Environmental Engineering Reference
In-Depth Information
TABLE 3.14
Correlation Equations for Activity Coefficients in a Binary Liquid Mixture
Based on Excess Gibbs Free Energy
Description
Equation
Ax
2
RT
,
Ax
1
RT
One constant Margules
a
γ
1
=
γ
2
=
Two constant Margules
b
RT
ln
γ
1
= α
1
x
2
+ β
1
x
2
,
RT
ln
γ
2
= α
2
x
1
+ β
2
x
1
α
1
+
(
α
/
β
)(x
1
/x
2
)
2
, ln
γ
2
=
β
1
+
(
β
/
α
)(x
2
/x
1
)
2
van Laar
c
ln
γ
1
=
ln
γ
1
=
ln
φ
1
x
1
1
−
2
, ln
γ
2
=
ln
φ
2
1
m
Flory-Huggins
d
1
+
φ
2
+ χφ
+
(m
−
1
)
φ
1
+χφ
x
2
ln
γ
1
=−
ln
(x
1
+ Λ
12
x
2
)
+
x
2
Λ
12
x
1
+ Λ
12
x
2
−
Λ
21
Λ
21
x
1
+
x
2
Wilson
e
τ
21
G
21
x
1
+
x
2
G
21
2
τ
12
G
12
(x
2
+
x
1
G
12
)
2
NRTL
f
ln
γ
1
=
x
2
+
a
Only applicable for liquid mixtures when the constituents (1 and 2) are both of equal size, shape, and
chemical properties.
b
α
i
=
A
+
(
3
−
1
)
i
+
1
B
and
β
i
=
4
(
−
1
)
i
B
;
i
=
1or2.
c
α =
2
q
1
a
12
,
β =
2
q
2
a
12
.
a
12
is the van Laar interaction parameter,
q
1
and
q
2
are liquid molar volumes,
and are listed in Perry's
Chemical Engineer's Handbook
. Note also that as
x
1
→
0, ln
γ
1
=
ln
γ
1
,
and as
x
2
→
0, ln
γ
2
=
ln
γ
2
.
d
ϕ
1
=
x
1
/(x
1
+
mx
2
)
and
ϕ
2
=
mx
2
/(x
1
+
mx
2
)
are volume fractions,
m
=
v
2
/v
1
.
χ
is the Flory
parameter.
e
Λ
12
and
Λ
21
are given in (1964).
Journal of the American Chemical Society
86, 127.
f
G
12
,
G
21
,
τ
12
, and
τ
21
are given in (1968).
AIChE Journal
14, 135.
activity coefficients of a vast number of binary and ternary liquid-liquid and vapor-
liquid systems already exists. At the same time there also exists a sound theory of
liquid mixtures based on the Guggenheim quasi-chemical approximation originating
instatisticalthermodynamics.ThisiscalledUNIversalQUAsiChemical(UNIQUAC)
equation. This theory incorporates the solvent cavity formation and solute-solvent
interactions and is useful for establishing group contribution correlations where the
independent variables are not the concentrations of the molecules themselves but
those of the functional groups. The basic tenet of this theory was combined with the
large database on activity coefficients to obtain group contribution parameters for a
variety of molecular groups. This approach was called UNIversal Functional group
Activity Coefficient (UNIFAC) method. Although developed by chemical engineers,
it has of late found extensive applications in environmental engineering.
The UNIFAC method is especially suitable for activity coefficients of complex
mixtures. It was developed for nonelectrolyte systems and should be used only for
such systems. Most of the data for group contributions were obtained at high mole
fractions, and therefore extrapolation to very small mole fractions (infinitely dilute)
for environmental engineering calculations should be made with caution. Arbuckle
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