Mixtures and Mineral Reactions

The chemical potential (μ_i and fugacity (f_i) are alternative but related measures of the escaping tendency of a substance. The necessary condition of equilibrium between two phases, α and β, with respect to the diffusion of any component i, which is free to move between these phases, can be expressed as either or .

The functional forms of the expressions of activity coefficients in terms of composition have been developed from both macroscopic and atomistic reasonings. The macroscopic models are derived from considerations of the appropriate forms of ΔG^XS as a function of composition, and operating on these empirical functions according to the relationships between \( {\tilde{\gamma }_i} \) and ΔG^XS described in Chapter 1.II. Of these, the most generally useful formulation of ΔG^XS(X) is due to Guggenheim (1937), who suggested the following polynomial form for a binary solution, which satisfies the required property that ΔG^XS = 0 at the end-member compositions.

In this section, we first develop the general property of the G vs. X relation at the extreme compositions, which has important consequences in the problem of the stability of a solution. For a binary solution As X₂→0, we have, according to laws of dilute solution (Chap. l.III), \( {\tilde{X}_2} \), and \( {a_1} = {\tilde{X}_1} \), where \( {\tilde{X}_i} \) is an appropriate compositional parameter with the property that at X₂ = 0, \( {\tilde{X}_2} = {X_2} \) and \( {\tilde{X}_1} = {X_1} \) [see Eq. (1.9) and discussion]. For the sake of simplicity, let us assume that \( {\tilde{X}_2} = {X_2} \) and \( {\tilde{X}_1} = {X_1} \). Then, differentiating Eq. (3.1) with respect to X₂, and substituting the dilute solution properties of a₁ and a₂, we find that as X₂→0, where \( {K'_H} \) is a constant [see Eq. (1.36)].

Consider the following reaction among the end-member components of py­roxene (Px), feldspar (Fs), and quartz (Q).

Calculation of phase-equilibrium relations critically depends on the accuracy of thermochemical data and models of solutions. An extensive discussion of the solution models has been presented in Chapter 2. As reviewed in this chapter, we note that different solution models may be fitted to the same experimental data with similar success in reproducing the phase diagrams (e.g., for aluminous pyroxene see Wood and Holloway 1984; Ganguly and Ghose 1979; Saxena 1982). However, the behavior of these model components may differ when extrapolated to pressures and temperatures beyond the range of experimental results. If several models can be fitted to the same experimental data, it is of course preferable to use the simplest formulation for interpolation. However, the success of extrapolation significantly beyond the experimental condition and prediction of thermochemical properties would depend on how closely the adopted solution model approximates the atomic configuration of the solution. The problem has been discussed by Green (1970a, b) and Cohen (1986).

The distribution of an element between coexisting phases depends on the P-T condition to which they have equilibrated and, in general, also on the composition of the mineral phases. For thermodynamic analysis, it is convenient to treat these distributions in terms of what is known as exchange equilibrium, the basic concept of which was first introduced by Ramberg and DeVore (1951) in the geologic literature. A general discussion of the theory was presented by Kretz (1961), who also made the first systematic study (Kretz 1959) of element distribution between coexisting minerals in natural rocks. For the purpose of developing the theory, let us consider the fractionation of Fe²⁺ and Mg between coexisting garnet and biotite, which can be expressed through the following exchange reaction: At equilibrium at P and T where Alm (almandine) ≡ Fe₃Al₂Si₃O₁₂, Py (pyrope) ≡ Mg₃Al₂Si₃O₁₂, Ph (phlogopite) ≡ KMg₃AlSi₃O₁₀(OH)₂, and Ann (annite) ≡ KF₃AlSi₃O₁₀(OH)₂.

Atomic ordering in minerals has been a subject of interest among earth scientists since the classical work of Goldschmidt (1954). The interest stems primarily from the recognition of the effects of ordering on the energetic and compositional properties of minerals (also see Chap. 2.VIII), and the relationship between the quenced ordering state of a mineral and the cooling history of rocks (see Sect. 7.VIII).³⁷

The thermodynamic properties are ultimately macroscopic manifestations of interactions in an atomic scale. However, detailed calculations and integration of these interactions are formidable tasks, especially for complex ionic solid solutions. Consequently, most works on the theoretical calculation of the thermodynamic properties of minerals and mineral solid solutions are semiempirical in nature. We present below a selected review of these works, which rely on relatively simple theoretical analyses. A formal discussion of the relationship between microscopic or atomic interactions and thermodynamic properties is beyond the scope of this work. However, we have attempted to provide some idea of the nature of the microscopic interactions that govern the thermodynamic properties of minerals, insofar as these concepts can be developed within the overall framework of this book.⁴¹