Environmental Engineering Reference
In-Depth Information
y
i
(Vapor)
x
i
(Liquid)
Figure 3.5
A batch two-phase system in thermodynamic equilibrium.
The number of independent equations (
E
) is:
2
(sum of mole fractions equals unity in each phase)
C
(or similar thermodynamics equilibrium relationship for each
component)
C
+
2
DF
=
V
−
E
=
2
C
+
2
−
(
C
+
2)
=
C
.
So, if the
C
component mole fractions in one phase are specified, one can uniquely solve
for the other variables.
Now, let's expand the previous example to the case of P phases.
The number of independent variables (
V
)is
P
C
(component mole fractions)
T
P
2
The number of independent equations (
E
) is:
P
P
C
+
(sum of mole fractions equals one in each phase)
C
(P
−
1)
(equilibrium relationships. Note that in previous example we
had two phases and
C
(2
1) equilibrium relationships.
Another way to think of this sum is that we start with the mole
fraction in one phase and use these relationships to calculate
the mole fraction in the other (P
−
−
1) phases)
P
+
C
(P
−
1)
DF
=
V
−
E
=
P
C
+
2
−
[P
+
C
(P
−
1)]
=
C
−
P
+
2
.
This is the Gibbs phase rule. This result demonstrates that the Gibbs phase rule is a specific
case of this analysis and should only be applied for this specific situation.
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