Helgeson and Kirkham

Based on the analytical data of K-mica, epidote and K-feldspar and using thermochemical data on these minerals (Helgeson and Kirkham, 1974 Helgeson et al., 1978 Bird and Helgeson, 1981), the /coz range for the propylitic alteration was estimated (Fig. 1.78). 

Helgeson (1969 see also Helgeson and Kirkham, 1974) presented an activity model based on an equation similar in form to the Davies equation. The model, adapted from earlier work (see Pitzer and Brewer, 1961, p. 326, p. 578, and Appendix 4, and references therein), is parameterized from 0°C to 300 °C for solutions of up to 3 molal ionic strength in which NaCl is the dominant solute. The model takes it name from the B-dot equation,... 

Fig. 8.2. Values of A, B, and B for the B-dot (modified Debye-Huckel) equation at 0 °C, 25 °C, 60 °C, 100 °C, 150 °C, 200 °C, 250 °C, and 300 °C (squares) and interpolation functions (lines). Values correspond to I taken in molal and a in A. Data from the LLNL database, after Helgeson (1969) and Helgeson and Kirkham (1974).

The work described by Marshall (16), together with the vapour pressure studies on 1 1 and 1 2 electrolytes up to 300 C reported by Lindsay and Liu (17) and recent theoretical work by Silvester and Pitzer (21) and by Helgeson and Kirkham (22) provide a good understanding of the behaviour of simple electrolytes over wide ranges of temperature and concentration. However, as just seen, the behaviour under SVP conditions above 300 C becomes decreasingly well defined towards the critical point. 

Helgeson and Kirkham, 1974). In equations 8.32.1 and 8.32.2, p is solvent density, e is the dielectric constant of the solvent, and a, is the effective diameter of ion i in solution. The latter parameter must not be confused with the ionic diameter of the ion in a condensed aggregation state, but represents the range of electrostatic interaction with the solvent molecules (see section 8.11.3). 

Table 8.4 Debye-Huckel constants for H2O in T range 0 to 300 °C at saturation P (from Helgeson and Kirkham, 1974). A is in kg X mole and B is in kg X mole X cm. ...

The next better approximation is to allow the linear coefficient to be a function of the ionic strength (Pitzer and Brewer, 1961) and was utilized by Helgeson and Kirkham (1974a,b, 1976), Helgeson et al. (1981), and Tanger and Helgeson (1988) in the development of the Helgeson-Kirkham-Flowers (HKF) equations, which include consideration of the Born function and ion hydration. 

An extensive dissertation on the prediction of thermodynamic properties for aqueous electrolytes can be found in the work of Helgeson and Kirkham (18). 

There have been four recent evaluations of the dielectric constant for water. The earliest is that of Helgeson and Kirkham... 

Fig. 8.3. Entropy of water as a function of temperature and pressure. Contours are labelled in calK moP. The slope is always positive, showing that isentropic expansion will always cool the fluid. [After Helgeson and Kirkham (1974a)].

When data at high temperatures and pressures began to be available, it was realized that the Born model was also capable of accounting for the large negative values of various partial molar properties of electrolytes at high temperatures (such as the partial molar volume, 10.2.4), and Helgeson and Kirkham (1976) used it in combination with other terms in their equation of state for aqueous species (Chapter 15). 

Other approaches can be used based on corrections to this equation (e.g., Helgeson and Kirkham, 1976), but in recent years the tendency has been to use the Pitzer equations (Chapter 15). Determining the intercept of this equation, or any nonlinear equation, at m = 0 places great emphasis on measurements of very dilute solutions, where they are most difficult. Clearly, some theoretical knowledge of what the slope at the intercept (the limiting slope ) should be is important, and all modern treatments of data of this type use the... 

Helgeson and Kirkham (1974a) reworked all the data available at that hme, and include extensive references to earlier work. They used the data of Burnham et al. as well as other data, and produced equations for many thermodynamic properties not considered by Burnham et al including the dielectric constant. They covered the region up to 900 °C and 10 kbar. 

Table 13.1 Some critical and triple point parameters for water, many from Helgeson and Kirkham (1974a).

Because it applies mostly to electrolytes, it is discussed in Chapter 15. Briefly, Helgeson models the behavior of solutes by developing equations for the standard state partial molar volume (Helgeson and Kirkham 1976) and standard state partial molar heat capacity (Helgeson et al. 1981) as a function of P and T, with adjustable constants such that they can be applied to a wide variety of solutes. If you know these quantities (V°, C°p), you can calculate the variation of the standard state Gibbs energy, and that leads through fundamental relationships to equilibrium constants, enthalpies, and entropies. 

The parameters po Cq refer to the density and dielectric constant of pure water at the T and P of interest, and z are the valences of the cation and anion constituents of the salt, and a is the D-H distance of closest approach in units of angstroms. The product kB usually approximates unity. The D-H A and B parameters, calculated over a wide range of P and T, are tabulated by Helgeson and Kirkham (1974b, pp. 1202 and 1256). I is the molal ionic strength, defined by the following sum over aU anions and cations,... 

Helgeson and co-workers (Helgeson and Kirkham, 1974, 1976 Helgeson et al, 1981) developed an equation of state for aqueous electrolytes based on this continuum model. The model, known as HKF, has two contributions to the standard partial volume an electrostatic part given by Equation (2.78) and the nonelectrostatic part having an intrinsic term, temperature and pressure independent, and a short-range term related to the electrostriction of water around the ion, equivalent to a change of density and dielectric constant of the continuum near the ion. This last contribution was considered to be dependent on temperature and pressure. 

Waldbaum (1968). It is, however, quite easily calculated, for example by first calculating the entropy of reaction at 1 atm. and T from tables and adding to this the slope of (Gp - atm) at T from Burnham et al. (1969) or Helgeson and Kirkham (1974, table 29), where T here refers to the reaction equilibrium temperature at the chosen pressure. Alternatively the complete free energy expression can be evaluated at two temperatures near the equilibrium temperature, preferably above and below it, and the slope calculated from these numbers. The standard error of the... 

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