Solid-solution Model

An equation of state can be used to calculate. (P, T). The expression for the PR-EOS for the calculation of (P, T) is provided by Eq. (3.22) of Chapter 3. Then with A/i, T[, and Ac/jp one can calculate fpure i( P)-order to proceed with the wax-precipitation calculations, one needs to dssume a proper solid model. Currently, there are two types of solid models. One is the solid-solution model, and the other is the multisolid-phase model. These models are presented below. 

Solid solution model. If one assumes that the precipitation forms a solid solution, then 

The fugacities in the vapor and liquid phases may be obtained from an EOS. Alternatively, an activity-coefficient model can be used to describe the liquid phase to estimate f. However, as discussed in Chapter 1, activity coefficient models, in general, may not be suitable for reservoir fluids because they are based on the assumption of no change in volume due to mixing. At equilibrium, when liquid and solid phases are present. 

With an activity coefficient model for the liquid phase, 

we provide both the equilibrium and the material balance equations for wax precipitation calculations for solid-liquid equilibria. At fixed temperature and pressure, for every component i, the multisolid-phase model must satisfy 

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Walshe, J.L. (1986) A six-component chlorite solid solution model and the conditions of chlorite formation in hydrothermal and geothermal systems. Econ. Geol, 81, 681-703. 

Many minerals are solid solutions (e.g., clays, amphiboles, and plagioclase feldspars). Solid-solution models are either not available or appropriate algorithms have not been incorporated into computer codes. 

Bourcier, W. L., 1985, Improvements to the solid solution modeling capabilities of the EQ3/6 geochemical code. Lawrence Livermore National Laboratory Report UCID-205 87. 

In the ideal solid solution model used, the enthalpy and entropy of oxidation are independent of composition. 

Solid-Solution Models. Compared with the liquid phase, very few direct experimental determinations of the thermochemical properties of compound-semiconductor solid solutions have been reported. Rather, procedures for calculating phase diagrams have relied on two methods for estimating solid-solution model parameters. The first method uses semiem-pirical relationships to describe the enthalpy of mixing on the basis of the known physical properties of the binary compounds (202,203). This approach does not provide an estimate for the excess entropy of mixing and thus... 

Ballard, A., A Non-Ideal Hydrate Solid Solution Model for a Multi-Phase Equilibria Program, Ph.D. Thesis, Colorado School of Mines, Golden, CO (2002). 

Equation 5.23 is considered to be an ideal solid solution model. If we choose to extend our equations from one hydrate crystal to a large number Na (Avogadro s number) of crystals, we must replace the Boltzmann constant k with the universal gas constant R (=kN ). Ballard (2002) defined the chemical potential of water in hydrates as... 

Figure 2 Condensation of major rock-forming phases from a gas of solar composition (Anders and Grevesse, 1989) at a total pressure of 10 atm. This calculation was done with the best currently available internally consistent thermodynamic data for solid and gaseous phases and includes nonideal solid solution models for melilite, Ca-rich pyroxene, feldspar, and metal. This calculation is the same as the one shown in Yoneda and Grossman (1995, table 1...

Trolard, F. and Y. Tardy (1989). An ideal solid solution model for calculating solubility of clay minerals. Clay Minerals 24, 1-21. 

The solid solution model implemented by Farley et al. (1985) allows for surface complexation at low surface coverages and co-precipitation of the metal hydroxide phase as a solid solution containing sorbing and sorbent ions. In an ideal solid solution, the solid phase activities are given by ... 

Fig. 7-9. TLM solid solution model calibration and predictions for Co(ll) sorption to a-AUO, at low and high surface coverage (A) model calibration to high coverage data, t/f) Surface complexatiou and solid solution model predictions of representative pi I sorption edges. (O surface complexatiou and solid solution model predictions of a sorption isotherms at pi I / t>. and ) surface couiplexa-lion and solid solution. a sorption isotherm at pi 11> (alter Kal/ Hayes, 19 ISh).

J. Sci.. in press). Indeed, in the case of a solid-solution with a small difference in the size of the substituting ions (relative to the size of the non-substituting ion), the first parameter, ao, is usually sufficient (8). Equations 5 and 6 then become identical to those of the "regular" solid-solution model of Hildebrand (9). For the case where both ao and ai parameters are needed, equations 5 and 6 become equivalent to those of the "subregular" solid-solution model of Thompson and Waldbaum (10). a model much used in high-temperature work. Equations 5 and 6 can also be shown equivalent to Margules activity coefficient series (11). 

Kgg values are evaluated from Thorstenson and Plummer s (3) equation 22, modified assuming a regular solid-solution model ... 

Despite the above problems, mixing parameters estimated from miscibility gap information will still be an improvement over the assumption of an ideal solid-solution model, ao parameters estimated from data in Palache et al. (20) and Busenberg and Plummer (21) are presented in table I for a few low-temperature mineral groups. Because of the large uncertainties in the data and in the estimation procedure, a sub-regular model is usually unwarranted. As a result, these estimated ao values presented should be used only for solid-solution compositions on a single side of the miscibility gap, i.e. only up to the given miscibility fraction. 

Solid Solutions. EQ3/6 includes provision for dealing with solid solutions, both as "reactant" and "product" phases. Both ideal and non-ideal solid solution models have been incorporated. The compositions of "product" solid solutions continually readjust to remain in equilibrium with the changing fluid composition. In the "flow-through" open system mode, "product" solid solutions are removed from the reacting system as they form, resulting in zoning... 

Application of EQ3/6 to many important problems will depend critically on the ability to model compositional variation in clays and zeolites. Thermodynamic data for 2 1 clays, and to a lesser extent zeolites, are not sufficiently abundant or of high enough quality to construct definitive solid solution models for these phases. Nevertheless, incorporation of reasonable models into EQ3/6 is a prerequisite to assessing the sensitivity of the geochemical modeling results to different solid solution approaches and for testing predictions against experimental and field observations. 

In the 3245 version of EQ3/6, the solid solution models are restricted to the molecular-mixing type. Included for example is a 12 component model for dioctahedral smectite, partly based on thermodynamic estimation techniques (46,5J)- Models are also included for some zeolites, based on similar methods (54). 

This is referred to as the quasi-chemical model. More detailed solid solution models can be found in textbooks on metallurgical thermodynamics, for example, Chemical Thennodynamics of Materials, by C. H. P. Lupis (Elsevier Science Publishers, Amsterdam, 1983). 

Where both cobalt and manganese are present in solution, the coprecipitation of hausmannite and C03O4 might be expected. Sinha et al. (25) found that random substitution of cobalt for Mn " or Mn could occur in such material occurrence of Co in manganese oxide crystal-lattice positions was noted by Bums (4). Apparently there are no thermodynamic data for mixed cobalt + manganese oxides, but the behavior of the ions can probably be represented over a considerable range of solid composition by a solid—solution model based on the equilibrium between the pure end members. Thus... 

It is observed frommFigure 7 that, first,Pthe value of r icu is not unity and, second, that there exists a partial cancellation of the composition dependence in the liquid phase activity coefficient product by that found in the solid solution. The second observation suggests that the liquid and solid solution model selection process should be insensitive with respect to liquidus and solidus data alone. Indeed, the assumption of ideal solution behavior in both phases closely predicts the correct distribution coefficient, yet experimental measurements of the solution thermochemical properties clearly indicate moderate negative deviations from ideal behavior. 

Assuming a regular solid solution model (Prieto 2009 Ruiz-Hernandez et al. 2010), the enthalpy of mixing at low zirconium content was fitted with a polynomial of the form ... 

We conclude this chapter by discussion of the superceU use for the solid-solution modeling. In this case the supercells of different size allow different percentages of doping in solid solution to be modeled. 

Hillert, M. (1961), A solid-solution model for inhomogeneous systems, Acta Metallurgica 9, 525-535. 

Fig. 5. The ideal and regular solid solution models that predict surface segregations of the constituents with lower surface free energy...

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