The Pitzer Equations

In the 1970s, Kenneth Pitzer and his associates developed a theoretical model for electrolyte solutions combined the D-H equation with additional terms in the form of a virial equation. This has proven to be extraordinarily successful at fitting the behavior of both single- and mixed-salt solutions to high concentrations. Recent summaries of this model are provided by Pitzer (1979, 1987), Harvie and Weare (1980), and Weare (1987), and much of the following discussion is adapted from these articles, particularly those by Harvie and Weare (1980) and Pitzer (1987). 

The Pitzer model adds a virial expansion to a simplified version of the D-H equation and begins by describing the total excess free energy of an electrolyte solution as 

Here is kilograms of water and mj is molality of species i. The term /(/) is a version of the D-H equation dependent only on ionic strength (with no adjustable 3i parameter). The quantities and Hijk are second and third virial coefficients 

Equation (17.38) can therefore be rewritten in terms of activity coefficients for specific ions and the osmotic coefficient of the solvent by taking derivatives with respect to the number of moles m, of each ionic constituent per Kg water (see Pitzer, 1987). These derivatives contain several functions B, C, and 5 ) defined below, and are as follows. 

In these equations, rric is the molality of cation c, which has charge (and analogous quantities nia, a, and apply to anions). The original Pitzer model takes no explicit account of ion association, so all molalities rrii used in the above expressions refer to the total analytic concentration of species i. Subscripts M, c, and d refer to cations, and X, a, and a to anions M and X to the ion being considered, c and a to ions being indexed or summed, and d and a to ions other than c and a. Thus, the summation is taken over all cation species, but the double summation 

Much of the theory underlying the Pitzer equations was developed for gases and extended to electrolyte solutions largely by Joseph Mayer (Mayer and Mayer, 1940 Mayer, 1950), and especially McMillan and Mayer (1945). For a brief summary see McQuarrie (2000, Chapter 15) and for a comprehensive summary see Friedman, (1962). If you do consult these references, your knowledge of statistical mechanics had better be pretty good. Mazo and Mou (1991) describe the technical details in McMillan and Mayer as rather intricate.  

The model relates the excess total Gibbs energy of a system to an equation similar in principle to the virial equation we saw for gases in 13.5, in which the first term is not the ideal gas expression, but a simplified form of the Debye-Htickel equation. The general equation used by Pitzer (from which many others are derived by differentiation) is 

The first virial coefficient /(/) is some function of the ionic strength and is not 0 as it would be for an ideal solution, but is in fact a version of the Debye-Htickel equation, which represents departure from ideality in very dilute solutions. The following term is a function of the interactions of all pairs of ions, and the third term a function of the interactions of ions taken three at a time. The second coefficient. Ay, is a function of ionic strength, but the third coefficient ju-y - is considered to be independent of ionic strength and equals zero if /, j, and k are all anions or cations. Later extensions to the model published by Pitzer and co-workers allow for an ionic strength dependence to the third coefficient. Pitzer (1987) and Harvie and Weare (1980) note that higher virial coefficients are required only for extremely concentrated solutions, so the series is stopped at the third coefficient. 

To relate this to activity coefficients we know, from (10.53) and (10.54), that the partial derivatives of G in (15.31) are 

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Can the species activity coefficients be calculated accurately An activity coefficient relates each dissolved species concentration to its activity. Most commonly, a modeler uses an extended form of the Debye-Hiickel equation to estimate values for the coefficients. Helgeson (1969) correlated the activity coefficients to this equation for dominantly NaCl solutions having concentrations up to 3 molal. The resulting equations are probably reliable for electrolyte solutions of general composition (i.e., those dominated by salts other than NaCl) where ionic strength is less than about 1 molal (Wolery, 1983 see Chapter 8). Calculated activity coefficients are less reliable in more concentrated solutions. As an alternative to the Debye-Hiickel method, the modeler can use virial equations (the Pitzer equations ) designed to predict activity coefficients for electrolyte brines. These equations have their own limitations, however, as discussed in Chapter 8. 

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. 

Recently, there have been a number of significant developments in the modeling of electrolyte systems. Bromley (1), Meissner and Tester (2), Meissner and Kusik (2), Pitzer and co-workers (4, ,j5), and" Cruz and Renon (7j, presented models for calculating the mean ionic activity coefficients of many types of aqueous electrolytes. In addition, Edwards, et al. (8) proposed a thermodynamic framework to calculate equilibrium vapor-liquid compositions for aqueous solutions of one or more volatile weak electrolytes which involved activity coefficients of ionic species. Most recently, Beutier and Renon (9) and Edwards, et al.(10) used simplified forms of the Pitzer equation to represent ionic activity coefficients. 

In this paper, two new models for the activity coefficients of ionic and molecular species in electrolyte systems are presented. The first is an extension of the Pitzer equation and is covered in more detail in Chen, et al. (11). 

Previous Applications of the Pitzer Equation to Weak Electrolytes... 

Recently, the Pitzer equation has been applied to model weak electrolyte systems by Beutier and Renon ( ) and Edwards, et al. (10). Beutier and Renon used a simplified Pitzer equation for the ion-ion interaction contribution, applied Debye-McAulay s electrostatic theory (Harned and Owen, (14)) for the ion-molecule interaction contribution, and adoptee) Margules type terms for molecule-molecule interactions between the same molecular solutes. Edwards, et al. applied the Pitzer equation directly, without defining any new terms, for all interactions (ion-ion, ion-molecule, and molecule-molecule) while neglecting all ternary parameters. Bromley s (1) ideas on additivity of interaction parameters of individual ions and correlation between individual ion and partial molar entropy of ions at infinite dilution were adopted in both studies. In addition, they both neglected contributions from interactions among ions of the same sign. 

In this study the Pitzer equation is also used, but a different, more straightforward approach is adopted in which the drawbacks just discussed do not arise. First, terms are added to the basic virial form of the Pitzer equation to account for molecule-ion and molecule-molecule interactions. Then, following Pitzer, a set of new, more observable parameters are defined that are functions of the virial coefficients. Thus, the Pitzer equation is extended, rather than modified, to account for the presence of molecular solutes. The interpretation of the terms and parameters of the original Pitzer equation is unchanged. The resulting extended Pitzer equation is... 

To test the validity of the extended Pitzer equation, correlations of vapor-liquid equilibrium data were carried out for three systems. Since the extended Pitzer equation reduces to the Pitzer equation for aqueous strong electrolyte systems, and is consistent with the Setschenow equation for molecular non-electrolytes in aqueous electrolyte systems, the main interest here is aqueous systems with weak electrolytes or partially dissociated electrolytes. The three systems considered are the hydrochloric acid aqueous solution at 298.15°K and concentrations up to 18 molal the NH3-CO2 aqueous solution at 293.15°K and the K2CO3-CO2 aqueous solution of the Hot Carbonate Process. In each case, the chemical equilibrium between all species has been taken into account directly as liquid phase constraints. Significant parameters in the model for each system were identified by a preliminary order of magnitude analysis and adjusted in the vapor-liquid equilibrium data correlation. Detailed discusions and values of physical constants, such as Henry s constants and chemical equilibrium constants, are given in Chen et al. (11). 

In general, data are fit quite well with the model. For example, with only two binary parameters, the average standard deviation of calculated lny versus measured InY of the 50 uni-univalent aqueous single electrolyte systems listed in Table 1 is only 0.009. Although the fit is not as good as the Pitzer equation, which applies only to aqueous electrolyte systems, with two binary parameters and one ternary parameter (Pitzer, (5)), it is quite satisfactory and better than that of Bromley s equation (J). 

The interaction parameters are weak, linear functions of temperature, as shown in Table 5, Table 6 and Figure 6. These tables and figure show the results of isothermal fits for activity coefficient data of aqueous NaCl and KBr at various temperatures. The Pitzer equation parameters are, however, strongly dependent on temperature (Silvester and Pitzer, (23)). 

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. 

For the industrially important class of mixed solvent, electrolyte systems, the Pitzer equation is not useful because its parameters are unknown functions of solvent composition. A local composition model is developed for these systems which assumes that the excess Gibbs free energy is the sum of two contributions, one resulting from long-range forces between ions and the other from short-range forces between all species. 

Chen, C., H. I. Britt, J. F. Boston, and L. B. Evans, "Extension and Application of the Pitzer equation for Vapor-Liquid Equilibrium of Aqueous Electrolyte Systems with Molecular Solutes," AIChE J., 1979, 25, 820. 

Generally, agreement has been found between our correlations and those of Pitzer, and others (1972, 1973, 1974, 1975, 1976) and Rard, and others (1976, 1977). Many of our correlations agree fairly well with Robinson and Stokes, (1965) and Harned and Owen, (1958) but in most cases a much larger data base and more recent measurements have been incorporated into the evaluations. It has been observed that agreement with Pitzer s equations is found below moderate concentrations (several molal), but often deviate at higher concentrations where the Pitzer equations do not contain enough parameters to account for the behavior of the activity (or osmotic) coefficient. 

The Pitzer equation for single electrolytes with the value of parameters collected in several publication may be used as a compact source of activity coefficient data. From the values y° yj. thus obtained, one may calculate, for example, (/, / ) values using Eq. (6.1). 

One method takes into account the individual characteristics of the ionic media by using a medium-dependent expression for the activity coefficients of the species involved in the equilibrium reactions. The medium dependence is described by virial or ion interaction coefficients as used in the Pitzer equations and in the specific ion interaction model. 

Numerous studies on the thermodynamics of calcium chloride solutions were published in the 1980s. Many of these were oriented toward verifying and expanding the Pitzer equations for determination of activity coefficients and other parameters in electrolyte solutions of high ionic strength. A review article covering much of this work is available (7). Application of Pitzer equations to the modeling of brine density as a function of composition, temperature, and pressure has been successfully carried out (8). 

Pitzer s solution to the problem was the development of a set of analytical equations that are thermodynamically consistent after transformations through the Gibbs-Duhem equation. These equations are known as the Pitzer equations, in recognition of the major role that he played in developing them and the major contributions he made in the understanding of electrolyte solutions through a lifetime of work. We will now summarize these equations and describe their usefulness. For details of the derivation we refer the reader to Pitzer s original paper.6... 

When Pitzer developed these equations, the ultimate form for describing the interaction terms was based on both theoretical models and experimental data. On the other hand, the number of terms to include in the equations is left to the user s discretion. For example, are neutral-neutral species interaction terms needed In some applications, yes in other applications, no. See Harvie et al. (1984), He and Morse (1993), and Pitzer (1995) for examples where different terms were selected. In what follows, we will specify the exact form of the Pitzer equations used in the FREZCHEM model. For a discussion of the connection between these equations (2.39 to 2.42) and Eq. 2.38, see Pitzer (1991, 1995). 

Note that the equations for estimating the pressure dependencies of 7 and aw (Eqs. 2.87 and 2.90) depend on the Pitzer equations (Eqs. 2.76, 2.80, and 2.81) but this is not the case for the pressure dependence of the equilibrium constants (Eq. 2.29) the latter equation is based entirely on partial molar volumes at infinite dilution, which are independent of concentration. Also, compared to the pressure-dependent equation for the equilibrium constant (Eq. 2.29), the pressure equations for activity coefficients (Eq. 2.87) and the activity of water (Eq. 2.90) do not contain compressibilities (K) because the database for these terms and the associated Pitzer parameters are lacking at present (Krumgalz et al. 1999). The consequences of truncating Eqs. 2.80 and 2.81 for ternary terms and Eqs. 2.87 and 2.90 for compressibilities will be discussed in Sect. 3.6 under limitations. 

This assumption limits application of the latter chemistries to low pressures. Activity coefficients for aqueous-phase gases (CO2, O2, and CH4) are calculated using the Pitzer equation for neutral species (Eq. 2.42). Activity coefficients for aqueous acids are calculated using the Pitzer equations for ions (Eqs. 2.40 and 2.41). For the case of HC1, the Henry s law constant is given by... 

Pierrot D, Millero FJ (2000) The apparent molal volume and compressibility of seawater fit to the Pitzer equations. J Soln Chem 29 719-742... 

For higher ionic strength, e g. highly saline waters the PITZER equation can be used (Pitzer 1973). This semi-empirical model is based also on the DEBYE-HUCKEL equation, but additionally integrates virial equations (vires = Latin for forces), that describe ion interactions (intermolecular forces). Compared with the ion dissociation theory the calculation is much more complicated and requires a... 

In the following only a simple example of the PITZER equation is briefly described. For the complete calculations and the necessary data of detailed parameters and equations the reader is referred to the original literature (Pitzer 1973, Pitzer 1981, Whitfield 1975, Whitfield 1979, Silvester and Pitzer 1978, Harvie and Weare 1980, Gueddari et al. 1983, Pitzer 1991). 

If the ionic strength exceeds 6 mol/L, the PITZER equation is no longer applicable though. 

Fig. 4 to Fig. 8 show the severe divergence for activity coefficients such as given here for calcium, chloride, sulfate, sodium and water ions, calculated with different equations. The activity coefficients were calculated applying Eq. 13 to Eq. 17 for the corresponding ion dissociation theories, whereas the values for the PITZER equations were gained using the program PHRQPITZ. The limit of validity of each theory is clearly shown. 

Fig. 6 Comparison of the activity coefficient of SO42" in relation to the ionic strength as calculated using a Na2(S04) solution (aS04-2= 5.31, bS04-2= -0 07 Table 4) and different theories of ion dissociation and the PITZER equation, dashed lines signify calculated values outside the validity range of the corresponding ion dissociation equation.

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