Solving for the equilibrium state

In Chapter 3, we developed equations that govern the equilibrium state of an aqueous fluid and coexisting minerals. The principal unknowns in these equations are the mass of water n w, the concentrations m,- of the basis species, and the mole numbers n/c of the minerals. 

Geochemists, however, seem to have reached a consensus (e.g., Karpov and Kaz min, 1972 Morel and Morgan, 1972 Crerar, 1975 Reed, 1982 Wolery, 1983) that Newton-Raphson iteration is the most powerful and reliable approach, especially in systems where mass is distributed over minerals as well as dissolved species. In this chapter, we consider the special difficulties posed by the nonlinear forms of the governing equations and discuss how the Newton-Raphson method can be used in geochemical modeling to solve the equations rapidly and reliably. 

The governing equations are composed of two parts mass balance equations that require mass to be conserved, and mass action equations that prescribe chemical equilibrium among species and minerals. Water Aw, a set of species, 4/, the min- 

The remaining aqueous species are related to the basis entries by the reaction 

The mass balance equations corresponding to the basis entries are 

If the governing equations were linear in these unknowns, we could solve them directly using linear algebra. However, some of the unknowns in these equations appear raised to exponents and multiplied by each other, so the equations are nonlinear. Chemists have devised a number of numerical methods to solve such equations (e.g.. van Zeggeren and Storey, 1970 Smith and Missen, 

All the techniques are iterative and, except for the simplest chemical systems, require a computer. The methods include optimization by steepest descent (White et al., 1958 Boynton, 1960) and gradient,descent (White, 1967), tback substitution (Kharaka and Barnes, 1973 Truesdell and Jones, 1974), and progressive narrowing of the range of the values allowed for each variable. (.the monotone sequence method Wolery and Walters, 1975). 

Geochemists, however, seem to have reached a consensus (e.g, Karpov and Kaz min. 1972 Morel and Morgan, 1972 Crerar, 1975 Reed, 1982 Wolery, 

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How can we express the equilibrium state of such a system A direct approach would be to write each reaction that could occur among the system s species, minerals, and gases. To solve for the equilibrium state, we would determine a set of concentrations that simultaneously satisfy the mass action equation corresponding to each possible reaction. The concentrations would also have to add up, together with the mole numbers of any minerals in the system, to give the system s bulk composition. In other words, the concentrations would also need to satisfy a set of mass balance equations. 

Second, at pressures below about 4.8 kPa there is no equilibrium liquid phase (i.e., / = 0) 4.8 kPa is the dew point pressure of this reacting mixture at 25 C. Consequently, to solve for the equilibrium state of this system at pressures lower than 4.8 kPa, only vapor-phase chemical equilibrium needs to be considered without any phase equilibrium constraints. In this case, the mass balance equations are... 

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