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Chapter 4
Equilibrium and Phase Stability in
OneComponent Systems
The purpose of this chapter is to provide answers to the following questions:
(
i
)
under what
conditions are the two phases of a pure substance in equilibrium with each other?,
(
ii
)
at a
given temperature and pressure, in which form does a pure substance exist  solid, liquid, or
gas?
4.1 EQUILIBRIUM CRITERIA
For a closed system, the second law of thermodynamics, Eq. (1.68), states that
TdS
−
δQ
≥
0
(4.11)
From the
f
rst law of thermodynamics
dU
=
δQ
−
PdV
(4.12)
Elimination of
δQ
between Eqs. (4.11) and (4.12) leads to
TdS
−
dU
−
PdV
≥
0
(4.13)
4.1.1 Maximum Entropy
Under the conditions of constant internal energy and volume, Eq. (4.13) simpli
f
es to
dS
U,V
≥
0
(4.14)
where the subscripts
U
and
V
imply that these quantities are kept constant during a process.
Under these conditions, any spontaneous process tends to increase the entropy of a system as
shown in Figure 4.1. When the system is at equilibrium, entropy reaches its maximum value
and
dS
U,V
=0
At equilibrium
(4.15)
Direction of a spontaneous process
0
=
dS
State of System
Equilibrium
Constant U & V
Entropy
Figure 4.1
The equilibrium condition for constant internal energy and volume.
77
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View Full Document4.1.2 Minimum Internal Energy
When entropy and volume are kept constant, Eq. (4.13) simpli
f
es to
dU
S,V
≤
0
(4.16)
Therefore, the internal energy of a system decreases as a result of a spontaneous process under
the conditions of constant entropy and volume. This phenomenon can be explained as follows.
Spontaneous processes lead to an increase in entropy. In order to keep entropy constant, heat
must be transferred to the surroundings with a concomitant decrease in internal energy. When
the system is at equilibrium, internal energy reaches its minimum value and
dU
S,V
=0
At equilibrium
(4.17)
4.1.3 Minimum Helmholtz Energy
In di
f
erential form, Helmholtz energy is given by
dA
=
d
(
U
−
TS
)=
dU
−
SdT
−
TdS
(4.18)
The use of Eq. (4.18) in Eq. (4.13) gives
dA
+
SdT
+
PdV
≤
0
(4.19)
When temperature and volume are kept constant, Eq. (4.19) simpli
f
es to
dA
T,V
≤
0
(4.110)
Therefore, the Helmholtz energy of a system decreases as a result of a spontaneous process
under the conditions of constant temperature and volume. When the system is at equilibrium,
Helmholtz energy reaches its minimum value and
dA
T,V
=0
At equilibrium
(4.111)
4.1.4 Minimum Gibbs Energy
In di
f
erential form, Gibbs energy is given by
dG
=
d
(
H
−
TS
)=
d
(
U
+
PV
−
TS
)
=
dU
+
PdV
+
VdP
−
TdS
−
SdT
(4.112)
The use of Eq. (4.112) in Eq. (4.13) gives
dG
+
SdT
−
VdP
≤
0
(4.113)
When temperature and pressure are kept constant, Eq. (4.113) simpli
f
es to
dG
T,P
≤
0
(4.114)
Under these conditions, any spontaneous process tends to decrease the Gibbs energy of a system
as shown in Figure 4.2. When the system is at equilibrium, Gibbs energy reaches its minimum
value and
dG
T,P
=0
At equilibrium
(4.115)
78
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 Fall '10
 ozbelge
 Equilibrium, pH

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