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Two-parameter model for correlating liquid phase activity coefficients of binary systems
Fluid Phase Equitibria, 20 (1985)183-188 Elsevier SciencePublishers B.V.,Amsterdam -Printed
in The Netherlands
TWO-PARAMETER
LIQUID
ACTIVITY
MODEL
FOR
COEFFICIENTS
F. Mato
and
F.
183
CORRELATING
OF
BINARY
PHASE
SYSTEMS
A. Mato
Department of Chemical Engineering. (Spa in) 47005 Val ladolid
University
of Valladolid
ABSTRACT A new two-parameter model for the activity coefficients of binary liquid systems is derived from the general expansion of Wohl. The model admits curves that exhibit maxima or minima in the activity coefficients and predicts well the liquid-vapor equilibrium from a restricted information : azeotropic condition,activity coefficients at infinite dilution or mutual solubilities.
The
general
the
excess
for
a binary
Q =
free
and
order
composition
(2a 122122
-
f
3a1122:22
+
assumption
z
introducing
parameter on Wohl
derived n
the
in the
liquid
3a122z1Zi)(q1x1
By
ding
equation,relating phase
system,is
gERT
energy
third
= molecular
two
the
Gibbs
of Wohl(lq46)
= qixi/Xjqjxj , aij i = effective molar volume.
where 'i
form
the
binary assumed
Some
by
of
various
for
the
the
n index,can
these
well
authors.
Equation
ql/q2
for
equations values
equation. earlier
general
force
=
(1)
92x2)
constant
activity
known
+
, and
(A/B)n,different coefficients,depen-
be
obtained
equations
from
have
been
Thus, lnY2
lnYl
.~.. 0
Margules
X:
[A +
2(~-A)
1
Van
x;A/[x2
+x1]'
lnV1/V2 lnA/B
Laar
Scatchard Hamer
0378-3812/85/$03.30
#;[A
+
V 2(B$
XJ
x;
[B
+
x;B/[xl
-A)fil] 2
0 1985 Elsevier SciencePublishers B.V.
@f[B
f
2(A-B)
x2]
++
x212
2(A>-B)fi2] 1
184 For
the
particular
value
of n = l/2,
Q=6 and
in
it
can
such
sus
be
a manner
great
is has
many
nary ge
positive
te
the
thirteen Ness(1973)
are
of
data
by
binary
(2)
. .
.
\/IQ1 x1/x2
when
are
and
shown
coefficients
the
excess
authors Gibbs
a wide
range
deviations
from
fitting
in
Gmehling,Onken
phase
to
to
results
by
the
systems
summarized
ver-
the
Table
I,
of
and
in
correlating
energy
behavior,
data
Byer,Gibbs
the
isobaric
for
the
Figure
of from
ideality.To
by
Schulte(lq80)
graphically
free
isothermal
measured
for
bila_r
illustra(3OPC) and
Van
data
system
of
(700
diethyl
1.
equations
differentiating
liquid
of
negative
obtained
ether-halotane
By
successfully
representing
large
nonideal
Activity
used
systems
results
line
reduced
plotted.
been
to
is
zizz seadly transformed
is
a straight
experimental
liquid
Hg)
that
that
dm Eq.(3)
mm
shown
Eq.(l)
the
activity
Eq.(Z),the
coefficients
following can
1 +
be
equations
for
the
obtained
(i_JAIEs)Xl
= Ax;
lnYl
cx2 I- x,G)3 (4) 1 -e (1-m)~~ = Bx2
lnY2
(Xl +
l where
the
vity
coefficients
From one to
constants
Eqs.(4)
hundred agree
the
Margules
milar
to
Van
tion.Unlike that
exhibit
B are
A =
1nYT
by applied
of
with
A and
binary
to
or the Laar Van
maxima
and
and or
to
vapor-liquid
data
and Wilson
minima
infinite-dilution
acti-
B = 1nYy. equilibrium
two quite
Scatchard-Hamer
equation
Laar
related
systems,this
experimental
~,a)~
parameter frequently
model better
equations.Their
correctly
predicts
equations,the in the
data
activity
model
of
was
shown
than
behavior azeotrope can
about
fit
coefficients.
either is
si-
formacurves This
185
DiETHYL
ETHER
l
- HALOTANE
EXPERIMENTAL
(700
mmHg)
(Gmchling
et al:
0.6-
0.t-
m
0.2-
0
Fig.
1.
= 10942
A3
= 0.0012
01
0.2
Graphical - halotane
TABLE I Correlation
of
0.3
0.4
0.5
excess
0.9
Gibbs
energy
data
by
= Tetrahydrofuran
acetate
ether
Eq.(3)
m
Carbontetrachloride?CHFa Chloroform-THF Dichloromethane'THF Carbontetrachloride'furan Chloroform-furan THF-furan Dichloromethane-methyl Dichloromethane-acetone Dichloromethane-1,4_dioxane Chloroform-1,4'dioxane Pyridine-acetone Pyridine-chloroform Pyridine-dichloromethane
THF
0.8
0.7
correlation of the binary system Diethyl Trifluoro-2-bromo-2-chloroethane). (l,l,l'
System
a
0.6
-0.4386 -1.2966 -0.9540 0.5182 -0.3541 -0.6006 -0.8289 -0.8672 -0.9810 -1.3008 0.4466 -0.8881 -0.6770
w
0.8457 1.0603 0.9676
0.9714 1.1828 0.9627 1.1989 1.1436 1.2309 1.4212 1.0259 0.7911 0.8815
Carrel. Coeff.
0.9993 0.9988 0.9995 1.0000 0.9989 0.9998 0.9970 0.9999 1.0000 0.9979 0.9998 0.9975 0.9999
186
condition
is
Calculation By
encountered
of
solving
when
l/2
o
or
&z>2.
parameters Eqs.(4)
\/B/A=
simultaneously,
x1(1
+
l/x2)lnYl
+
x21nY2
x2(1
+
l/xl)lnY2
+
xllnY1
(5)
(x1
+
&Zi)"lnY2
x2
(6)
B= x
from
which
parameters to
a
set
of
Parameter For partially
the
parameters
can
be
combining for
B
can
be
applying
data
coexisting miscible
and by
from
of
mutual
equilibrium
calculated.
the
linear
vapor-liquid
function
Eq.(3)
equilibrium.
solubility liquid
Likewise,the
data phases
(I
and
II)
of
a
pair, II
and y1
these
activity
II
z
11
Eqs.(iZ)
A
experimental
yIxl By
(1-a)x2]
obtained
prediction two
1 +
x1
II Y2x2
liquid-liquid
II =
Y2
II x2
equilibrium
coefficients,one
obtains
conditions the
following
with equa-
tions
(x:)"[r
+
(r-1)x:]
+ rx ; ) 3
(x;
(x:I)'[r -
(XC1
+
(r-l)xtI]
f
II rx2 )
+
(1-r)xt']
+
II 3 rx2 )
3 (7)
=
(xy[
1 +
(1-r)xz]
(xt')'[l -
+
(x;
B
=
(x;,2L r
(x;
where
r =\/B/A
,
rx ; ) 3
+
+
(,;I
(Xa1J2Lr+
(r_l)xz] rx ; ) 3
-
(x;I
+
(r-l~xI,T II 3 rx2 )
187 From
these
iterative
procedure
Even
if
tive
value
meters
equations
Eq.(7)
A and
B,calculated
obtaining
that
in Table
B can
result.
way
for
be
the The
entire
obtained.An from
\/B/A
solutions,only
suitable
in
A and
Eq. (7).
smaller
values
posi-
of
range
of
paramutual
II.
II
Parameters I x1
A and
II x1
0.01
0.05 0.20
0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 0.05
0.10 0.20
0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95
The
mutual
seen
that
tion.From
-Hamer
mutual
3.041 3.331 3.678 3.935 4.133 4.292 4.420 4.525 4.608 4.671 4.691 4.689
0.024 0.061 0.172 0.326 0.523 0.763 1.052 1.403 1.856 2.548 3.195 4.689
2.819
0.166
2.942 3.052 3.138 3.205 3.255 3.289 3.306 3.298 3.272
0.355 0.571. 0.809 1.067 1.351 II.674 2.073 2.684 3.272
of
the
solubilities the
model
a study
that
liquid
from
the phase
solubilities I xl
B
predictions
from
showed
B
A
0.10
gle
for
multiple
is the
shown
parameters
required
have
~/B/A
solubilities,are
TABLE
is
may
of
the
0.10
Van
Laar
regions
0.60 0.70 0.80 0.90 0.30 0.40 0.50 0.60 0.70 0.80 0.40 0.50 0.60
2.208
0.70 0.45 0.50 0.55 0.60
2.118 2.~89 2.136 2.082 2.027
0 .3'0
0.20
0.30
0.40
infinite-dilution are
shown
equation and
is
behaves superior
binary well to
equations.
0378-3812/85/$03.30
0.529 0.773 1.020 1.272 1.536 1.828 2.188 2.747 1.076 1.309 1.532 1.758 2.004 2.310 1.523 1.715 1.907 2.118 1.783 1.865 1.944 2.027
coefficients
III,where
behavior
aqueous
3
activity
in Table
a similar five
A
2.753 2.784 2.810 2.826 2.832 2.825 2.802 2.747 2.569 2.526 2.485 2.440 2.385 2.310 2.366 2.287
0.20
0.30 0.40
exhibits of these
II x1
0 1985 Elsevier SciencePublishers B.V.
to
it
Van
can
Laar
be equa-
systems,Brian(l965) over
Margules
the and
entire
sin-
Scatchard
188 TABLE
III
Infinite-dilution solubilities
of
activity aqueous
coefficients
binary
1
Aniline Tsobutyl l-B&an01 Phenol Propylene
T
CC
100 alcohol
oxide
90 90 43.4 36.3
I x1
0.01475 0.0213 0.0207 0.02105 0.166
from
mutual
systems
Mutual Solubilities Component
determined
Van Laar Equation II x1
0.628 0.4025 0.3640 0.2675 0.625
= 5
68.2 45.0 44.3 40.1 12.9
This
work
Eqs. (7)&(8)
m y2
Q) 5
y"2
4.04 2.52 2.32 1.94 5.97
63.0 37.6 36.3 30.6 12.6
3.32 1.88 1.72 1.41 5.79
NOTATION A
>B
g n r xyy AY ; fl
= parameters in Eqs.(4) = Gibbs free energy = index in fraction (A/B) =m in liquid and vapor phases = mole fraction = mean deviation in vapor phase mole fraction = effective volume fraction = activity coefficient = volume fraction = absolute value
Subscripts 1,2,i,j
= component
1,2,i,j
Superscripts E aD
= excess property = infinite dilution
REFERENCES Brian,P.L.T.,Ind. Eng. Chem. Fundam., 4 (1965) 100-101. 19 (1973) Byer, S.M., R.E. Gibbs and H.C. Van Ness, A.I.Ch.E.J., 245-2.51. J. Chem. Eng. Data, 25 Gmehling, J., U. Onken and H.W. Schulte, (1980) 29-32. Wohl K., Trans. Am. Inst. Chem. Engrs., 42 (1946) 215.
in The Netherlands
TWO-PARAMETER
LIQUID
ACTIVITY
MODEL
FOR
COEFFICIENTS
F. Mato
and
F.
183
CORRELATING
OF
BINARY
PHASE
SYSTEMS
A. Mato
Department of Chemical Engineering. (Spa in) 47005 Val ladolid
University
of Valladolid
ABSTRACT A new two-parameter model for the activity coefficients of binary liquid systems is derived from the general expansion of Wohl. The model admits curves that exhibit maxima or minima in the activity coefficients and predicts well the liquid-vapor equilibrium from a restricted information : azeotropic condition,activity coefficients at infinite dilution or mutual solubilities.
The
general
the
excess
for
a binary
Q =
free
and
order
composition
(2a 122122
-
f
3a1122:22
+
assumption
z
introducing
parameter on Wohl
derived n
the
in the
liquid
3a122z1Zi)(q1x1
By
ding
equation,relating phase
system,is
gERT
energy
third
= molecular
two
the
Gibbs
of Wohl(lq46)
= qixi/Xjqjxj , aij i = effective molar volume.
where 'i
form
the
binary assumed
Some
by
of
various
for
the
the
n index,can
these
well
authors.
Equation
ql/q2
for
equations values
equation. earlier
general
force
=
(1)
92x2)
constant
activity
known
+
, and
(A/B)n,different coefficients,depen-
be
obtained
equations
from
have
been
Thus, lnY2
lnYl
.~.. 0
Margules
X:
[A +
2(~-A)
1
Van
x;A/[x2
+x1]'
lnV1/V2 lnA/B
Laar
Scatchard Hamer
0378-3812/85/$03.30
#;[A
+
V 2(B$
XJ
x;
[B
+
x;B/[xl
-A)fil] 2
0 1985 Elsevier SciencePublishers B.V.
@f[B
f
2(A-B)
x2]
++
x212
2(A>-B)fi2] 1
184 For
the
particular
value
of n = l/2,
Q=6 and
in
it
can
such
sus
be
a manner
great
is has
many
nary ge
positive
te
the
thirteen Ness(1973)
are
of
data
by
binary
(2)
. .
.
\/IQ1 x1/x2
when
are
and
shown
coefficients
the
excess
authors Gibbs
a wide
range
deviations
from
fitting
in
Gmehling,Onken
phase
to
to
results
by
the
systems
summarized
ver-
the
Table
I,
of
and
in
correlating
energy
behavior,
data
Byer,Gibbs
the
isobaric
for
the
Figure
of from
ideality.To
by
Schulte(lq80)
graphically
free
isothermal
measured
for
bila_r
illustra(3OPC) and
Van
data
system
of
(700
diethyl
1.
equations
differentiating
liquid
of
negative
obtained
ether-halotane
By
successfully
representing
large
nonideal
Activity
used
systems
results
line
reduced
plotted.
been
to
is
zizz seadly transformed
is
a straight
experimental
liquid
Hg)
that
that
dm Eq.(3)
mm
shown
Eq.(l)
the
activity
Eq.(Z),the
coefficients
following can
1 +
be
equations
for
the
obtained
(i_JAIEs)Xl
= Ax;
lnYl
cx2 I- x,G)3 (4) 1 -e (1-m)~~ = Bx2
lnY2
(Xl +
l where
the
vity
coefficients
From one to
constants
Eqs.(4)
hundred agree
the
Margules
milar
to
Van
tion.Unlike that
exhibit
B are
A =
1nYT
by applied
of
with
A and
binary
to
or the Laar Van
maxima
and
and or
to
vapor-liquid
data
and Wilson
minima
infinite-dilution
acti-
B = 1nYy. equilibrium
two quite
Scatchard-Hamer
equation
Laar
related
systems,this
experimental
~,a)~
parameter frequently
model better
equations.Their
correctly
predicts
equations,the in the
data
activity
model
of
was
shown
than
behavior azeotrope can
about
fit
coefficients.
either is
si-
formacurves This
185
DiETHYL
ETHER
l
- HALOTANE
EXPERIMENTAL
(700
mmHg)
(Gmchling
et al:
0.6-
0.t-
m
0.2-
0
Fig.
1.
= 10942
A3
= 0.0012
01
0.2
Graphical - halotane
TABLE I Correlation
of
0.3
0.4
0.5
excess
0.9
Gibbs
energy
data
by
= Tetrahydrofuran
acetate
ether
Eq.(3)
m
Carbontetrachloride?CHFa Chloroform-THF Dichloromethane'THF Carbontetrachloride'furan Chloroform-furan THF-furan Dichloromethane-methyl Dichloromethane-acetone Dichloromethane-1,4_dioxane Chloroform-1,4'dioxane Pyridine-acetone Pyridine-chloroform Pyridine-dichloromethane
THF
0.8
0.7
correlation of the binary system Diethyl Trifluoro-2-bromo-2-chloroethane). (l,l,l'
System
a
0.6
-0.4386 -1.2966 -0.9540 0.5182 -0.3541 -0.6006 -0.8289 -0.8672 -0.9810 -1.3008 0.4466 -0.8881 -0.6770
w
0.8457 1.0603 0.9676
0.9714 1.1828 0.9627 1.1989 1.1436 1.2309 1.4212 1.0259 0.7911 0.8815
Carrel. Coeff.
0.9993 0.9988 0.9995 1.0000 0.9989 0.9998 0.9970 0.9999 1.0000 0.9979 0.9998 0.9975 0.9999
186
condition
is
Calculation By
encountered
of
solving
when
l/2
o
or
&z>2.
parameters Eqs.(4)
\/B/A=
simultaneously,
x1(1
+
l/x2)lnYl
+
x21nY2
x2(1
+
l/xl)lnY2
+
xllnY1
(5)
(x1
+
&Zi)"lnY2
x2
(6)
B= x
from
which
parameters to
a
set
of
Parameter For partially
the
parameters
can
be
combining for
B
can
be
applying
data
coexisting miscible
and by
from
of
mutual
equilibrium
calculated.
the
linear
vapor-liquid
function
Eq.(3)
equilibrium.
solubility liquid
Likewise,the
data phases
(I
and
II)
of
a
pair, II
and y1
these
activity
II
z
11
Eqs.(iZ)
A
experimental
yIxl By
(1-a)x2]
obtained
prediction two
1 +
x1
II Y2x2
liquid-liquid
II =
Y2
II x2
equilibrium
coefficients,one
obtains
conditions the
following
with equa-
tions
(x:)"[r
+
(r-1)x:]
+ rx ; ) 3
(x;
(x:I)'[r -
(XC1
+
(r-l)xtI]
f
II rx2 )
+
(1-r)xt']
+
II 3 rx2 )
3 (7)
=
(xy[
1 +
(1-r)xz]
(xt')'[l -
+
(x;
B
=
(x;,2L r
(x;
where
r =\/B/A
,
rx ; ) 3
+
+
(,;I
(Xa1J2Lr+
(r_l)xz] rx ; ) 3
-
(x;I
+
(r-l~xI,T II 3 rx2 )
187 From
these
iterative
procedure
Even
if
tive
value
meters
equations
Eq.(7)
A and
B,calculated
obtaining
that
in Table
B can
result.
way
for
be
the The
entire
obtained.An from
\/B/A
solutions,only
suitable
in
A and
Eq. (7).
smaller
values
posi-
of
range
of
paramutual
II.
II
Parameters I x1
A and
II x1
0.01
0.05 0.20
0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 0.05
0.10 0.20
0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95
The
mutual
seen
that
tion.From
-Hamer
mutual
3.041 3.331 3.678 3.935 4.133 4.292 4.420 4.525 4.608 4.671 4.691 4.689
0.024 0.061 0.172 0.326 0.523 0.763 1.052 1.403 1.856 2.548 3.195 4.689
2.819
0.166
2.942 3.052 3.138 3.205 3.255 3.289 3.306 3.298 3.272
0.355 0.571. 0.809 1.067 1.351 II.674 2.073 2.684 3.272
of
the
solubilities the
model
a study
that
liquid
from
the phase
solubilities I xl
B
predictions
from
showed
B
A
0.10
gle
for
multiple
is the
shown
parameters
required
have
~/B/A
solubilities,are
TABLE
is
may
of
the
0.10
Van
Laar
regions
0.60 0.70 0.80 0.90 0.30 0.40 0.50 0.60 0.70 0.80 0.40 0.50 0.60
2.208
0.70 0.45 0.50 0.55 0.60
2.118 2.~89 2.136 2.082 2.027
0 .3'0
0.20
0.30
0.40
infinite-dilution are
shown
equation and
is
behaves superior
binary well to
equations.
0378-3812/85/$03.30
0.529 0.773 1.020 1.272 1.536 1.828 2.188 2.747 1.076 1.309 1.532 1.758 2.004 2.310 1.523 1.715 1.907 2.118 1.783 1.865 1.944 2.027
coefficients
III,where
behavior
aqueous
3
activity
in Table
a similar five
A
2.753 2.784 2.810 2.826 2.832 2.825 2.802 2.747 2.569 2.526 2.485 2.440 2.385 2.310 2.366 2.287
0.20
0.30 0.40
exhibits of these
II x1
0 1985 Elsevier SciencePublishers B.V.
to
it
Van
can
Laar
be equa-
systems,Brian(l965) over
Margules
the and
entire
sin-
Scatchard
188 TABLE
III
Infinite-dilution solubilities
of
activity aqueous
coefficients
binary
1
Aniline Tsobutyl l-B&an01 Phenol Propylene
T
CC
100 alcohol
oxide
90 90 43.4 36.3
I x1
0.01475 0.0213 0.0207 0.02105 0.166
from
mutual
systems
Mutual Solubilities Component
determined
Van Laar Equation II x1
0.628 0.4025 0.3640 0.2675 0.625
= 5
68.2 45.0 44.3 40.1 12.9
This
work
Eqs. (7)&(8)
m y2
Q) 5
y"2
4.04 2.52 2.32 1.94 5.97
63.0 37.6 36.3 30.6 12.6
3.32 1.88 1.72 1.41 5.79
NOTATION A
>B
g n r xyy AY ; fl
= parameters in Eqs.(4) = Gibbs free energy = index in fraction (A/B) =m in liquid and vapor phases = mole fraction = mean deviation in vapor phase mole fraction = effective volume fraction = activity coefficient = volume fraction = absolute value
Subscripts 1,2,i,j
= component
1,2,i,j
Superscripts E aD
= excess property = infinite dilution
REFERENCES Brian,P.L.T.,Ind. Eng. Chem. Fundam., 4 (1965) 100-101. 19 (1973) Byer, S.M., R.E. Gibbs and H.C. Van Ness, A.I.Ch.E.J., 245-2.51. J. Chem. Eng. Data, 25 Gmehling, J., U. Onken and H.W. Schulte, (1980) 29-32. Wohl K., Trans. Am. Inst. Chem. Engrs., 42 (1946) 215.