficient modeling equations presented in Chapters IV and V. The Pitzer formalism, in particular, is frequently used. 

Chemical equilibrium in a closed system at constant temperature and pressure is achieved at the minimum of the total Gibbs energy, min(G) constrained by material-balance and electro-neutrality conditions. For aqueous electrolyte solutions, we require activity coefficients for all species in the mixture. Well-established models, e.g. Debye-Htickel, extended Debye-Hiickel, Pitzer, and the Harvie-Weare modification of Pitzer s activity coefficient model, are used to take into account ionic interactions in natural systems [15-20]. 

Nonelectrolyte G mcxlels only account for the short-range interaction among non-charged molecules (—One widely used G model is the Non-Random-Two-Liquid (NRTL) theory developed in 1968. To extend this to electrolyte solutions, it was combined with either the DH or the MSA theory to explicitly account for the Coulomb forces among the ions. Examples for electrolyte models are the electrolyte NRTL (eNRTL) [4] or the Pitzer model [5] which both include the Debye-Hiickel theory. Nasirzadeh et al. [6] used a MSA-NRTL model [7] (combination of NRTL with MSA) as well as an extended Pitzer model of Archer [8] which are excellent models for the description of activity coefficients in electrolyte solutions. Examples for electrolyte G models which were applied to solutions with more than one solvent or more than one solute are a modified Pitzer approach by Ye et al. [9] or the MSA-NRTL by Papaiconomou et al. [7]. However, both groups applied ternary mixture parameters to correlate activity coefficients. Salimi et al. [10] defined concentration-dependent and salt-dependent ion parameters which allows for correlations only but not for predictions or extrapolations. 

Archer DG, Phys J (1991) Modification of the Pitzer Model to Calculate the Mean Activity-Coefficients of Electrolytes In a Water-Alcohol Mixed-Solvent Solution. Thermodynamic properties of the NaBr -l-H20 system. Chem Ref Data 20 509-555... 

In actual experiments, as indicated above, ionization quotients Q are usually measured in a solution at finite ionic strength made up by the addition of supporting electrolytes such as NaCl, KCl, or NaCFsSOs. Therefore, activity coefficient models are needed to extrapolate the Q values to infinite dilution for such equilibria. All of these models are based on some version of the Debye-Hiickel equation, which determines the initial slope the logio0 versus ionic strength dependence, with additional empirical ionic strength terms which are typically derived from those used in the Pitzer ion interaction model (Pitzer, 1991). An example of this empirical approach is given in Equation (3.29). 

Typical behaviour of osmotic and activity coefficients as calculated using Eqs. (5.36) and (5.37), is illustrated for trisodium citrate and tripotassium citrate in Fig. 5.15. It can be observed, that values of the (/w) and y+(/w) coefficients after a strong fall in very dilute solutions depend rather weakly on the citrate concentration. Since a T-,m) values are nearly temperature independent, the same is observed in the case osmotic and activity coefficients. It is worthwhile to mention that the Pitzer model was also used by Schunk and Maurer [163] when they determined water activities at 25 °C in ternary systems (citric acid + inorganic salt). The interactions parameters between ions, which were applied to represent activities in ternary systems, were calculated by taking into account the dissociation steps of citric acid and the formation of bisulfate ions for solutions with sodium sulfate. 

The Pitzer model includes a modified Debye-Hiickel-like contribution and a virial term to take short range interactions into account. Only two parameters having physical meaning must be adjusted. Accurate results have been obtained for the properties of electrolyte solutions attaining molalities up to 6 moles/kg of solvent. Activity coefficients have been calculated from this model for solutions containing different salts. They have been correctly predicted for solutions of NaCl, KCl and CaCl2 (from [DEM 91]). 

Rard, J.A., Palmer, D.A., and Albright, J.G. (2003) Isopiestic determination of the osmotic and activity coefficients of aqueous sodium trifluoromethanesulfonate at 298.15 K and 323.15 K, and representation with an extended ion-interaction (Pitzer) model. / Ghent. Eng. Data, 48, 158-166. 

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