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The problem of deriving an optimal set of thermodynamic properties of minerals from a diverse experimental data base is reviewed and a preferred methodology proposed. Mathematical programming(MAP) methods extend the linear programming (LIP) approach first presented by Gordon (1973), and make it possible to account for the type of information conveyed, and the uncertainties attending both phase equilibrium data and direct measurements of phase properties. For phase equilibrium data which are (in most cases) characterized by non-normal error distributions across experimental brackets, the midpoint of a bracket is no more probable than other points, and the data are best treated by considering the inequality in the change in Gibbs free energy of reaction at each half-bracket. Direct measurements of phase properties can be assumed to have approximately normal error distributions, and the MAP technique optimizes agreement with these values by using the principles of least squares in the definition of an objective function. The structure of this problem, treatment of uncertainties in various types of experimental data, and method of optimizing final solutions are discussed in some detail.The method is applied to experimental data in the MgO-SiO 2 -H 2 O system, where inconsistencies among the data are resolved and an optimal set of thermodynamic properties is presented. The derived standard state entropies and volumes agree with all direct measurements (within their uncertainties), as do enthalpies of formation from the elements except for those of talc (+16 kJ mol" •), anthophyllite (+ 14 kJ mol~'), and brucite (-1 kJ mol~'). Stable phase relations in the system have the topology predicted by Greenwood (1963Greenwood ( , 1971, with quartz-and forsterite-absent invariant points at 683 "C-6-4 kb and 797 °C-12 kb respectively, repeating at 552 °C-120 b and 550 °C-55 b. The thermodynamic analysis indicates little remaining flexibility in the phase relations, which, when combined with suitable activity models for solid solution, should allow for accurate determination of the conditions of metamorphism of ultramafic rocks.

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