Consider the following situation: I have a bottle of volume V, with ns moles of sugar at the bottom. I fill it to the top with water and then cap it with a lid. The lid is rigid, but permeable to water and impermeable to sugar. Now I take the bottle and immerse it completely in a large vat of water. We take the pressure in the vat at the position of the bottle to be Pv, the pressure in the bottle to be Pb, and the temperature of both the bottle mixture and the vat to be a constant T. We are given the fundamental relation for the sugar-water mixture in the bottle as U=U(S,V,nw,ns) with mole numbers nw and ns being for the water and sugar in the bottle respectively.
a. Which variables should subject to a Legendre transformation?
b. Comment on why. For concreteness, consider that I want to compute the pressure inside the bottle. What has the transformation done for me?