Pitzer equation limitations

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. 

The semi-empirical Pitzer equation for modeling equilibrium in aqueous electrolyte systems has been extended in a thermodynamically consistent manner to allow for molecular as well as ionic solutes. Under limiting conditions, the extended model reduces to the well-known Setschenow equation for the salting out effect of molecular solutes. To test the validity of the model, correlations of vapor-liquid equilibrium data were carried out for three systems the hydrochloric acid aqueous solution at 298.15°K and concentrations up to 18 molal the NH3-CO2 aqueous solution studied by van Krevelen, et al. 

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... 

A number of limitations of the FREZCHEM model can be broadly grouped under Pitzer-equation parameterization, modeling (mathematics, convergence, and coding), and applications. The first two limitations are discussed in this chapter. Application limitations are discussed in Chap. 5 after presentation of multiple applications. 

In Chap. 3 (Sect. 3.6), we discussed limitations of the FREZCHEM model that were broadly grouped under Pitzer-equation parameterization and mathematical modeling. There exists another limitation related to equilibrium principles. The foundations of the FREZCHEM model rest on chemical thermodynamic equilibrium principles (Chap. 2). Thermodynamic equilibrium refers to a state of absolute rest from which a system has no tendency to depart. These stable states are what the FREZCHEM model predicts. But in the real world, unstable (also known as disequilibrium or metastable) states may persist indefinitely. Life depends on disequilibrium processes (Gaidos et al. 1999 Schulze-Makuch and Irwin 2004). As we point out in Chap. 6, if the Universe were ever to reach a state of chemical thermodynamic equilibrium, entropic death would terminate life. These nonequilibrium states are related to reaction kinetics that may be fast or slow or driven by either or both abiotic and biotic factors. Below are four examples of nonequilibrium thermodynamics and how we can cope, in some cases, with these unstable chemistries using existing equilibrium models. 

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. 

The parameters for these equations are tabulated in the appropriate tables in reference (4). Activity coefficients for these charge types may also be calculated from the Pitzer equations for the uni-univalent and uni-bi and bi-univalent salts. In these cases, the Pitzer equations are sometimes applicable to a more limited concentration range. If the concentration being investigated is beyond the range of validity specified by Pitzer, the Hamer-Wu, Lietzke-Stoughton equations are recommended. 

The Pitzer-equation computations for Figures 3 and 4 are based upon experimentally derived 25°C ion-pair and interaction coefficients taken from the literature. From the extensive prior work validating the theory and parameters, these curves should deviate from experiment by less than 20%. However, as Figures 1-4 show, solubility calculations are very sensitive to variations in activity coefficients and the approximations made in eqs. (l)-(9) limit the accuracy of the solubility curves which can be calculated. When higher-order terms are included, Pitzer s equations accurately oredict solubility in the CaSO -MgSO system up to... 

Details of the Pitzer equations and definitions of the notations utilized in this paper and in PHRQPITZ are given in the literature (3-lOL As the focus of this report is on the capabilities and limitations of PHRQPITZ in relation to its application to geochemical problems, only selected aspects of the implementation of the Pitzer equations in PHRQPITZ are presented. 

Other approaches can be used based on corrections to this equation (e.g., Helgeson and Kirkham, 1976), but in recent years the tendency has been to use the Pitzer equations (Chapter 15). Determining the intercept of this equation, or any nonlinear equation, at m = 0 places great emphasis on measurements of very dilute solutions, where they are most difficult. Clearly, some theoretical knowledge of what the slope at the intercept (the limiting slope ) should be is important, and all modern treatments of data of this type use the... 

The Debye-Htickel theory is a cornerstone of electrolyte theory. It is always used in extrapolating data to infinite dilution, and must be embedded in any generalized treatment of activity coefficients as a function of concentration, as it is in the Pitzer equations. However, at concentrations beyond the validity of the limiting law (Equation 15.26), all attempts at predicting electrolyte behavior at higher concentrations are more or less empirical. 

In a binary electrolyte solution such as this one, terms containing A, 0, or tf/ are zero, since these involve interactions with two dissimilar anions or cations. In most such cases, the parameter is unnecessary, because it is invoked to account for exceptionally strong ion-ion interactions. In fact, Pitzer shows that should approach -K/2 in the limit of infinite dilution, where K is the association constant for the ion-pair. The work of Harvie and Weare (1980), Eugster, Harvie and Weare (1980), and Harvie, Eugster and Weare (1982), who modeled solubility equilibria in the multicomponent oceanic salt system is considered a milestone in the application of the Pitzer equations, and the set of parameters in Harvie, Mller and Weare (1984) is considered a sort of standard for modeling of seawater evaporitic systems. 

Inaccuracies arise at modest electrolyte concentrations because the concentrations of ions predicted within an ionic atmosphere are imrealistic if a finite ionic radius is not considered. The assumption of a finite ionic radius leads to the extended Debye-Hiickel limiting law. This is still inaccurate for concentrated ionic solutions because the depletion of solvent molecules due to solvation shells makes the assumption of zero ion-solvent interaction highly inaccurate. Only empirical formulas such as the Robinson-Stokes or Pitzer equations are able to address this issue. Most well-supported electrolyte solutions have behaviours in this latter regime. 

We have applied Pitzer s equations at T = 298.15 K, but they are not limited to that temperature and can be applied at any temperature where the coefficients are known.k Table I8.l (and Table A7.1 of Appendix 7) gives the Debye-Hiickel coefficients AA, Ah, and Aj as a function of temperature, but the coefficients specific to the electrolyte are tabulated in Appendix 7 only at T = 298.15 K. The usual solution to this problem is to express the coefficients as... 

In this appendix, we summarize the coefficients needed to calculate the thermodynamic properties for a number of solutes in an electrolyte solution from Pitzer s equations.3 Table A7.1 summarizes the Debye-Huckel parameters for water solutions as a function of temperature. They provide the leading terms for Pitzer s equations, and can also be used to calculate the Debye-Huckel limiting law values from the equations... 

The symbols have the same physical significance as those in Pitzer s previous papers (1,2,3,4,5), which are based on a different theoretical framework from that of Scatchard and use the ions of the mixed electrolytes as components. The 0mn (the doublet cation-cation interaction) represents the interactions between H+ and (Et)4N+, whereas mnx (the triplet ion interaction) indicates the interactions between H+, Br", and (Et)4N+. Thus, the quantities 0, 0, and are properties characteristic of the mixture, whereas By, BCY, and C are the properties of the single electrolyte solution, and are functions of the ionic strength. Equation 10 can be further reduced after imposing the conditions that Mnx = 0, 0 Mn = 0, and y2 (at the limit) = 0 ... 

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