Problem 5.29. Aluminum silicate, AliO 5, has three different crystalline forms: liyanite, andalusite, and sillimanite. Because each is stable under a different set of temperature-pressure conditions, and all are commonly found in metamorphic rocks, these minerals are important indicators of the geologic history of rock bodies. (a) Referring to the thermodynamic data at the back of this book, argue that at 298 K the stable phase should be kyanite, regardless of pressure. (b) Now consider what happens at xed pressure as we vary the temperature. Let AG be the dilTerence in Gibbs free energies between any two phases, and similarly for AS. Show that. the T dependence of AG is given by T2 AGtTg) : AGEFl) AS(T) dT. - Ti Although the entropy of any given phase will increase significantly as the temperature increases, above room temperature it is often a good apln'ox- imation to take AS, the difference in entropies l'Jetween two phases, to be independent of T. This is because the temperature dependence of S is a function of the heat capacity (as we saw in Chapter 3), and the heat ca- pacity of a solid at high temperature depends, to a good apm'oximation, only on the number of atoms it contains. (c) Taking A8 to be independent of T, nd the range of temperatures over which kyanite, andalusite, and sillimanite should be stable (at 1 bar). ((1) Referring to the room-temperature heat capacities of the three forms of 3112 SiO5, discuss the accuracy the approximation AS : constant