7. Consider the three solid phases of aluminium silicate Al,SiO,, Kyanite, Andalusite and Sillimanite, as seen in the phase diagram below (not to scale) Relevant thermodynamic properties are given in the data table below. P kyanite sillimanite andalusite 298 T(K) (T = 298 K, P = 1 atm) AG (kJ/mol) | S (J/K.mol) | V (cm /mol) AlzSiO, - Kyanite -2443.88 83.81 44.09 Al-SiO- - Andalusite -2442.66 93.22 51.53 Al-SiO, - Sillimanite -2440.99 96.11 49.90 (a) Explain why Sillimanite is the least stable phase at T = 298 K and 1 atmosphere. (b) Explain why Sillimanite is not the least stable phase at T = 298 K at all pressures above 1 atmosphere (Hint: use a relevant thermodynamic identity and the Table.)(c) Consider the derivation of the Clausius-Clapeyron relation dP AS L dT AV TAV (1) for the phase transition between Kyanite (subscript k) and Andalusite (subscript a). (i) At an arbitrary point on the phase boundary why is the Gibbs Free Energy G equal in the two phases, i.e., GR = G.? (ii) Move a small distance dG = dG: = dG, along the boundary, where G* + dGk = Ga + dG., use the Thermodynamic Identity for the Gibbs Free Ener- gies in both phases, and rearrange to derive the leftmost form of the Clausius- Clapeyron relation. (d) Justify the signs of the slopes of the phase boundaries between Andalusite and the other two phases near the triple point using the Clausius-Clapeyron relation and data in the table. (e) Beginning with Andalusite, which phase transition would heat the material? As- sume the transition is adiabatic and justify your