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Thermodynamic Background • the mass of each mineral or gas phase; • the concentration of basis species beyond those controlled by the mineral and gas phases.
An example will help to clarify this (see the Box on page 55). However, if one finds this confusing, you are not alone, and it is not terribly important because most modelers do not actually start with the Phase Rule in constructing models, in spite of the fact that it does tell us how many data are required. If the modeler was starting from scratch, it would undoubtedly be useful, but when using well established programs, it is often more efficient to bypass the Phase Rule, and rely on error messages from the modeling program to get things right. It is useful, though, to know that the Phase Rule used or implied in geochemical modeling is somewhat different from the one derived by Gibbs. Duhem’s Theorem The term “Extensive Phase Rule” is our own terminology, and may prove confusing to geochemists more used to seeing it referred to as Duhem’s Theorem. As expressed by Prigogine and Defay (1965), p. 188, Duhem’s Theorem says Whatever the number of phases, of components or of chemical reactions, the equilibrium state of a closed system, for which we know the initial masses m◦1 . . . m◦c , is completely determined by two independent variables. By “completely determined”, the masses of all phases is meant to be included. In the general case, where the system is not invariant or univariant, the “two independent variables” can be T and P , and this amounts to saying (from the Theorem) that in addition we need to know the mass of every traditional component, if we are not concerned with ionic speciation, or every basis species, if we are.
3.8 Redox Many elements in natural systems occur in more than one state of oxidation, or valence state. Thus iron occurs as either Fe2+ or Fe3+ ; arsenic occurs as either As3+ or As5+ ; sulfur occurs in many valence states between S2− and S6+ ; and so on. If all these valence states have concentrations that are independent of one another, then each represents an additional component, and each must be specified or constrained somehow in setting up a geochemical model. However, if chemical equilibrium prevails in the system, these states are not all independent – each is dependent on the oxidation–reduction (redox) state of the system. Specifying this redox state is sufficient to specify the ratio of the activities of each pair of valence states, e.g., aFe3 + /aFe2+ , aAs5+ /aAs3+ , and so on. This ratio, together with the total amount of the element (total Fe, total As) in the system, is then sufficient to determine the activities of each species separately. Therefore, in setting up equilibrium models of systems containing elements which occur in more than one valence state, one additional parameter is required – a measure of the redox state of the system. In some programs, the user is also allowed to specify a