Pitzers Generalized Correlations

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)  

Generalized correlations are developed here for the residual enthalpy and residual entropy from Eqs. (4-48) and (4-49). Substitution for Z by Eq. (2-63) puts Eq. (4-48) into generalized form  

If the first terms on the right sides of Eq. (4-117) and of this equation (including the minus signs) are represented by H y /RT, and (S ) /R and if the second terms, excluding co but including the minus signs, are represented by (H Y/RT, and (S )VP, then 

Pitzer s original correlations for Z and the derived qnantities were determined graphically and presented in tabnlar form. Since then, analytical refinements to the tanles have been developed, with extended range and accnracy. The most popnlar Pitzer-type correlation is that of Lee and Kesler [AIChE J. 21 510-527 (1975) see also Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 5th, 6th, and 7th eds., App. E, McGraw-Hill, New York (1996, 2001, 2005)]. These tables cover both the liqnid and gas phases and span the ranges 0.3 T, 4.0 and 0.01 Pr 10.0. They list values of Z , Z , (H /RT, (H y/RT, (S ) /R, and (S Y/R. 

Lee and Kesler also included a Pitzer-type correlation for vapor pressures  

---

The generalized correlations of Pitzer provide an alternative to the use of a cubic equation of state for the calculation of thermodynamic properties. However, no adequate general method is yet known for the extension of the Pitzer correlations based on the compressibility factor to mixtures. Nevertheless, Z, as given by... 

Phase rule, 37-39 362-363, 529-532 Pitzer correlations (see Generalized correlations)... 

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. ... 

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 ... 

The virial equation is truncated in application. Figure 4.14 shows the comparison of B-virial equation and B h- C virial equation with generalized correlation of Pitzer [1] at T, = 0.8, 1.0, and... 

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 liquid-phase fugacity coefficient = /f/P may be calculated from a generalized correlation in terms of reduced temperature and pressure such as those of Lydersen et al.42 and Curl and Pitzer.15 Chao and Seader used a modified form of the Curl and Pitzer correlation. The correlation was modified by use of experimental data such that appropriate values of could be computed for the case where a component does not exist as a liquid and for the case of low temperatures. The following expression was proposed for the calculation of the fugacity coefficient for any component / in the liquid phase... 

Chao and Sender developed an empirical expression for vfi in terms of T, Pr, and <0 using the generalized correlation of Pitzer et al., which is based on the equation of state given as (4-33). For hypothetical liquid conditions (P < PJ or T > Tc), the correlation was extended by back calculating r°L from vapor-liquid equilibrium data. The C-S equation for is... 

The following generalized correlations developed by Lee-Kesler and Pitzer are based on the modified form of the Benedict-Webb-Rubin equation of state. The correlations provide reliable results for nonpolar or slightly polar gases. 

In the same period . Prausnitz (1969) C. Tsonopoulos(1974, 1975) and J. O Connell (1975), following the pioneering work of K. Pitzer in the middle 50 s, presented generalized correlations for the second virial coefficient. The theoretically based virial equation introduced by K. Onnes in 1901, could be now used in the prediction of the volumetric behavior of nonpolar and often polar gases and vapors, but only at low pres-... 

Can the species activity coefficients be calculated accurately An activity coefficient relates each dissolved species concentration to its activity. Most commonly, a modeler uses an extended form of the Debye-Hiickel equation to estimate values for the coefficients. Helgeson (1969) correlated the activity coefficients to this equation for dominantly NaCl solutions having concentrations up to 3 molal. The resulting equations are probably reliable for electrolyte solutions of general composition (i.e., those dominated by salts other than NaCl) where ionic strength is less than about 1 molal (Wolery, 1983 see Chapter 8). Calculated activity coefficients are less reliable in more concentrated solutions. As an alternative to the Debye-Hiickel method, the modeler can use virial equations (the Pitzer equations ) designed to predict activity coefficients for electrolyte brines. These equations have their own limitations, however, as discussed in Chapter 8. 

Generally, agreement has been found between our correlations and those of Pitzer, and others (1972, 1973, 1974, 1975, 1976) and Rard, and others (1976, 1977). Many of our correlations agree fairly well with Robinson and Stokes, (1965) and Harned and Owen, (1958) but in most cases a much larger data base and more recent measurements have been incorporated into the evaluations. It has been observed that agreement with Pitzer s equations is found below moderate concentrations (several molal), but often deviate at higher concentrations where the Pitzer equations do not contain enough parameters to account for the behavior of the activity (or osmotic) coefficient. 

The truncated virial equation (331), with a value of B from the generalized Pitzer correlation,... 

Alternatively, fluid characteristics other than Zc can be used as the additional parameter in the generalization of the simple corresponding-states principle. In fact, since for many substances the critical density, and hence Zc, is known with limited accuracy, if at all, there is some advantage in avoiding the use of Zc-Pitzer has suggested that for nonspherical molecules the acentric factor co be used as the third correlative parameter, where co is defined to be... 

These forms of a generalized equation of state only require the critical temperature and the critical pressure as substance-specific parameters. Therefore, these correlations are an example for the so-called tsvo-parameter corresponding-states principle, which means that the compressibility factor and thus the related thermodynamic properties for all substances should be equal at the same values of their reduced properties. As an example, the reduced vapor pressure as a function of the reduced temperature should have the same value for all substances, provided that the regarded equation of state can reproduce the PvT behavior of the substance on the basis of the critical data. In reality, the two-parameter corresponding-states principle is only well-suited to reflect the properties of simple, almost spherical, nonpolar molecules (noble gases as Ar, Kr, Xe). For all other molecules, the correlations based on the two-parameter corresponding-states principle reveal considerable deviations. To overcome these limitations, a third parameter was introduced, which is characteristic for a particular substance. The most popular third parameter is the so-called acentric factor, which was introduced by Pitzer ... 

Electron Correlation Effects. -PITZER has suggested that London, or van der Waals, dispersion forces between atoms or groups in the same molecule may lead to an appreciable stabilization of the molecule. Such London forces are large when both groups are highly polarizable. It seems plausible to generalize and state that additional stability due to London forces will always exist in a complex formed between a polarizable acid and a polarizable base. In this way the affinity of soft acids for soft bases can be accounted for. 

---