EQUATIONS FOR SPECIATION AND FORWARD MODELING

In this section of the report, the algebraic equations used to define thermodynamic activities of aqueous species, ion-exchange species, surface-complexation species, gas-phase components, solid solutions, and pure phases are presented. First, thermodynamic activities and mass-action equations are described for aqueous, exchange, and surface species. Then, a set of functions, denoted , are defined that must be solved simultaneously to determine equilibrium for a given set of conditions. Many of these functions are derived from mole-balance equations for each element or element valence state, exchange site, and surface site or from mass-action equations for pure phases and solid solutions. Additional functions are derived for alkalinity, activity of water, aqueous charge balance, gas-phase equilibria, ionic strength, and surface-complexation equilibria. Each function is reduced to contain a minimum number of variables, such that the number of functions equals the number of variables. The program uses a modified Newton-Raphson method to solve the simultaneous nonlinear equations. This method uses the residuals of the functions and an array of partial derivatives of each function with respect to the set of master unknowns or master unknowns. For clarity, the set of variables used in partial differentiation are referred to as "master unknowns". The total derivatives of each function, , will be presented without derivation. In the following equations, lack of a subscript or the subscript "(aq)" will refer to entities in the aqueous phase, "(e)" refers to exchangers, "(g)" refers to gases, "(s)" refers to surfaces, "(ss)" refers to solid solutions, and "(p)" refers to phases.

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