Homogeneous equilibrium in solutions

We will follow the layout of the problem as described by Morel and Hering (1993) and use their conventions. A solution may be defined as a solvent, most frequently water, and solutes, which can be neutral species such as 02 or, more frequently, ions such as Ca2+ or OH-. Charged particles, such as electrons and protons, are not present in solutions but nevertheless may be handled in the same way as other charged species (Stumm and Morgan, 1981). Solvent concentration overwhelms solute concentrations. 

For reasons of roundoff errors due to water being the dominant species, a simplification is introduced for dilute solutions (Morel and Hering, 1993). One unknown and one equation are simultaneously eliminated from the set of conservation equations which make the recipe. The unknown species HzO is expressed as a function of the unknowns OH- and H+, which assigns OH- a — 1 H+ coefficient, and the OH- conservation equation (first column in Table 6.1) is left out. 

We assume that no precipitation occurs, although this assumption will later be proved to be inadequate. Using a non-linear system is a complicated way of solving this particular problem, but this example is quite illustrative and can be extended to any number of components. Although ionized atoms like H+ and Ca2 + are natural components of the dilute solution under consideration, carbon and oxygen do not appear as such in natural systems. Since the group C032- is not destroyed in any reaction it will therefore be taken as the carbon host. The component matrix B is shown in Table 6.1. As explained above, the H20 row is subtracted from the OH-row, which is left with — 1 in the H + column, which produces the new component matrix of Table 6.2. 

The system to be solved involves five unknown concentrations which, for sake of illustration, are made equal to the corresponding activities. An identical number of equations must be found that include component conservation plus a number of mass action laws corresponding to the formation of as many species as the excess of species over components. We first write the recipe, i.e., the mass balance for the components, not including the components of water. Calcium mass balance reads 

This matrix can be derived from Table 6.1 by expressing the unknown species H20 as a function of OH- and H+, then removing the OH conservation equation, i.e., removing the corresponding column. 

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