Addendum :

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, z0 at the bottom of the
column and zh 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.