Starting from the molar energy, u(s, v), we defined the Legendre transform, a su-(Qu/as), s=u-Tis in class. We called this Legendre transform the molar Helmholtz free energy, and used it to derive a useful Maxwell relationship. In this problem, we will derive an expression for a Legendre transform of the molar Helmholtz free energy, a(Tv). (a) Let us define the Legendre transform, g = a - (da/dur. v. This transform is called the molar Gibbs free energy. Show that g=a+P.v. (b) Show that the differential for the molar Gibbs free energy is given by: dg = -sdT + vdP. (c) Derive expressions for the first derivatives, (ag/at)p and (@g/aP)T. (d) By equating the second derivatives of g, derive the following Maxwell relationship: (as/ap)r = -(u/at)p