Pitzer parameters, 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 log K values shown in Figure 18.10 are the values that best reproduce all of the heat of mixing curves.v The J1 values are obtained by estimating initial values using the activity coefficients for NaCl(aq).16 These initial values of Jy are then readjusted, as the value for Km is optimized, by adjusting the coefficients of Pitzer s equations, whose form is described in the previous section. Pitzer s equations are, of course, internally consistent so that adjustments to the activity or osmotic coefficient parameters result in adjustments to the thermal parameters (L, L2, 4>J, or J2), and hence, to the heat effects. 

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 three appendices in this volume give selected sets of thermodynamic data (Appendix 5), review the statistical calculations covered in Principles and Applications (Appendix 6), and summarize the equations and parameters required to calculate the properties of electrolyte solutions, principally from Pitzer s equations (Appendix 7). 

In this work the temperature dependencies of Pitzer parameters and equilibrium constants were largely determined from isothermal data sets that were then fit to the following equation ... 

So there is an underlying basis related to the standard enthalpy of reaction (or heat capacities, Eq. 2.26) for the equation form used in this work to characterize the temperature dependence of equilibrium constants and Pitzer parameters (cf. Eqs. 2.70 and 2.73). 

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. 

Equations 2.87 (activity coefficient), 2.88 (density), and 2.90 (activity of water) are all indirectly dependent on the temperature and pressure dependence of B v, B v, BC2), and Cv (Eqs. 2.76, 2.80, and 2.81). Table B.10 (Appendix B) lists the temperature dependence of these volumetric Pitzer parameters. The pressure dependence of these parameters were evaluated with the density equation (Eq. 2.88). All three terms in the denominator of Eq. 2.88 are temperature and pressure dependent. The density of pure water (p°) as a function of temperature and pressure is evaluated with Eqs. 3.14-3.16 and 3.20. Similarly, the molar volume of ions as a function of temperature and pressure is calculated by... 

The Pitzer parameters have been fitted as a function of temperature to the following equations... 

In Equation 16, the Pitzer parameter 0 is truly independent of the common ion, and is equal to V2 (bAjB0,1) in the Scatchard treatment, or gM,x according to Friedman. Also, the Pitzer and Scatchard equations for uni-univalent, three-ion systems are of comparable forms, with bA,B0 2 =... 

The value of C has to be determined experimentally for a particular system but is typically 0.2-0.3. Pitzer presented equations that give activity coefficients for both binary solutions and mixed electrolytes for 7 up to 6 m, but there are several parameters that must be determined empirically. 

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... 

Pitzer s equations and available ion-pair parameters allow calculation of mean-ion activity coefficients Y+ in complex, concentrated electrolyte solutions with an accuracy estimated to be better than + 25% in the range 25° - 55°C. The accuracy of calculated activity coefficients is limited to about the same degree by uncertainties in the estimated parameters and by simplifications introduced in the theory both to reduce the number of parameters to be estimated and to reflect the uncertainties of the estimates. Because activity coefficients are determined to quite an extent by the form of Pitzer s equations and are not extremely sensitive to the exact values of parameters, ion-pair parameters only have to be estimated within a reasonable range. 

Figures 1 and 2 show the results of fitting the osmotic coefficient of the aqueous electrolytes sodium perchlorate and potassium chloride, respectively. Analysis of the variance in fitting the osmotic coefficient indicates that the fits are about as good as those obtained using Pitzer s equations, despite the fact that our equations have one less fitting parameter. For sodium perchlorate, the standi d deviation in our fit is 0.0011, whereas Pitzer ( ) reports 0.001 using his equation. For potassium chloride, the standard deviation in our fit is 0.00036, that in Pitzer s,...

The individual-ion activity coefficients for the free ions were based on the Macinnis (18) convention, which defines the activity of Cl to be equal to the mean activity coefficient of KCl in a KCl solution of equivalent ionic strength. From this starting point, individual-ion activity coefficients for the free ions of other elements were derived from single-salt solutions. The method of Millero and Schreiber (14) was used to calculate the individual-ion, activity-coefficient parameters (Equation 5) from the parameters given by Pitzer (19). However, several different sets of salts could be used to derive the individual-ion activity coefficient for a free ion. For example, the individual-ion activity coefficient for OH could be calculated using mean activity-coefficient data for KOH and KCl, or from CsOH, CsCl, and KCl, and so forth. 

Throughout the present review the SIT is used for ionic strength corrections. However, numerous computer codes for geochemical model calculations, in particular for calculations in concentrated chloride solutions, are based on the ion interaction equations of Pitzer [1991PIT]. Pitzer parameters reported in the literature to calculate activity coefficients for the Th" ion in chloride solutions are briefly discussed and summarised in Table VI-2. 

Collections of useful Pitzer parameters can be taken from Rosenblatt [11], Zemaitis et al. [12], and Pitzer [13]. Temperature-dependent interaction parameters should be used if a wide temperature range must be covered, as it is common practice for nonelectrolyte systems as well. It is important especially for strong acids and bases. The Pitzer equation is usually valid up to molalities of 6 mol/kg. 

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. 

Pitzer s equation - a three to four parameter model for mean molal activity coefficients... 

Pitzer - an equation using three to four parameters per ion pair, one... 

For cuprous chloride in HCl-HClOii solutions, the solubility data of Hikita et al. (C6) met the requirement for data taken at multiple ionic strengths and chlorine concentrations that Fritz needed in order to solve for the stability constants and activity coefficient equation parameters. Unfortunately, this data, like many sets of solubility data, was presented as molarities without the solution densities needed to convert them to the molalities required by Pitzer s equations. Consequently Fritz replaced the molality terms of the equations with molarities. He presented the following justifications (C2) ... 

In equations (15), (16) and (17), y is an adjustable parameter for each pair of anions or cations for each cation-cation and anion-anion pair, called triplet-ion-interaction parameter. The functions, 0 and 0 are fxmctions only of ionic strength and the electrolyte p>air type. Pitzer (1975) derived equations for calculating these effects, and Harvie and Weare (1981) summarized Pitzer s equations in a convenient form as following ... 

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