Pitzer expressions

The case of activity coefficients in solutions is easily but tediously implemented since well-constrained expressions exist, like those produced by the Debye-Hiickel theory for dilute solutions or the Pitzer expressions for concentrated solutions (brines). The interested reader may refer to Michard (1989) for a recent and still reasonably simple account. However simple to handle, activity coefficients introduce analytically cumbersome expressions incompatible with the size of a textbook. Real gas theory demands even more complicated developments. 

Since Gjn) = Wg] and the Gibbs excess free energy G was defined as a summation of terms, equation (7.44) is then related to the Pitzer expression, equation (4,57), seen earlier. The following definitions were made by equivalencing the terms of equation (7.44) to the interaction terms, X ... 

Hiickel coefficient, Zj is the absolute value of the ionic charge, p is the parameter of the Pitzer expression and Ix is the ionic strength, mole fraction base. 

Values expressed in this way, taken from a long list published by Latimer, Pitzer, and Smith, arc given in Table 25. It will be seen that... 

Table A4.7 summarizes the thermodynamics properties of monatomic solids as calculated by the Debye model. The values are expressed in terms of d/T, where d is the Debye temperature. See Section 10.8 for details of the calculations. Tables A4.5 to A4.7 are adapted from K. S. Pitzer, Thermodynamics, McGraw-Hill, New York, 1995.

In applying this equation to multi-solute systems, the ionic concentrations are of sufficient magnitude that molecule-ion and ion-ion interactions must be considered. Edwards et al. (6) used a method proposed by Bromley (J7) for the estimation of the B parameters. The model was found to be useful for the calculation of multi-solute equilibria in the NH3+H5S+H2O and NH3+CO2+H2O systems. However, because of the assumptions regarding the activity of the water and the use of only two-body interaction parameters, the model is suitable only up to molecular concentrations of about 2 molal. As well the temperature was restricted to the range 0° to 100 oc because of the equations used for the Henry1s constants and the dissociation constants. In a later study, Edwards et al. (8) extended the correlation to higher concentrations (up to 10 - 20 molal) and higher temperatures (0° to 170 °C). In this work the activity coefficients of the electrolytes were calculated from an expression due to Pitzer (9) ... 

About the same time Beutier and Renon (11) also proposed a similar model for the representation of the equilibria in aqueous solutions of weak electrolytes. The vapor was assumed to be an ideal gas and < >a was set equal to unity. Pitzer s method was used for the estimation of the activity coefficients, but, in contrast to Edwards et al. (j)), two ternary parameters in the activity coefficient expression were employed. These were obtained from data on the two-solute systems It was found that the equilibria in the systems NH3+ H2S+H20, NH3+C02+H20 and NH3+S02+H20 could be represented very well up to high concentrations of the ionic species. However, the model was unreliable at high concentrations of undissociated ammonia. Edwards et al. (1 2) have recently proposed a new expression for the representation of the activity coefficients in the NH3+H20 system, over the complete concentration range from pure water to pure NH3. it appears that this area will assume increasing importance and that one must be able to represent activity coefficients in the region of high concentrations of molecular species as well as in dilute solutions. Cruz and Renon (13) have proposed an expression which combines the equations for electrolytes with the non-random two-liquid (NRTL) model for non-electrolytes in order to represent the complete composition range. In a later publication, Cruz and Renon (J4J, this model was applied to the acetic acid-water system. 

The function f(I) expresses the effect of long-range electrostatic forces between ions. It is a function of ionic strength, temperature and solvent properties. The empirical form chosen by Pitzer for f(I) is... 

Recognizing the ionic strength dependence of the effect of short range forces in binary interactions, Pitzer was able to develop an empirical relation for B,-a(I). The expression for systems containing strong electrolytes with one or both ions univalent is... 

A is a Debye-Hiickel parameter (cf. Appendix II) and I is the ionic strength. Pitzer found that binary interaction parameter Xdepends on ionic strength and may conveniently be expressed as ... 

The mean activity coefficient is the standard form of expressing electrolyte data either in compilations of evaluated experimental data such as Hamer and Wu (2) or in predictions based on extensions to the Debye-Huckel model of electrolyte behavior. Recently several advances in the prediction and correlation of mean activity coefficients have been presented in a series of papers starting in 1972 by Pitzer (3, Meissner 04), and Bromley (5) among others. 

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] ... 

Baes and Mesmer [3] use the function F I ) proposed by Pitzer to express the ionic strength dependence of the ion interaction coefficient fimx in Guggenheim s equations. For a single electrolyte... 

One method takes into account the individual characteristics of the ionic media by using a medium-dependent expression for the activity coefficients of the species involved in the equilibrium reactions. The medium dependence is described by virial or ion interaction coefficients as used in the Pitzer equations and in the specific ion interaction model. 

The rotational contribution is expressed in Pitzer s classical form 6 = torsion angle see footnote on page 43.) ... 

We have applied Pitzer s equations at T = 298.15 K, but they are not limited to that temperature and can be applied at any temperature where the coefficients are known.k Table I8.l (and Table A7.1 of Appendix 7) gives the Debye-Hiickel coefficients AA, Ah, and Aj as a function of temperature, but the coefficients specific to the electrolyte are tabulated in Appendix 7 only at T = 298.15 K. The usual solution to this problem is to express the coefficients as... 

Starting with Pitzer s equations, internally consistent generalized equations can be written to express the thermodynamic properties of aqueous electrolytes as a function of pressure, as well as temperature and molality. For example, Archer10 gives the equations for calculating 7 , , L, 4>CP, V, K and E for aqueous NaCl solutions1 as a function of p, T, and m over the temperature range from... 

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. 

The expression for the excess Gibbs energy is built up from the usual NRTL equation normalized by infinite dilution activity coefficients, the Pitzer-Debye-Hiickel expression and the Born equation. The first expression is used to represent the local interactions, whereas the second describes the contribution of the long-range ion-ion interactions. The Bom equation accounts for the Gibbs energy of the transfer of ionic species from the infinite dilution state in a mixed-solvent to a similar state in the aqueous phase [38, 39], In order to become applicable to reactive absorption, the Electrolyte NRTL model must be extended to multicomponent systems. The model parameters include pure component dielectric constants of non-aqueous solvents, Born radii of ionic species and NRTL interaction parameters (molecule-molecule, molecule-electrolyte and electrolyte-electrolyte pairs). 

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

Thus, Pitzer and coworkers proposed a second correlation, which expresses the quantity BPJ RTC as... 

The Pitzer model can be used to obtain activity coefficients for solutes in low (<0.1 mol L ), intermediate (0.1-3.5 mol L ) and high (>3.5 mol L ) ionic strength solutions. The Pitzer equations include terms for binary and ternary interactions between solute species as well as a modified DH expression. The general formula is... 

Although naturally occurring brines and some high ionic strength contaminated waters may require the more complicated expressions developed in the Davies, SIT, or Pitzer models, the use of Equations (3.3)-(3.5) is justified for the ionic strengths of many freshwaters. 

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