Model Pitzer approach

The statistical thermodynamic approach of Pitzer (14), involving specific interaction terms on the basis of the kinetic core effect, has provided coefficients which are a function of the ionic strength. The coefficients, as the stoichiometric association constants in our ion-pairing model, are obtained empirically in simple solutions and are then used to predict the activity coefficients in complex solutions. The Pitzer approach uses, however, a first term akin to the Debye-Huckel one to represent nonspecific effects at all concentrations. This weakens somewhat its theoretical foundation. 

As most other methods, Including our 1on-pairing model which was described earlier, the Pitzer approach 1s empirical in practice. The interaction coefficients in this case are determined by curve fitting in single electrolyte solutions. 

Implementation of the Pitzer approach is through the FREZCHEM (FREEZING CHEMISTRY) model, which is at the core of this work. This model was originally designed to simulate salt chemistries and freezing processes at low temperatures (—54 to 25 °C) and 1 atm pressure. Over the years, this model has been broadened to include more chemistries (from 16... 

The chemical equilibrium model, FREZCHEM, requires calculation of solute activity coefficients (7) and the osmotic coefficient ((f)) in concentrated solutions (Chap. 3). In this work, the Pitzer approach is used to calculate these quantities. 

To our knowledge, no one has ever worked out the mathematics for directly estimating the pressure dependence of the osmotic coefficient (or aw) using the Pitzer approach. However, Monnin (1990) developed an alternative model based on the Pitzer approach that allows calculation of the pressure dependence for the activity of water (aw). The density of an aqueous solution (p) can be calculated with the equation... 

There are five reactions that deal with ion associations (numbers 8 to 12 in Table 3.3). There are, of course, many more such associations in concentrated electrolyte solutions. But the Pitzer approach allows one to either explicitly identify an ion association (Table 3.3) or to implicitly include the interaction effect in the interaction coefficients (B, C, minor components of the aqueous phase. 

The FREZCHEM model was basically designed for estimating the aqueous properties of concentrated electrolyte solutions, which is why we used the Pitzer approach. Nevertheless, it is still necessary that these models accurately describe dilute solutions. The comparisons in Table 3.5 demonstrate that the FREZCHEM model is reasonably accurate for both dilute and concentrated electrolyte solutions. 

The thermodynamic data compiled in the present review (see Chapters III and IV and Appendix E) refer to the reference temperature of 298.15 K and to standard conditions, cf. Section II.3. For the modelling of real systems it is, in general, necessary to recalculate the standard thermodynamic data to non-standard state conditions. For aqueous species a procedure for the calculation of the activity factors is thus required. This review uses the approximate specific ion interaction method (SIT) for the extrapolation of experimental data to the standard state in the data evaluation process, and in some cases this requires the re-evaluation of original experimental values (solubilities, emf data, etc.). For maximum consistency, this method, as described in Appendix B, should always be used in conjunction with the selected data presented in this review. However, some solubility data for highly soluble selenates were evaluated in the original papers by the Pitzer approach. No attempt was made to re-evaluate these data by the SIT method. 

Pitzer s equations can be used for mixtures of electrolyes. Thermodynamic functions are obtained in the usual way as the derivatives of the chemical potential with respect to temperature or pressure. However, a considerable number of empirically adjusted parameters is needed to obtain satisfactory data description. The Pitzer approach is used as a self-standing data-reduction method, but it is also embedded by engineers in the so-called NRTL (nonrandom two liquid) electfolyte models. 

As to the ion-interaction model, it is a semiempirical statistical thermodynamics model. In this model, the Pitzer approach begins with a virial expansion of the excess free energy of the form to consider the three kinds of existed potential energies on the ion-interaction potential energy in solution. 

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. 

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). 

There have been numerous attempts to treat torsional motion in molecular statistical mechanics and we do not attempt to review this vast literature here. We do note, however, the ground breaking early work of Pitzer and Gwinn who proposed a separable hindered rotor model. This approach, and closely related variations, is still widely used today as the most common method to go beyond the HO-RR model. We also point out that several non-separable methods have recently been developed that appear to be promising. ... 

Equilibrium constants calculated from the composition of saturated solutions are dependent on the accuracy of the thermodynamic model for the aqueous solution. The thermodynamics of single salt solutions of KC1 or KBr are very well known and have been modeled using the virial approach of Pitzer (13-15). The thermodynamics of aqueous mixtures of KC1 and KBr have also been well studied (16-17) and may be reliably modeled using the Pitzer equations. The Pitzer equations used here to calculate the solid phase equilibrium constants from the compositions of saturated aqueous solutions are given elsewhere (13-15, 18, 19). The Pitzer model parameters applicable to KCl-KBr-l O solutions are summarized in Table II. 

The most fertile approaches so far, from the standpoint of predicting y+ in complex solutions from data in simple ones, have been that of Pitzer (H), and the ion-pair approach of Pytkowicz and Kester (2) and of Johnson and Pytkowicz (3). The lattice model of Pytkowicz and Johnson (19) is, at thTs time, an explanatory rather than a predictive tooT. 

Caylor et al. [134] performed ellipsometric measurements at the liquid-liquid interface of Pitzer s system. The experiments yielded a decrease in ellipticity, when approaching the critical point. This is in clear contrast to a (/ — v) divergence predicted theoretically [135] and observed experimentally [136] for nonionic Ising mixtures. The results point toward an anisotropic interface caused by an orientation of ion pairs perpendicular to the interface. A rough model for such an interface captured some of the observed features. 

In addition to knowing the TP dependence of equilibrium constants (Eqs. 2.25 and 2.28), we must also know the T-P dependence of solute activity coefficients and the osmotic coefficient of the solution. A theoretical model, such as Pitzer s approach, is necessary for this purpose because activity coefficients and the osmotic coefficient must be defined at finite concentrations and not simply for the infinitely dilute state, which suffices for equilibrium constants (Eqs. 2.25 and 2.28). 

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 most common approach used by geochemical modeling codes to describe the water-gas-rock-interaction in aquatic systems is the ion dissociation theory outlined briefly in chapter 1.1.2.6.1. However, reliable results can only be expected up to ionic strengths between 0.5 and 1 mol/L. If the ionic strength is exceeding this level, the ion interaction theory (e.g. PITZER equations, chapter 1.1.2.6.2) may solve the problem and computer codes have to be based on this theory. The species distribution can be calculated from thermodynamic data sets using two different approaches (chapter 2.1.4) ... 

The difference between the extended Debye-Hiickel equation and the Pitzer equations has to do with how much of the nonideahty of electrostatic interactions is incorporated into mass action expressions and how much into the activity coefficient expression. It is important to remember that the expression for activity coefficients is inexorably bound up with equilibrium constants and they must be consistent with each other in a chemical model. Ion-parr interactions can be quantified in two ways, explicitly through stability constants (lA method) or implicitly through empirical fits with activity coefficient parameters (Pitzer method). Both approaches can be successful with enough effort to achieve consistency. At the present, the Pitzer method works much better for brines, and the lA method works better for... 

Pitzer and co-workers (1973, 1974) have proposed a more detailed, but at the same time more complex, approach. Whitfield (1973, 1975) has applied these equations to seawater and has shown that this model gives good agreement with available experimental data for the osmotic coefficient and for the mean ion activity coefficient of the major electrolyte components. The results obtained yield numerical results similar to the predictions of the ion association model (see Table A6.2). 

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... 

The same approach can be applied to investigate the explosivity conditions of the H20-NaCl system. We have selected the Anderko-Pitzer (AP) equation of state,which is based on realistic physical hypotheses. It describes H20-NaCl by means of statistical thermodynamic models developed for dipolar hard spheres. This assumption is reasonable at high temperatures, where NaCl is known to form dipolar ion pairs. However, for this reason, this equation of state is only applicable above 573 K, 300°C. 

Upon comparing Pitzer-theory calculations for typical scrubber and model solutions with the association-equilibrium, extended Debye-Hiickel code in current use for FGD systems, one sees differences which reflect the differences in concentration range and applicability to mixtures of the two approaches. 

Further efforts based upon the Pitzer equation approach should allow one to model reasonably accurately the complex thermodynamics occurring in flue-gas-desulfurization aqueous scrubbers. Tasks to be pursued to this end include (1) Replacing important estimated parameters by ones based upon experiment, particularly for sulfites. (2) Including higher-order terms (three-body, electrostatic, temperature dependence) where data are available. (3) Extending the treatment to include weak electrolytes. 

The ion-interaction theory in contrast to the above was developed by Pitzer (42) as an outgrowth of work done by Guggenheim (44). This phenomenological methodology was based on the concept that ions electrostaticly interact in solution and that these interactions were based on a statistical likelihood of collision, hence the ionic strength dependency. Several papers in this volume discuss aspects of the importance of this approach to modeling the chemistry of complex systems. 

Ion Interaction. Ion-interaction theory has been the single most noteworthy modification to the computational scheme of chemical models over the past decade this option uses a virial coefficient expansion of the Debye-Huckel equation to compute activities of species in high ionic strength solutions. This phenomenological approach was initially presented by Pitzer ( ) followed by numerous papers with co-workers, and was developed primarily for laboratory systems it was first applied to natural systems by Harvie, Weare and co-workers (45-47). Several contributors to the symposium discussed the ion interaction approach, which is available in at least three of the more commonly used codes SOLMNEQ.88, PHRQPITZ, and EQ 3/6 (Figure 1). 

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