Pitzer Parameter Derivatives

From K.S. Pitzer, Theory Ion Interaction Approach , Activity 

Coefficients in Electrolyte Solutions, vol. 1, Ricardo M. Pytkowlcz, ed., CRC Press, Boca Raton, Fla. (1979) 

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The truly remarkable thing about the Pitzer equations (17.40) to (17.42) is that the above parameters derived from one- and two-salt systems can be used with extraordinary success to predict behavior in systems containing many more ionic components. Use of the Pitzer equations to predict activities in very complex salt solutions requires data for the single-salt parameters f MX I MX C mx for possible com-... 

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 ... 

The derivative equations for osmotic and activity coefficients, which are presented below, were applied to the experimental data for wide variety of pure aqueous electrolytes at 25°C by Pitzer and Mayorga (23) and to mixtures by Pitzer and Kim (11). Later work (24-28) considered special groups of solutes and cases where an association equilibrium was present (H PO and SO ). While there was no attempt in these papers to include all solutes for which experimental data exist, nearly 300 pure electrolytes and 70 mixed systems were considered and the resulting parameters reported. This represents the most extensive survey of aqueous electrolyte thermodynamics, although it was not as thorough in some respects as the earlier evaluation of Robinson and Stokes (3). In some cases where data from several sources are of comparable accuracy, a new critical evaluation was made, but in other cases the tables of Robinson and Stokes were accepted. 

Table HI compiles MC results obtained over the years for the critical temperature and critical density of the RPM. Table in includes also results from the cluster calculations of Pitzer and Schreiber [141]. In a critical assessment of earlier work [40, 141, 179-181, 246], Fisher deduced in 1994 that T = 0.052-0.056 and p = 0.023-0.035 represent the best values [15]. Since then, however, the situation has substantially changed. Caillol et al. [53,247] performed simulations of ions on the surface of a four-dimensional hypersphere and applied finite-size corrections. Valleau [248] used his thermodynamic-scaling MC for systems with varying particle numbers to extract the infinite-size critical parameters. Orkoulas and Panagiotopoulos [52] performed grand canonical simulations in conjunction with a histogram technique. All studies indicate an insufficient treatment of finite-size effects in earlier work. While their results do not agree perfectly, they are sufficiently close to estimate T = 0.048-0.05 and p = 0.07-0.08, as already quoted in Eq. (6). Critical points of some real Coulombic systems match quite well to these figures [5]. The coexistence curve derived by Orkoulas and Panagiotopoulos [52] is displayed in Fig. 9.

The problem with this procedure is that it adds more parameters, although not as many as might be expected, since derivatives of equations (18.46) to (18.48) using equations (18.33), (18.36), and (18.37) give the thermal coefficients / (°)L, Bj x, and C x. Silvester and Pitzer use a total of 16 parameters to... 

For dilute solutions. Equation (1.18) and Equation (1.19) are sufficient to describe the density and enthalpy. For more concentrated solutions, corrections must be applied. The corrections to the density are obtained from the pressure derivative of an activity-coefficient expression, while those for enthalpy are obtained from a temperature derivative. Details are given by Zemaitis et al. [72] and Pitzer [70] the latter also tabulates parameters for temperature dependence in the Pitzer activity-... 

For vapor-liquid equilibrium calculations up to moderate pressures, the B equation is suitable and convenient for the vapor phase for its applicability and simple form. Formulas have been derived from statistical theory for the calculation of virial coefficients, including B, from intermo-lecular potential energy functions, but intermolecular energy functions are hardly known quantitatively for real molecules. B is found for practical calculations by correlating experimental B values. Pitzer [1] correlated B of normal flnids in a generalized form with acentric factor to as the third parameter. 

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... 

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. 

Few parameters for ion-neutral interactions are available as yet G.,5,9). However, we have recently derived parameters for NH3-salt interactions, using partial pressure, liquid phase partitioning and salt solubility data (, which may serve as a model for the treatment of other weak and nonelectrolytes. The results of this work, and the application of the Pitzer equations to the calculation of neutral species solubility, are discussed below. 

Equation 25 was developed from an empirical representation of thg second virial coefficient correlation of Pitzer and Curl (I) parameter b was left unchanged at its classical value of 0.0866. Because of the substantial improvement in the prediction of and its temperature derivatives for nonsimple fluids, the Barner modification of the RK equation gave improved estimates of enthalpy deviations for nonpolar vapors and for vapor-phase mixtures of hydrocarbons. However, the new equation was unsuitable for fugacity calculations. 

The same methods used to describe activity coefficients here can also be applied to other thermodynamic properties such as excess volumes, enthalpies, entropies, heat capacities, and so on by manipulating the defining equation (17.38) appropriately (see Pitzer, 1987). Experimental data useful in deriving the ion-interaction parameters of... 

The first two terms are derived from Debye-Hiickel theory, and the third and fourth terms express short-range interactions (e.g., ion-molecular interactions). A( ) can be calculated as a function of temperature using the polynomial equation given by Clegg et al. (1994), which is based on the study of Archer and Wang (1990). Pitzer and Mayorga (1973) determined three parameters mx)... 

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. 

Felmy et al, (18) investigated foe solubility of Pu(OH)3 under reducing conditions in deioni water and brine solution. They deriv a much lower solubility product (log K = -26.2) (see Table I) than foe value (log K = -19.6) reported in foe literature (iP). However, foe solubility in brines [I 6 2Uid I - 10] was found to be larger than foat in deionized (I = 0) waters. The solubility of Pu(OH)3 in brines was accurately predicted with foe Pitzer ion-interaction model using only foe parameters for binary interactions between Pu and Cl". 

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