Pitzer-type equations

The most widely used corresponding states approach is that due to Pitzer and his co-workers [8], often called Pitzer-type equations. The common form of their approach is... 

The terms contribute 14% of the answer. The common z charts Uke Figure A.4 are made up for an average substance, so that if we use them without the corrections for Zc not being equal to 0.27, we often make a small error. The same is not the case for the Pitzer-type equations, tn was defined to be zero for argon, which is not a typical substance, so if we used only the z° term for a typical substance we would make a serious error. 

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... 

Perhaps the most useful of all Pitzer-type correlations is the one for the second virial coefficient. The basic equation (see Eq. [2-68]) is... 

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. 

Stokes and Robinson (1) fit their equation to data reported for 35 pure aqueous electrolytes of the 1 1 and 2 1 types and obtained good results to fairly high concentrations. For example, for NaCl, the reported average difference in In y was only 0.002 for over a concentration range of 0.1-5.0 m (1 = 0.1-5.0 m) for CaCl2, 0.001 from 0.01-1.4 m (I = 0.06 to 8.4 m). These statistics can not be directly compared with those reported for fitting with Pitzer s equations (e.g., ref. 9) as the... 

Critical temperature and pressure are required and can be estimated from the methods of this section. Vapor pressure is predicted by the methods of the next section. Experimental values should be used if available. The acentric factor is used as a third parameter with Tc and Pc in Pitzer-type corresponding states methods to predict volumetric properties and in cubic equations of state such as the Redlich-Kwong-Soave and Peng-Robinson equations. For simple spherical molecules, the acentric factor is essenti y zero, rising as branching and molecu-... 

Criss CM, MUlero FJ (1996) Modeling of the heat capacities of aqueous 1-1 electrolyte solutions with Pitzer s equations. J Phys Chem 100 1288-1294 Criss CM, Mdlero FJ (1999) Modeling of heat capacities of high-valence type electrolyte solutions with Pitzer s equations. J Solution Chem 28 849-867 Danielewicz-Ferchmin 1, Banachowicz E, Ferchmin AR (2003) Protein hydration and the huge electrostriction. Biophys Chem 106 147-153... 

A new equation of state in the acentric factor system developed by Schreiber and Pitzer [4] is used in this work. It is an extended Benedict-Webb-Rubin (eBWR) type equation... 

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 short-range contributions are described with a virial-type equation for the excess Gibbs energy that was adapted from Pitzer [105]. It is applied here neglecting ternary and higher interactions between solute species ... 

Pitzer s Corresponding-States Correlation A three-parameter corresponding-states correlation of the type developed by Pitzer, K.S. Thennodynamic.s, 3ded., App. 3, McGraw-HiU, New York, 1995) is described in Sec. 2. It has as its basis an equation for the compressibility factor ... 

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. 

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. 

DH-type, low ionic-strength term. Because the DH-type term lacks an ion size parameter, the Pitzer model is also less accurate than the extended DH equation in dilute solutions. However, a.ssuming the necessary interaction parameters (virial coefficients) have been measured in concentrated salt solutions, the model can accurately model ion activity coefficients and thus mineral solubilities in the most concentrated of brines. 

For the NBS report (4), the activity and osmotic coefficients of 2 2 charge type electrolyte (MgS04, CaS04, and MnS04) have been calculated from the Pitzer equations (7). 

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. 

A system of equations for electrolytes based on the reference states expressed in Equations 3 and 4 was developed in detail for singly-charged ions by Pitzer and Simonson (. Although they considered both types of reference states for the solute, most of their working equations are for the pure liquid reference state. This reference state was used by Pitzer and Li (32) for a study of the NaCl-H20 system extending to 550 C. For the present research limited to 350 C, however, it seemed better to use the infinitely dilute reference state, and the equations below are derived on that basis. The short-range... 

Errors in Potential Fxmctions, Equation 7 will yield the correct value of only if the potential energy functions making up A, and A12 are correctly stated there. For example, the question about the use of and 12 has already been mentioned. If, in addition, types of force other than dispersion, induction, and dipole-dipole orientation make significant contributions to the surface energy. Equation 7 will be in error and the results invalidated to the extent of the other contributions. (The Sinanoglu-Pitzer treatment of dispersion forces, which involves a third-order perturbation treatment of three interacting bodies, has not as yet been put in suitable form for application to complex molecules. Hence this effect was not included in the treatment above or in [18].)... 

In 1973 Kenneth Sanborn Pitzer (1914-1997) imdertook an attempt to take into accoimt these interactions in the solution s composition. He included binary interaction cation-anion, anion-anion, cation-cation, cation-neutral component, anion-neutral component, neutral component-neutral component and triple interaction cation-cation-anion, anion-anion-cation, etc., for which he expanded first member of equation (1.78) into a series of addends with virial coefficients (Pitzer, 1973). Each of these addends characterizes one type of interaction. His model of more detailed accounting of the interaction between components of water solution is sometimes called the Pitzer model. According to it, equation (1.78) acquired the format of a virial equation of the state of solution, or Pitzer equation with virial coefficients ... 

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... 

All coefficients iii the Pitzer equations (15.35)-(15.37) and (15.41) vary with ionic strength I. For electrolytes of valence 1-1 and 1-2 (e.g., NaCl and Na2S04) they are written in terms of two regression parameters specific to the electrolyte, and a parameter a which depends on the type of electrolyte (for 1-1, 1-2 and 2-1 salts, a = 2.0), and the ionic strength ... 

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