HKF model

Actually, the various equations listed in this section are insufficient to perform the complete calculation since one would first calculate the density of H2O through eq. 8.12 or 8.14. Equation 8.14 in its turn involves the partial derivative of the Helmholtz free energy function 8.15. Moreover, the evaluation of electrostatic properties of the solvent and of the Bom functions (o, Q, Y, X involve additional equations and variables not given here for the sake of brevity (eqs. 36, 40 to 44, 49 to 52 and tables 1 to 3 in Johnson et ah, 1991). In spite of this fact, the decision to outline here briefly the HKF model rests on its paramount importance in geochemistry. Moreover, most of the listed thermodynamic parameters have an intrinsic validity that transcends the model itself... 

HKF model (see section 8.11). The main observations on which the generalizations were based are as follows. 

The Born coefficients of the HKF model for the various organic species are simply related to their standard partial molal entropies at 25 °C and 1 bar through the effective charge (Shock et al., 1989). 

Figure 8.27 Experimental values of standard partial molal heat capacity of sodium acetate (A) and ethylene (B) in water as a function of P(°C) at pressures corresponding to water-vapor equilibrium. Interpolating curves generated by the HKF model equations. Reprinted from E. L. Shock and H. C. Helgeson, Geochimica et Cosmochimica Acta, 54, 915-946, copyright 1990, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

Table 8.22 Partial molal thermodynamic properties of organic aqueous species at 25 HKF model, after Shock and Helgeson (1990). 

Discrete values of the HKF model parameters for various organic aqueous species are listed in table 8.22. Table 8.23 lists standard partial molal thermodynamic properties and HKF model parameters for aqueous metal complexes of monovalent organic acid ligands, after Shock and Koretsky (1995). 

Table 9.14 Standard partial molal properties and coefficients of HKF model for neutral molecules in aqueous solution. and in cal/mole Sp p in cal/(mole X K)) Vp p in cm /mole... 

In table 9.14, for the sake of completion, we list the thermodynamic parameters of the HKF model concerning neutral molecules in solution (Shock et al., 1989). Calculation of partial molal properties of solutes (see section 8.11), combined with calculation of thermodynamic properties in gaseous phases (Table 9.5), allows rigorous estimates of the various equilibrium constants at all P and T of interest. 

For solutes no general expression exists to substitute for the volume integral. To integrate partial molar volumes as a function of pressure one must have an analytical expression for the effect of P on V, either derived empirically or from some model (see HKF model. Chapter 17). We are left with the general expression... 

Note too that our use of um in the top equation assumes complete ionization. Helge-son et al. (1981) use m, which allows for incomplete ionization (see discussion of the HKF model later in this chapter). 

HKF model, after the three authors of Part IV of the series of papers just mentioned, Helgeson, Kirkham, and Howers. 

The HKF model is semi-empirical, in the sense that it uses a number of empirical parameters within a framework suggested by fundamental physics and thermodynamics. The variation of the Gibbs free energy of individual ions with T, P, and composition can be represented by writing the total differential of the (partial molar) free energy of the jth ion, giving... 

A number of theoretical difficulties in equating the Bom function with this process are believed to be accommodated in the parameter, which in the Bom model is the ion radius, but in the HKF model is an adjustable parameter called the effective ionic radius. The Vej parameters were originally related to crystallographic ionic radii... 

The (conventional) entropy and volume changes of solvation according to the Born model are then derived by differentiation, which in the original HKF model was a simple procedure because Vej was independent of T, P, which meant that Wj was also independent of T, P. Thus in the original model. 

The re,j term in the HKF model essentially takes the place of the term in the D-H model, and because it is different for each ion, the overall model suffers from the problem mentioned in 17.4, i.e., that dGsolution is not an exact differential. However, the authors consider that the error introduced is acceptable in view of other sources of uncertainty. 

In the original HKF model, 0 was a fit parameter for each ion having values usual ly ranging from about 200 to 260 K. Recent studies of supercooled water reviewed by Angell (1982, 1983 references in Tanger and Helgeson, 1988) however show that 228 3 K is a singular temperature at which several properties approach oo, and in the revised model 0 takes on the fixed value of 228 K. The vF parameter is also fixed at 2600 bars. 

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