Chapter 5.3, Problem 29P

Aluminum silicate, ${\text{Al}}_{\text{2}}{\text{SiO}}_{\text{5}}$, has three different crystalline forms: kyanite, 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 fixed pressure as we vary the temperature. Let $△G$ be the difference in Gibbs free energies between any two phases, and similarly for $△S$. Show that the T dependence of $△G$ is given by $△G\left(T_2\right)=△G\left(T_1\right)-{\int }_{T_1}^{T_2}△S\left(T\right)dT$

Although the entropy of any given phase will increase significantly as the temperature increases, above room temperature it is often a good approximation to take $△S$, the difference in entropies between 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 capacity of a solid at high temperature depends, to a good approximation, only on the number of atoms it contains.

(c) Taking $△S$ to be independent of T, find the range of temperature over which kyanite, andalusite, and sillimanite should be stable (at 1 bar).

(d) Referring to the room–temperature heat capacities of the three form of ${\text{Al}}_{\text{2}}{\text{SiO}}_{\text{5}}$, discuss the accuracy the approximation $△S=\text{constant}$.