Thermodynamics and lattice vibrations of minerals: 5. Applications to phase equilibria, isotopic fractionation, and high‐pressure thermodynamic properties

In previous papers in this series, a model that uses available elastic, structural, and spectroscopic data on minerals has been used to predict the thermodynamic functions Cv (heat capacity), S (entropy), E (internal energy), and F (Helmholtz free energy). In this paper, four applications to problems of current geochemical and geophysical interest are presented: (1) interpretation of complex trends of calorimetric data; (2) calculation of phase equilibria; (3) calculation of oxygen-isotopic fractionation factors; and (4) estimation of the effect of pressure on thermodynamic functions. The model demonstrates that trends in high-temperature thermodynamic properties of silicates are determined by the position and relative numbers of high-frequency modes, generally antisymmetric (Si, Al)-O stretching modes. The position of these modes varies systematically with degree of polymerization of tetrahedra, and therefore high-temperature calorimetric behavior is relatively systematic as a function of crystal structure and mineral composition. Trends at low frequency are much more complex because the low-frequency optic modes that most strongly influence the low-temperature thermodynamic functions depend in a complex way on the size, coordination, and mass of cations and various polyhedra in the minerals. The heat capacity curves of kyanite, andalusite, and sillimanite and of quartz, coesite, and stishovite show crossovers that cannot be explained by Debye theory, which accounts only for acoustic mode behavior, but can be explained by the model spectra proposed because proper account is taken of the changing low- and high-frequency optic modes upon polymorphic transformations. The proposed model is sufficiently accurate that phase equilibrium problems can be addressed: the quartz-coesite-stishovite equilibrium curves, the kyanite-andalusite-sillimanite triple point, and the breakdown of albite to jadeite-plus-quartz are cited as specific examples. For each example, predicted slopes of equilibrium curves agree moderately well to excellently with slopes determined experimentally. The calculated slopes are sensitive to spectroscopic parameters, particularly to the distribution of optic modes in the far infrared; this sensitivity is discussed in detail for the albite breakdown reaction. The model can be used for prediction of isotopic fractionation factors if spectra of the isotopic forms of the mineral are known or postulated. A simple set of ‘rules’ for generating hypothetical spectra of 18O minerals from measured spectra of the 16O forms is given. Reduced partition functions are calculated for 13 minerals. At 298°K the model values of reduced partition function, 1000 ln α, of these minerals decrease in the order quartz > calcite ≳ albite > muscovite > clinoenstatite ≈ anorthite > diopside > pyrope > grossular > zircon > forsterite > andradite > rutile, in good agreement with experimental data. At 1000°K the first six minerals show small crossovers so that the order becomes calcite, muscovite ≈ albite, quartz, anorthite, and clinoenstatite; the differences in 1000 ln α at high temperature for these minerals are so small that the model probably cannot address the deviations from experimental trends. The model clearly defines the region in which the fractionation factors do not follow a 1/T² trend and should be useful for extrapolation of experimental data to low temperatures. Finally, a modified Gruneisen parameter model is proposed for shift of the lattice vibrational frequencies under compression, and thermodynamic properties to 1000 kbar, 1000°K, are given for nine minerals. At 1 Mbar, the predicted decrease in entropy at 298°K ranges from 54% (of the 1-bar value) for periclase to 25% for stishovite.