Derivation of Internally-Consistent Thermodynamic Data by the Technique of Mathematical Programming: a Review with Application the System MgO-SiO2-H2O

Abstract

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.

Development of schistosity by dissolution–crystallization in a Californian serpentinite gouge

“…Viti and Mellini, 1998;Mével, 2003) and therefore may have been present prior to deformation. Antigorite, which is known as a 'higher grade' type of serpentine (Berman et al, 1986;Ulmer and Trommsdorff, 1995) that is typically found in subducted rocks, has not been observed in the studied sample. The microtextures vary within the sample and a clear distinction is made between the clasts and the schistosity planes.…”

Internally consistent thermodynamic data for magnesium sulfate hydrates

“…In this study, we have employed the technique of mathematical programming to derive an internally consistent thermodynamic database for the magnesium sulfate hydrates (cf. Berman et al, 1986;Chatterjee, 1991;Grevel, 2004).…”

Phlogopite in the contact-metamorphosed dunite-harzburgite complex at Yanomine, eastern Chugoku, Japan

“…Na phlogopite hydrate I with a composition of NaMg 3 AlSi 3 O 10 (OH) 2 2H 2 O is, for example, stable below 380°C (Carman, 1974). The temperature is low enough to that of the olivine orthopyroxene equilibrium (600 650°C after Berman et al, 1986). Partial hydration of phlogopite in the olivine orthopyroxene zone probably implies that supplies of fluids were locally prolonged in the stages after the peak of the contact metamorphism.…”