Page 149 - Physical Chemistry

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Chapter 4 of substance j flow spontaneously from phase b to phase d. For this irreversible
Material Equilibrium
process, the inequality (4.15) gives dG SdT VdP. But dG for this process
a
a
is given by (4.81) as dG SdT VdP m dn . Therefore the inequality
a i i i
dG SdT VdP becomes
a
a
S dT V dP a a m dn 6 S dT V dP
i
i
a i
a
a
a a m dn 6 0
i
i
a i
For the spontaneous flow of dn moles of substance j from phase b to phase d, we
j
b
a
a
b
d
b
d
d
have m dn m dn m dn m dn m dn 0, and
a i i i j j j j j j j j
d
b
1m m 2 dn 6 0 (4.89)
j
j
j
b
d
b
d
Since dn is positive, (4.89) requires that m m be negative: m m . The sponta-
j j j j j
neous flow was assumed to be from phase b to phase d. We have thus shown that for
a system in thermal and mechanical equilibrium:
Substance j flows spontaneously from a phase with higher chemical potential M
j
to a phase with lower chemical potential M .
j
This flow will continue until the chemical potential of substance j has been equalized
in all the phases of the system. Similarly for the other substances. (As a substance
flows from one phase to another, the compositions of the phases are changed and hence
the chemical potentials in the phases are changed.) Just as a difference in temperature
is the driving force for the flow of heat from one phase to another, a difference in
chemical potential m is the driving force for the flow of chemical species i from one
i
phase to another.
d
b
d
b
If T T , heat flows spontaneously from phase b to phase d until T T . If
d
b
d
b
d
b
P P , work “flows” from phase b to phase d until P P . If m m , substance
j j
d
b
j flows spontaneously from phase b to phase d until m m . The state function T de-
j j
termines whether there is thermal equilibrium between phases. The state function P
determines whether there is mechanical equilibrium between phases. The state func-
tions m determine whether there is material equilibrium between phases.
i
One can prove from the laws of thermodynamics that the chemical potential m d
j
d
of substance j in phase d must increase when the mole fraction x of j in phase d is
j
increased by the addition of j at constant T and P (see Kirkwood and Oppenheim,
sec. 6-4):
d
d
10m >0x 2 d 7 0 (4.90)
j
j T,P,n i j
EXAMPLE 4.5 Change in M when a solid dissolves
i
A crystal of ICN is added to pure liquid water and the system is held at 25°C and
1 atm. Eventually a saturated solution is formed, and some solid ICN remains
undissolved. At the start of the process, is m greater in the solid phase or in the
ICN
pure water? What happens to m in each phase as the crystal dissolves? (See if
ICN
you can answer these questions before reading further.)
At the start of the process, some ICN “flows” from the pure solid phase into
the water. Since substance j flows from a phase with higher m to one with lower
j
m , the chemical potential m in the solid must be greater than m in the pure
j ICN ICN
water. (Recall from Sec. 4.6 that m is defined for the pure-water phase even
ICN
though there is no ICN in the water.) Since m is an intensive quantity and the