Page 150 - Physical Chemistry

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Section 4.7
temperature, pressure, and mole fraction in the pure-solid phase do not change Phase Equilibrium
as the solid dissolves, m ICN(s) remains constant during the process. As the crystal
dissolves, x ICN in the aqueous phase increases and (4.90) shows that m ICN(aq)
increases. This increase continues until m ICN(aq) becomes equal to m ICN(s) . The
system is then in phase equilibrium, no more ICN dissolves, and the solution is
saturated.

Exercise
The equilibrium vapor pressure of water at 25°C is 24 torr. Is the chemical po-
tential of H O(l) at 25°C and 20 torr less than, equal to, or greater than m of
2
H O(g) at this T and P? (Hint: The vapor pressure of water at temperature T is the
2
pressure of water vapor that is in equilibrium with liquid water at T.) (Answer:
greater than.)

Just as temperature is an intensive property that governs the flow of heat, chemi-
cal potentials are intensive properties that govern the flow of matter from one phase to
another. Temperature is less abstract than chemical potential because we have experi-
ence using a thermometer to measure temperature and can visualize temperature as a
measure of average molecular energy. One can get some feeling for chemical potential
d
by viewing it as a measure of escaping tendency. The greater the value of m , the
j
greater the tendency of substance j to leave phase d and flow into an adjoining phase
where its chemical potential is lower.
b
d
There is one exception to the phase-equilibrium condition m m , which we now
j j
examine. We found that a substance flows from a phase where its chemical potential is
higher to a phase where its chemical potential is lower. Suppose that substance j is ini-
tially absent from phase d. Although there is no j in phase d, the chemical potential m d j
is a defined quantity, since we could, in principle, introduce dn moles of j into d and
j
d
d
d
measure 10G >0n 2 d m d (or use statistical mechanics to calculate m ). If initially
j T,P,n i j j j
b d
m 7 m , then j flows from phase b to phase d until phase equilibrium is reached.
j j
d b
However, if initially m 7 m , then j cannot flow out of d (since it is absent from d). The
j j
system will therefore remain unchanged with time and hence is in equilibrium. Therefore
when a substance is absent from a phase, the equilibrium condition becomes
b
d
m m phase equilib., j absent from d (4.91)
j
j
for all phases b in equilibrium with d. In the preceding example of ICN(s) in equilib-
rium with a saturated aqueous solution of ICN, the species H O is absent from the pure
2
solid phase, so all we can say is that m H 2 O in the solid phase is greater than or equal to
m H 2 O in the solution.
The principal conclusion of this section is:
In a closed system in thermodynamic equilibrium, the chemical potential of any
given substance is the same in every phase in which that substance is present.
EXAMPLE 4.6 Conditions for phase equilibrium
Write the phase-equilibrium conditions for a liquid solution of acetone and water
in equilibrium with its vapor.
Acetone (ac) and water (w) are each present in both phases, so the equilib-
y
y
l
y
l
l
rium conditions are m m and m m , where m and m are the chemical
ac ac w w ac ac
potentials of acetone in the liquid phase and in the vapor phase, respectively.