The calculation of mineral compositions and modes of olivine-two pyroxene-spinel assemblages

Duhem's theorem states that the equilibrium position of a system of known bulk composition and size is completely determined by two independent variables. Given these data all other equilibrium parameters can be calculated. The method of calculation is set down in detail for olivine-two pyroxene-spinel assemblages which may serve as models for portions of the upper mantle. To make the calculations, equations for the Gibbs energy of the constituent minerals, olivine, spinel, orthopyroxene and clinopyroxene are formulated. The olivine is treated as an ideal, binary, cationic solid solution. The spinel is treated as a ternary, reciprocal solution between the four end members MgAl₂O₄, MgCr₂O₄, FeAl₂O₄ and FeCr₂O₄. The pyroxenes are treated as seven-component, simplified reciprocal systems. The eight oxides SiO₂, TiO₂, Al₂O₃, Cr₂O₃, FeO, MgO, CaO and Na₂O are thought sufficient for expressing the composition of the model mantle.The paucity and inconsistency of the required thermodynamic data preclude accurate calculations. To illustrate the calculation and to arrive at some highly tentative results, the required free energy data are estimated by combining analytical data from Iherzolite xenoliths with interpretations of experimental data.It seems likely that the Al content of pyroxenes cannot be used as a geobarometer without considering the bulk composition of the assemblage or, equivalently, the composition of the associated spinel phase. Temperature estimates based on the Ca to Ca+Mg ratio in Ca-rich pyroxenes may be approximately correct in the restricted composition range of mantle derived xenoliths.