Pitzers Method

In 1973, Pitzer el al. began presenting a series of papers (PI - P12) reporting their development of a system of equations for electrolyte thermodynamic properties. In expanding the Debye-Htickel method, terms were added to account for the ionic strength dependence of the short-range forces effect in binary interactions. 

While the Debye-Huckel derivation was based on a charging process which took into account the distance of closest approach, a, the kinetic effect of the hard core could not adequately be dealt with. In order to include these effects. Pitzer proposed using the osmotic pressure equation (PI)  

The charge distribution described by Debye-Huckel is found by expanding the exponential term as before. 

Pitzer then used the same kind of expansion for the gij(a) term for the hard core effect of equation (4.53). Here the second term of the expansion dropped out, but the third term was kept. He then found the osmotic coefficient to be  

The first term of equation (4.56) describes the electrostatic energy, and the second the hard core effects. This hard core effect term shows dependence on the ionic strength of the solution due to the K. This indicates the need for a series of virial coefficients that are functions of the ionic strength, and justifies Bromley and Guggenheim s additions to the Debye-Hiickel equation. 

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FIGt 2-35 Generalized compressibility factors—Pitzer Method, simple fluid term. 

Lee and Kesler fit the entire Pitzer method to equations, rewriting the basic Eq. (2-75) with respect to a heavy reference fluid (n-octane) as shown by Eq. (2-78). 

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] ... 

The difference between the extended Debye-Hiickel equation and the Pitzer equations has to do with how much of the nonideahty of electrostatic interactions is incorporated into mass action expressions and how much into the activity coefficient expression. It is important to remember that the expression for activity coefficients is inexorably bound up with equilibrium constants and they must be consistent with each other in a chemical model. Ion-parr interactions can be quantified in two ways, explicitly through stability constants (lA method) or implicitly through empirical fits with activity coefficient parameters (Pitzer method). Both approaches can be successful with enough effort to achieve consistency. At the present, the Pitzer method works much better for brines, and the lA method works better for... 

It would be difficult to find more comprehensive or more detailed studies on the physical chemistry of seawater than those done at the University of Miami (Millero, 2001). Several programs were developed for calculation of activity coefficients and speciation of both major ions and trace elements in seawater. The activity coefficient models have been influenced strongly by the Pitzer method but are best described as hybrid because of the need to use ion-pair formation constants (Millero and Schreiber, 1982). The current model is based on Quick Basic computes activity coefficients for 12 major cations and anions, 7 neutral solutes, and more than 36 minor or trace ions. At 25 °C the ionic strength range is 0-6 m. For major components, the temperature range has been extended to 0-50 °C, and in many cases the temperature dependence is reasonably estimated to 75 °C. Details of the model and the parameters and their sources can be found in Millero and Roy (1997) and Millero and Pierrot (1998). Comparison of some individual-ion activity coefficients and some speciation for seawater computed with the Miami model is shown in Section 5.02.8.6 on model reliability. 

With the arrival of the Pitzer method for calculating activity coefficients at high ionic strengths (si m), research by Harvie and Weare (1980) led to computations of equilibrium mineral solubilities for brines. They could calculate... 

One new feature added to SOLMINEQ.88 is the Pitzer method of calculating the activity coefficients of aqueous species as discussed by Kharaka et al. (1987), and detailed in Kharaka et al. 

In the Pitzer method, the compressibility factor is expressed in the form ... 

A useful empirical correlation for the second virial coefficient is based on the Pitzer method. In this method, the dimensionless ratio, BPc/RT, is expressed in the form... 

Figure 2-ll Calculated isotherms of ethylene at 40 °C using the ideal-gas law, the truncated virial equation, and the Pitzer method with the Lee-Kesler values of Z< ) and Z<>) (see Example 2.Q ). 

In the previous section we discussed the calculation of residual properties from cubic equations of state. The calculations are straightforward, though somewhat time consuming. A quicker alternative is to use generalized graphs. In Chapter 2 we discussed the Pitzer method for calculating the compressibility factor in terms of reduced temperature, reduced pressure, and acentric factor. Analogous equations can be obtained for the residual enthalpy and entropy. In this approach, the residual enthalpy, made dimensionless by the product RTc, is computed as... 

Fig. 2. A comparison of dJoiine 3s pseudo-orbitals generated with the Christiansen-Lee-Pitzer method and the Phillips-Kleinman pseudo-orbital and the all-electron Hartree-Fodc orbital.

With th results from equstion (4.104). Mesmer et al. used a least-squares equation fitting program to regress the and of the Pitzer method... 

The activity coefficients of KDP ions and urea molecule in KDP-water-urea system were modelled using the Pitzer method. The equations for activity coefficients yof KDP ions and urea molecule in the ternary KDP - urea - water system can be described as ... 

Located within the Project Configuration tab are two options for sea water osmotic pressure calculation methods. The designer can select either the TEOS 10 method or the more accurate Pitzer method. 

The two regions of the orbitals are matched at a matching radius, so that the derivatives are continuous at this radius. The shape-consistent pseudo-orbital is defined in the Christiansen-Lee-Pitzer method as... 

Figure 2 compares the pseudo-orbitals obtained from the shape-consistent Christiansen-Lee-Pitzer method with the Phillips-Kleinman, and the all-electron techniques for the chlorine 3s orbital. As seen from Figure 2, the maximum of the pseudo orbital obtained from the Phillips-Kleinman method, occurs at a shorter distance, compared to both the Christiansen-Lee-Pitzer orbital and the all-electron 3s Hartree-Fock orbital of the chlorine atom, which is the origin of under-estimation of the repulsive wall of the potential surface in the Phillips-Kleinman method. Furthermore, the shorter maximum in the Phillips-Kleinman method leads to shorter equilibrium bond distances using the Phillips-Kleinman potentials. 

Ermler, Christiansen, and co-workers have generated analytical Gaussian forms for both the averaged RECPs and spin-orbit operators, derived using the Christiansen-Lee-Pitzer method. The ECPs were fitted into Gaussians as in equation (39) ... 

Estimate the fugacity coefficient of n-butane at 340 K and 0.7 MPa using the virial, SRK, and Pitzer methods and conq>are them with the recommended value of 0.862. 

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