202 5 Phase EquilibriumThis result can be summarized:At constant temperature and pressure, any substance
tends to move spontaneously from a phase of higher chemical potential to a phase of
lower chemical potential.The name “chemical potential” was chosen by analogy with
the fact that a mechanical particle experiences a force pushing it in the direction of
lower potential energy.Exercise 5.1
Argue that a substance will move spontaneously from any phase in which its chemical potential
has a higher value to a phase in which it has a lower value if each phase is at constant volume
and if both phases are at the same constant temperature.Transport of Matter in a Nonuniform Phase
Consider a one-phase system that has a uniform temperature and pressure but a nonuni-
form composition. Imagine dividing the system into small regions (subsystems), each
one of which is small enough so that the concentration of each substance is nearly
uniform inside it. Any substance will move spontaneously from a subsystem of higher
value of its chemical potential to an adjacent subsystem with a lower value of its chem-
ical potential. The fundamental fact of phase equilibrium for nonuniform systems can
be stated:In a system with uniform temperature and pressure, any substance tends to
move from a region of higher chemical potential to a region of lower chemical potential.
A nonuniformity of the chemical potential is the driving force for diffusion, which we
will discuss in Chapter 10.PROBLEMS
Section 5.1: The Fundamental Fact of Phase Equilibrium
5.1 For water at equilibrium at 1.000 atm and 273.15 K, find the
values ofG(liq)m −G(solid)m ,A(liq)m −A(solid)m ,Hm(liq)−Hm(solid),
Um(liq)−Um(solid), andSm(liq)−Sm(solid).
5.2 For water at equilibrium at 1.000 atm and 373.15 K, find the
values ofG(gas)m −G(liq)m ,A(gas)m −A(liq)m ,Hm(gas)−Hm(liq),
Um(gas)−Um(liq), andS(gas)m −S(liq)m. State any assumptions.
5.3For water at equilibrium at 23.756 torr and 298.15 K,
find the values ofG(gas)m −G(liq)m ,A(gas)m −A(liq)m ,
Hm(gas)−H(liq)m ,U(gas)m −U(liq)m , andSm(gas)−Sm(liq).
State any assumptions.
5.4Repeat the calculation of Exercise 5.12 assuming thatCP, m
for the liquid is constant but thatCP, mfor the vapor is
given by the polynomial formula in Table A.6 in the
appendix.5.2 The Gibbs Phase Rule
The Gibbs phase rule gives the number of independent intensive variables in a simple
system that can have several phases and several components. The equilibrium ther-
modynamic state of a one-phase simple system withccomponents is specified by the
values ofc+2 thermodynamic variables, at least one of which must be an extensive
variable. All other equilibrium variables are dependent variables. Theintensive stateis
the state of the system so far as only intensive variables are concerned. Changing the
size of the system without changing any intensive variables does not affect the intensive
state. Intensive variables cannot depend on extensive variables, so specification of the