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?
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.
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).
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.
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.]