Pitzer Based Equations

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 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. 

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... 

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. 

Theoretical relationships for activity coefficients. The Setschenow equation is used to calculate the activity coefficients of aqueous molecular species in salt solutions. The Pitzer based methods may be used for binary or multicomponent solution activity coefflcient calculations for all species in the solution. 

DEBYE-HUCKEL CONSTANT FROM LEWIS RANDALL, CONVERTED FROM ACTVITY COEFFICIENT LOG 10 BASIS TO THE OSMOTIC COEFFICIENT NATURAL LOG BASED ONE USED BY PITZER S EQUATIONS... 

They calculated y,-, by a slightly modified version of Equation 9.24, based on Scatchard and Hildebrand s regular solution theory, and found (< i)pure liquid t by a Pitzer-type equation... 

Helgeson (1969 see also Helgeson and Kirkham, 1974) presented an activity model based on an equation similar in form to the Davies equation. The model, adapted from earlier work (see Pitzer and Brewer, 1961, p. 326, p. 578, and Appendix 4, and references therein), is parameterized from 0°C to 300 °C for solutions of up to 3 molal ionic strength in which NaCl is the dominant solute. The model takes it name from the B-dot equation,... 

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. 

Pitzer et al (1972, 1973, 1974, 1975, 1976) have proposed a set of equations based on the general behavior of classes of electrolytes. Pitzer (1973) writes equations for the excess Gibbs energy, AGex, the osmotic coefficient activity coefficient Y+ for single unassociated electrolytes as... 

K. S. Pitzer and L. Brewer used equation (18.12) as the base for tabulating a summary of the thermodynamic properties of aqueous electrolyte solutions at 298.15 K. They wrote equation (18.12) in the form that involves base 10 logarithms... 

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). 

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... 

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. 

Table 1.5. Pitzer equation activity coefficients (based on Nagy, 1988).

The LCM is a semi-theoretical model with a minimum number of adjustable parameters and is based on the Non-Random Two Liquid (NRTL) model for nonelectrolytes (20). The LCM does not have the inherent drawbacks of virial-expansion type equations as the modified Pitzer, and it proved to be more accurate than the Bromley method. Some advantages of the LCM are that the binary parameters are well defined, have weak temperature dependence, and can be regressed from various thermodynamic data sources. Additionally, the LCM does not require ion-pair equilibria to correct for activity coefficient prediction at higher ionic strengths. Thus, the LCM avoids defining, and ultimately solving, ion-pair activity coefficients and equilibrium expressions necessary in the Davies technique. Overall, the LCM appears to be the most suitable activity coefficient technique for aqueous solutions used in FGD hence, a data base and methods to use the LCM were developed. 

Davies = Eq. 1.21 Pitzer = Eq. 1.13 and correlation equations based on Young s rules special models = expressions like Eq. 1.23. 

Although use of an equation based on the two-parameter theorem of corresponding states provides far better results in general than the ideal-gas equation, significant deviations from experiment still exist for all but the simple fluids argon, krypton, and xenon. Appreciable improvement results from the introduction of a third corresponding-states parameter, characteristic of molecular structure the most popular such parameter is the acentric factor , introduced by K. S. Pitzer and coworkers.t... 

The generalized correlations of Pitzer provide an alternative to the use of a cubic equation of state for the calculation of thermodynamic properties. However, no adequate general method is yet known for the extension of the Pitzer correlations based on the compressibility factor to mixtures. Nevertheless, Z, as given by... 

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... 

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 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 ... 

Although the tables representing the Pitzer correlations are based on data for pure materials, they may also be used for the calculation of mixture properties. A set of recipes is required relating the parameters T P and co for a mixture to the pure-species values and to composition. One such set is given Iw Eqs. (2-80) through (2-82) in the Seventh Edition of Perry s Chemical Engineers Handbook (1997). These equations define pseudoparameters, so called because the defined values of Tpr, Pp and co have no physical significance for the mixture. 

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