Please do all parts since they are all inter connected. (Show work) thank you :)

Phase a 1. Consider a material that can exist in two distinct phases, a and , depending on the pressure and temperature. At any point along the coexistence line in the phase diagram, the Gibbs free energy of the two phases is the same: Ga (P, T) = GB (P, T). In this problem, you will derive some useful properties of the slope of the coexistence line. Pressure Phase = . Temperature a. a The coexistence line is a function P(T). Imagine the system is at coexistence. Write down the expression for how Ga changes if you make a small change in pressure and temperature, and do the same for GB. If you want to stay on the coexistence line after such a change, you cannot change P and Tindependently; you have to change them in a specific ratio. Calculate this ratio dP/dT in terms of the entropies and volumes of the two phases. b. As we have seen, a converting one phase a to another phase at the phase transition requires transfer of energy in form of the latent heat AHa. Rewrite the expression you found in part a) in terms of this latent heat AHa and the change in volume AVa = V8 - Va. c. a Let's assume that we are interested in the transition from a liquid (Phase a) to a gas (Phase B). The gas has a much greater than volume, so you can assume that AVAB VB. Use the ideal gas to simplify the expression for the slope of the coexistence line. a d. As we have seen in class, the coexistence line between the solid and liquid phases of water has a negative slope. Given your result from part b), what does that imply for the densities of ice and liquid water? Pressure Ice Liquid Temperature