Finding the stable phase assemblage

The remaining step is to compute the system s bulk composition, if it is not fully known, according to the mass balance equations. The mole numbers Mw, Mi, and are not known when the modeler has constrained the corresponding variable nw, mt, or tig. In these cases, the mole numbers are determined directly from Equations 4.3 1.5. Where gases appear in the basis, the mole numbers Mm of gas components are similarly calculated from Equation 4.6. 

The calculation described to this point does not predict the assemblage of minerals that is stable in the current system. Instead, the assemblage is assumed implicitly by setting the basis B before the calculation begins. A solution to the governing equations constitutes the equilibrium state of the system if two conditions are met  

A calculation procedure could, in theory, predict at once the distribution of mass within a system and the equilibrium mineral assemblage. Brown and Skinner (1974) undertook such a calculation for petrologic systems. For an -component system, they calculated the shape of the free energy surface for each possible solid solution in a rock. They then raised an n -dimensional hyperplane upward, allowing it to rotate against the free energy surfaces. The hyperplane s resting position identified the stable minerals and their equilibrium compositions. Inevitably, the technique became known as the crane plane method. 

Such a method has seldom been used with systems containing an aqueous fluid, probably because the complexity of the solution s free energy surface and the wide range in aqueous solubilities of the elements complicate the numerics of the calculation (e.g., Harvie el al., 1987). Instead, most models employ a procedure of elimination. If the calculation described fails to predict a system at equilibrium, the mineral assemblage is changed to swap undersaturated minerals out of the basis or supersaturated minerals into it, following the steps in the previous chapter the calculation is then repeated. 

Minerals that have become undersaturated are revealed in the iteration results by negative mole numbers ng. A negative mass, of course, is not meaningful physically beyond demonstrating that the mineral was completely consumed, perhaps to form another mineral, in the approach to equilibrium. 

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