Summary of the Numerical Model

In this section we briefly outline the numerical method used in a PATH program to compute reaction paths for mineral-water systems. For complete details we recommend the articles by Helgeson (1968) and (1979), and the notation in these articles is used below. 

The quantities n above are the reaction coefficients of each species s. For convenience, we will require that the system contain 1000 g water, and assign flKAisijOs = -1- Each rxs then represents the change in the number of moles of the subscripted species s with a reaction step d  

Reaction (19.91), to be completely rigorous, should include all species of significance, but some are ignored here for brevity. We might even have written some species that are really products as reactants, but this will ultimately become evident from the signs of each fig (negative for reactants, positive for products). The exact values of the 10 reaction coefficients fis in (19.92) depend on the extent of the reaction, and are therefore unknown at this stage. To solve for each individual we will require 10 equations in the 10 unknowns fig. 

We can begin assembling these equations by writing equilibrium constants for 5 independent reactions between the species in (19.91). 

The next step is to take the derivative of each of the above 5 equations with respect to the progress variable and rearrange. We will illustrate this just for the first equation (19.93) since the method is identical for the remaining four. The derivative of (19.93) with respect to i is 

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