Virial coefficients Pitzer

Pitzer s Generalized Correlations In addition to the corresponding-states coorelation for the second virial coefficient, Pitzer and coworkers [Thermodynamics, 3d ed., App. 3, McGraw-Hill, New York (1995)] developed a full set of generalized correlations. They have as their basis an equation for the compressibility factor, as given by Eq. (2-63) ... 

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 ... 

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... 

To make the basic Pitzer equation more useful for data correlation of aqueous strong electrolyte systems, Pitzer modified it by defining a new set of more directly observable parameters representing certain combinations of the second and third virial coefficients. The modified Pitzer equation is... 

In this study the Pitzer equation is also used, but a different, more straightforward approach is adopted in which the drawbacks just discussed do not arise. First, terms are added to the basic virial form of the Pitzer equation to account for molecule-ion and molecule-molecule interactions. Then, following Pitzer, a set of new, more observable parameters are defined that are functions of the virial coefficients. Thus, the Pitzer equation is extended, rather than modified, to account for the presence of molecular solutes. The interpretation of the terms and parameters of the original Pitzer equation is unchanged. The resulting extended Pitzer equation is... 

X = second virial coefficient of the basic Pitzer equation... 

An important series of papers by Professor Pitzer and colleagues (26, 27, 28, 29), beginning in 1912, has laid the ground work for what appears to be the "most comprehensive and theoretically founded treatment to date. This treatment is based on the ion interaction model using the Debye-Huckel ion distribution and establishes the concept that the effect of short range forces, that is the second virial coefficient, should also depend on the ionic strength. Interaction parameters for a large number of electrolytes have been determined. 

The Pitzer virial coefficient method, see section 6.2.2. Methods 1 and 2 are equivalent and differ only in the form of the denominator in the Debye-Huckel term. Method 3 requires more parameters for the description of the activity factors. These parameters are not available in many cases. This is generally the case for complex formation reactions. 

One may sometimes have access to the parameters required for the Pitzer approaches, e.g., for some hydrolysis equilibria and for some solubility product data, cf. Baes and Mesmer [3] and Pitzer [4]. In this case, the reviewer should perform a calculation using both the B-G-S and the P-B equations and the full virial coefficient methods and compare the results. 

The following text is only intended to provide the reader with a brief outline of the Pitzer method. This approach consists of the development of an explicit function relating the ion interaction coelScient to the ionic strength and the addition of a third virial coefficient to Eq. (6.1). For the solution of a single electrolyte MX, the activity coefficient may be expressed by Eq. (6.29) [15] ... 

A related method is typified by Pitzer and Weltner s paper on methanol (2156). They used heat capacity expressions corresponding to the virial equation to calculate virial coefficients. They use the calculated fourth virial coefficient to argue that H bonded tetramers arc present. A corresponding interpretation of B is not given. 

Pitzer (1973) re-examined the statistical mechanics of aqueous electrolytes in water and derived a different but semi-empirical method for activity coefficients, commonly termed the Pitzer specific-ion-interaction model. He fitted a slightly different function for behavior at low concentrations and used a virial coefficient formulation for high concentrations. The results have proved extremely fruitful for modeling activity coefficients over a very large range of molality. The general equation is... 

Generalized correlations find widespread use. Most popular are correlations of die kind developed by Pitzer and cowoikers for the compressibility factor Z and for the second virial coefficient B. ... 

Pitzer, K.S. (1955a). The Volumetric and Thermodynamic Properties of Fluids. I. Theoretical Basis and Virial Coefficients. J.Am.Chem.Soc., 77, 3427-3433. 

Generalized correlations for the compressibility factor, Z, as well as analytical expressions, based on the second virial coefficients, have been developed by Pitzer et a/. The correlation for Z takes the form ... 

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 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. 

Pitzer s formulation offers a satisfactory and desirable way to model strong electrolyte activity coefficients in concentrated and complex mixtures. When sufficient experimental data are available, one can make calculations which are considerably more accurate than those presented in this paper. Attaining high accuracy requires not only experimentally-based parameters but also that one employ third virial coefficients and additional mixing terms and include explicit temperature dependencies for the various parameters. 

Ion Interaction. Ion-interaction theory has been the single most noteworthy modification to the computational scheme of chemical models over the past decade this option uses a virial coefficient expansion of the Debye-Huckel equation to compute activities of species in high ionic strength solutions. This phenomenological approach was initially presented by Pitzer ( ) followed by numerous papers with co-workers, and was developed primarily for laboratory systems it was first applied to natural systems by Harvie, Weare and co-workers (45-47). Several contributors to the symposium discussed the ion interaction approach, which is available in at least three of the more commonly used codes SOLMNEQ.88, PHRQPITZ, and EQ 3/6 (Figure 1). 

PHRQPITZ A program adapted from the computer code PHREEQE which makes geochemical calculations in brines and other electrolyte solutions at high concentrations, using the Pitzer virial coefficient approach (see paper, this volume). 

---