Contrib Mineral Petrol (1988) 98:346-351 Contributions to Mineralogy and Petrology 9 Springer-Verlag1988 Thermodynamic projection and extrapolation of high-variance mineral assemblages Frank S. Spear Department of Geology, Rensselaer Polytechnic Institute, Troy, NY 12180, USA Abstract. A procedure is described whereby the effect of "extra" components on the plotting positions of minerals in projected phase diagrams may be accounted for rigorous- ly. The method employs the equilibrium constraints of the mineral assemblage to extrapolate the compositions of the minerals to where the values of the "extra" components approach 0. The same procedure may also be used to ex- trapolate the compositions of natural assemblages over iso- thermal, isobaric composition diagrams or polythermal, polybaric diagrams. Examples from typical garnet-bearing pelite assem- blages indicate that the "extra" components MnO and CaO dramatically shift the compositions of coexisting phases to lower Fe/Mg, even where the phase itself (e.g. chlorite or biotite) does not contain appreciable quantities of the extra component. Recognition of, and correction for, this effect is critical if projected phase diagrams are to be compared with experimentally calibrated phase diagrams in the chemi- cal subsystem. Introduction A common problem encountered in the analysis of the phase equilibria of metamorphic and igneous systems is the graphical representation of phase assemblages. In systems of low thermodynamic variance, projections through ubi- quitous phases are often employed (e.g. Thompson 1957; Greenwood 1975). In systems where the projection points are phases of fixed composition, the projected composition diagram may be interpreted as a valid phase diagram drawn at constant values of P, T and the chemical potentials of the projection points. In systems of high thermodynamic variance, projection points of fixed composition are generally insufficient to per- mit rigorous projective analysis of the phase relations. An example is the treatment of so-called "extra components" on a diagram such as the AFM diagram (Thompson 1957), as has been discussed by Albee (1968). Several solutions to the problem of "extra components" have been proposed, as will be discussed below. None is completely satisfactory because the plotting positions of phases in the projected systems are in general dependent on the bulk composition and hence, phase relations in the projected system may be- come distorted. The purpose of the present communication is to outline an approach whereby high variance mineral assemblages may be "projected" onto composition planes in such a way that the projected phase diagram is thermodynamically val- id and can thus be compared directly with experimental phase equilibria in the chemical subsystem. The problem As an illustration of the problem consider the assemblage garnet + biotite + kyanite + quartz + muscovite in the system SiO2- A1203 - MgO- FeO- MnO- KOH (MnKFMASH). This particular assemblage is independent of H20; hence H20 is not required as an independent sys- tem component. Quartz, kyanite and muscovite the as- sumed to be pure SiO2, A12SiO 5 and KA13Si3Olo(OH)2, respectively. Garnet and biotite are assumed to be ternary Fe- Mg- Mn solutions. The assemblages garnet + biotite + kyanite + quartz- +muscovite, when projected from muscovite and quartz into the tetrahedron A12Oj-FeO-MgO-MnO, appears as shown in Fig. la. In the MnKFMASH system, this as- semblage is trivariant and can be depicted as a set of 3-phase triangles (garnet + biotite + kyanite) that fan across the te- trahedral composition space. Further projection from kya- nite onto the plane FeO-MgO-MnO (Fig. 1 b) clearly re- veals the fan of garnet + biotite tie lines. The question to be addressed may be stated as follows: How can the compositions of coexisting garnet and biotite be represented in a projection onto the AFM plane in such a way that the phase relations are not seriously distorted by the presence of the "extra" componenet, MnO ? Three basic approaches have been suggested: (1) ignore MnO and renormalize to 100% FeO+MgO; (2) combine FeO + MnO into a single component; and (3) drop a perpen- dicular from the composition to be plotted onto the projec- tion plane (the least squares solution). The plotting position of a single garnet on the FeO-MgO join using each of these assumptions can be viewed in Fig. 2. Procedure (1) (cf. Thompson 1957) preserves the Fe/Mg of the garnet and the plotting position is found at the intersection of a line radiating from the MnO apex and passing through the gar- net composition with the FeO-MgO join. Procedure (2) preserves the MgO content of the garnet, and the plotting position is found along a line of constant MgO. Procedure (3), the last squares solution, finds the point closest to the original garnet composition (cf. Greenwood 1968; Spear et al. 1982b). It is apparent from Fig. 2 that the plotting position of the garnet on the AFM plane is a function of the way