Addendum : Hydrostatic Equilibrium and Chemical Potential.

1. Addendum : Hydrostatic Equilibrium and Chemical Potential. z Previously we found that between two incompressible fluids the equilibrium condition is satisfied by two equalities, temperature and chemical potential, where h The question of a static column of fluid whose free surface is in equilibrium with the vapor phase above it. We expect from our knowledge of statics that the pressure in the fluid at any particular point depends on the height of the water column above it and the reference pressure at the free surface. How can we reconcile this idea with the above definition of chemical potential, and the requirement that at equilibrium the chemical potential be the same everywhere in the fluid?

2. Can we relax the requirement that chemical potential be uniform at equilibrium? No, that would violate the fundamental postulate. Clearly, our definition of chemical potential must be broken! If we consider the definition of chemical potential and the source of the variation in pressure in the static column we can see the error we have fallen into. Notice that in introducing the static column, we implicitly opened our system to gravitational fields. A proper and rigorous treatment would therefore require a field theory; here we simply note that by considering our definition of chemical potential we can identify the missing term and repair our understanding with only a little effort.

3. Recall that chemical potential is the change in free energy of a system for the addition of a single molecule of a species to that system. In fact, this holds true regardless of the particular free energy under consideration (to understand why that should be recall that CP is always defined respective to some reference state). We therefore recognize that at every height of the fluid the gravitational field contributes a potential energy term to the free energy per molecule: Where z is the height, z=0 at the bottom of the column and z=h at the top (see figure on first slide).

4. The variation in potential energy is given by (holding T constant) but so It is now clear that the potential energy due to gravity balances the pressure due to the weight of the fluid above at every height, such that the equilibrium condition is satisfied.

5. Finally, what happens if one opens a port at the base of the column - if the water is in chemical equilibrium with the vapor phase and uniform throughout both phases, will it not move? Recall that in equilibrium between two systems, if one of them is compressible then the EQ condition consists of three independent statements: equal temperature, pressure, and chemical potential. At a port at the base, the pressure in the fluid in the port and in the vapor would come into equilibrium. We know flow will occur as we now have a gradient in chemical potential in the fluid: The liquid phase will invade the vapor phase until gravitational potential once more balances fluid pressure. [Note we are neglecting the gravitational gradient in the vapor phase since it is proportional to the density of the vapor and therefore small.]