# Activity–composition relations for phases in petrological calculations: an asymmetric multicomponent formulation

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DOI: 10.1007/s00410-003-0464-z

- Cite this article as:
- Holland, T. & Powell, R. Contrib Mineral Petrol (2003) 145: 492. doi:10.1007/s00410-003-0464-z

## Abstract

For petrological calculations, including geothermobarometry and the calculation of phase diagrams (for example,* P*–*T* petrogenetic grids and pseudosections), it is necessary to be able to express the activity–composition (*a*–*x*) relations of minerals, melt and fluid in multicomponent systems. Although the symmetric formalism—a macroscopic regular model approach to* a*–*x* relations—is an easy-to-formulate, general way of doing this, the energetic relationships are a symmetric function of composition. We allow asymmetric energetics to be accommodated via a simple extension to the symmetric formalism which turns it into a macroscopic van Laar formulation. We term this the asymmetric formalism (ASF). In the symmetric formalism, the* a*–*x* relations are specified by an interaction energy for each of the constituent binaries amongst the independent set of end members used to represent the phase. In the asymmetric formalism, there is additionally a "size parameter" for each of the end members in the independent set, with size parameter differences between end members accounting for asymmetry. In the case of fluid mixtures, for example, H_{2}O–CO_{2}, the volumes of the end members as a function of pressure and temperature serve as the size parameters, providing an excellent fit to the* a*–*x* relations. In the case of minerals and silicate liquid, the size parameters are empirical parameters to be determined along with the interaction energies as part of the calibration of the* a*–*x* relations. In this way, we determine the* a*–*x* relations for feldspars in the systems KAlSi_{3}O_{8}–NaAlSi_{3}O_{8} and KAlSi_{3}O_{8}–NaAlSi_{3}O_{8}–CaAl_{2}Si_{2}O_{8}, for carbonates in the system CaCO_{3}–MgCO_{3}, for melt in the melting relationships involving forsterite, protoenstatite and cristobalite in the system Mg_{2}SiO_{4}–SiO_{2}, as well as for fluids in the system H_{2}O–CO_{2}. In each case the* a*–*x* relations allow the corresponding, experimentally determined phase diagrams to be reproduced faithfully. The asymmetric formalism provides a powerful and flexible way of handling* a*–*x* relations of complex phases in multicomponent systems for petrological calculations.