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Fundamental equations of solution thermodynamics are given in
the preceding chapter.
In this chapter, experimental vapor/liquid equilibrium (VLE)
data are considered, from which the activity coefficient
correlations are derived.
2
OVERVIEW
4
LIQUID PHASE PROPERTIES FROM VLE DATA
The figure shows coexistence of a vapor mixture and liquid
solution in vapor/liquid equilibrium. T and P are uniform
throughout the vessel and can be measured with appropriate
instruments. Vapor and liquid samples may be withdrawn for analysis
and this provides experimental values for mole fractions in the
vapor {yi} and mole fractions in the liquid {xi}.
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For species i in the vapor mixture, eq. (11.52) is written
as
The criterion of vapor/liquid equilibrium, as given by eq.
(11.48), is that . Therefore,
VLE measurements are made at low pressure (P 1 bar) that the
vapor phase may be assumed an ideal gas. In this case .
Therefore,
The fugacity of species i (in both the liquid and vapor phase)
is equal to the partial pressure of species i in the vapor
phase.
5
Fugacity
l vi i if y P
l vi i if f y P
v vi i if y P
l vi if f
1vi
ii
i
f
y P (11.52)
Fugacity increases from zero at infinite dilution (xi = yi 0) to
Pisat for pure
species i. This is illustrated by the data of Table 12.1 for
methyl ethyl ketone(1)/toluene(2) at 50oC (323.15K). The first
three columns list a set of experimental P-x1-y1 data and columns 4
and 5 show
6
1 1 2 2 andf y P f y P
ii sat
i i
y P
x P
Eqn. (12.1)
The fugacities are plotted in Fig. 12.2 as solid lines. The
straight dashed lines represent the Lewis/Randall rule expressing
the composition dependence of the constituent fugacities in an
ideal solution,
This figure illustrates the general nature of relationships for
a binary liquid solution at constant T.
idi i if x f
7
idi i if x f
1 2 1 and vs. f f x
7
The equilibrium pressure P varies with compositions but has
negligible influence on the liquid phase values of
Therefore, a plot at constant T and P is as shown in Fig. 12.3
for species i (i = 1, 2) in a binary solution at constant T and
P.
8
1 2 and .f f
8
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The lower dashed line in Fig. 12.3 represent the Lewis/Randall
rule, which characterize ideal solution behavior.
Activity coefficient as defined by eq. (11.90) provide the
actual behavior (non-ideal) from the idealize one:
The activity coefficient of a species in solution is the ratio
of its actual fugacity to the value given by Lewis/Randall rule at
the same T, P and composition. For the calculation of experimental
values, both are eliminated in favor of measurable quantities:
This is restatement of eq. (10.5), modified Raoults law, and
allow calculation of activity coefficient from VLE data as shown in
the last two columns of Table 12.1.
i i
i idi i i
f f
x f f
Activity coefficient
1 2i ii sati i i i
y P y P (i , , ....N)
x f x P
and idi if f
(12.1)
9
The solid lines in Fig. 12.2 and 12.3, representing experimental
values of , become tangent to the Lewis/Randall rule lines at xi =
1.
In the other limit, becomes zero. The ratio is indeterminate in
this limit, and application of lHopital rule yields
Eq. (12.2) defines Henrys constant, Hi as the limiting slope of
the
curve at xi = 0.
As shown by Fig. 12.3, this is the slope of a line drawn tangent
to the curve at xi = 0. The equation of this tangent line expresses
Henrys law:
Henrys law as given by eq. (10.4) follows immediately from this
equation when i.e. when has its ideal gas value.
10
if
0,i ix fi if x
00
limi
i
i ii
xi i x
f df
x dx
(12.2)
vs i if x
i i if x (12.3)
,i if y P if
10.4i i iy P x H
Henrys law is related to the Lewis/Randal rule through the
Gibbs/Duhem equation.
Writing eq. (11.14) for a binary solution and replacing
Differentiate eq. (11.46) at constant T and P yields:
Therefore,
Divide by dx1 becomes
This is the special form of the Gibbs/Duhem equation.
11
1 21 21 1
ln ln0 const T,P
d f d fx x
dx dx (12.4)
byi i iM G
0i ii
x d M
(11.14) 1 1 2 2 0 const , x d x d T P
lni i iT RT f (11.46)
lni id RTd f
1 1 2 2 ln ln 0 const , x d f x d f T P
Substitute dx1 by dx2 in the second term produces
In the limit as x1 1 and x2 0,
Because when x1 = 1, this may be rewritten as
According to eq. (12.2), the numerator and denominator on the
right side of this equation are equal.
12
1 1 2 21 21 2
1 2 1 1 2 2
ln lnor
d f dx d f dxd f d fx x
dx dx f x f x
1 2
1 1 2 2
1 01 1 2 2
lim lim
x x
d f dx d f dx
f x f x
1 1f f
2
12
2 201
1 1 2 210
1
lim
x
xx
d f dxdf
f dx f x
00
limi
i
i i
xi i x
f df
x dx
(12.2)
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Therefore the exact expression of Lewis/Randall rule applied to
real solutions is:
It also implies that eq. (11.83) provides approximately correct
values of when xi 1:
13
Henrys law applies to a species as it approaches infinite
dilution in a
binary solution, and the Gibbs/Duhem equation insures validity
of
the Lewis/Randall rule for the other species as it approaches
purity.
1
11
1 1
x
dff
dx
(12.5)
if
idi i i if f x f
14
When the second species is methanol,
acetone exhibits positive deviations
from ideality and with chloroform it
exhibits negative deviations.
The fugacity of pure acetone, facetone
is the same regardless of the identity
of the second species.
Henrys constants are represented by
the slopes of the two dotted lines.
For a binary system,
15
Excess Gibbs Energy
1 1 2 2ln lnEG
x xRT
(12.6)
Properties
of liquid
phase
ii sat
i i
y P
x P
Experiment
al data
16
Activity coefficient of a species in solution becomes unity as
the species becomes pure,
each ln i (i=1,2) tends to zero as xi 1.
As xi 0, species i becomes infinitely dilute, ln i approaches a
finite limit, namely ln i.
For i 1 and ln i 0
P-x1 data points all lie
above dash line (Raoults
law) positive deviation.
Raoults law
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The most important step in determining composition of species in
solution (that is in equilibrium with vapor) is to calculate the
species activity coefficients. There are a number of models that
can be used to determine the activity coefficients depending on the
type of species.
Excess Gibbs free energy model e.g. Margules and Van Laar
Local composition model e.g. Wilson, NRTL and UNIQUAC
17
18
Model Equations
Margules
Van Laar
Wilson
NRTL
21 2 12 21 12 1ln 2x A A A x
22 1 21 12 21 2ln 2x A A A x
(12.10a)
(12.10b)
21 1 12 2
1 2
EGA x A x
x x RT (12.9a)
' '
12 21
' '
1 2 12 1 21 2
EG A A
x x RT A x A x
(12.16)
2'
' 12 11 12 '
21 2
ln 1A x
AA x
2'
' 21 22 21 '
12 1
ln 1A x
AA x
(12.17a)
(12.17b)
12 211 1 2 12 21 2 12 2 1 21
ln ln x x xx x x x
1 1 2 12 2 2 1 21ln ln
EGx x x x x x
RT
12 212 2 1 21 11 2 12 2 1 21
ln ln x x xx x x x
21 21 12 12
1 2 1 2 21 2 1 12
EG G G
x x RT x x G x x G
2
2 21 12 121 2 21 2
1 2 21 2 1 12
lnG G
xx x G x x G
2
2 12 21 212 1 12 2
2 1 12 1 2 21
lnG G
xx x G x x G
exp (i j)j ij
ij
i
V a
V RT
12 12 21 21
12 2112 21
exp( ) exp( )
G G
b b
RT RT
(12.18)
(12.24)
(12.19a)
(12.19b)
(12.20) (12.21a)
(12.21b)
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Set of points in Fig. 12.5(b) provide linear relation for
GE/x1x2RT
where A21 and A12 are constant.
Alternatively,
From eqn. (12.10a) & (12.10b), the limiting conditions of
infinite dilution (at xi = 0)
21 1 12 2 1 2EG
A x A x x xRT
20
Margules Equation
21 1 12 2
1 2
EGA x A x
x x RT (12.9a)
(12.9b)
1 12 1 2 21 2ln at 0 and ln at 0A x A x
Fig. 12.5(b)
A12 = 0.372
A21 = 0.198
From Fig. 12.5(b), the intercepts
at x1 = 0 and x1 =1 of the
straight line GE/x1x2RT gives the
parameters A12 and A21.
21 2 12 21 12 1ln 2x A A A x
22 1 21 12 21 2ln 2x A A A x
(12.10a)
(12.10b)
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Linear relation for GE/x1x2RT
where A21 and A12 are constant.
Alternatively,
From eqn. (12.17a) & (12.17b), the limiting conditions of
infinite dilution
(at xi = 0)
Values of A12 and A21 are obtained from the intercept of plot
GE/x1x2RT vs. x1 at
x1 = 0 and x1=1. Or from plot of x1x2RT/GE vs. x1, intercept at
x1=0 is 1/A12 and
at x1=1 is 1/A21.
Van Laar Equation
21
' '
12 21
' '
1 2 12 1 21 2
EG A A
x x RT A x A x
(12.16)
' '
1 2 12 1 21 2 1 2
' ' ' '
12 21 21 12
E
x x RT A x A x x x
G A A A A
2'
' 12 11 12 '
21 2
ln 1A x
AA x
2'
' 21 22 21 '
12 1
ln 1A x
AA x
(12.17a) (12.17b)
1 12 1 2 21 2ln ' at 0 and ln ' at 0A x A x
22
Wilson Equation
12 211 1 2 12 21 2 12 2 1 21
ln ln x x xx x x x
1 1 2 12 2 2 1 21ln lnEG
x x x x x xRT
12 212 2 1 21 11 2 12 2 1 21
ln ln x x xx x x x
exp (i j)j ij
ij
i
V a
V RT
(12.18)
(12.24)
(12.19a)
(12.19b)
The limiting conditions of infinite dilution (at xi = 0),
1 12 21 1 2 21 12 2ln ln 1 at 0 and ln ln 1 at 0x x
23
NRTL Equation
21 21 12 12
1 2 1 2 21 2 1 12
EG G G
x x RT x x G x x G
2
2 21 12 121 2 21 2
1 2 21 2 1 12
lnG G
xx x G x x G
2
2 12 21 212 1 12 2
2 1 12 1 2 21
lnG G
xx x G x x G
12 12 21 21
12 2112 21
exp( ) exp( )
G G
b b
RT RT
(12.20)
(12.21a)
(12.21b)
1 21 12 12 2 12 21 21ln exp( ) and ln exp( )
The limiting conditions of infinite dilution (at xi = 0),
All models provide eqns. for ln 1 and ln 2. This allow
construction of a correlation of the original P-x1-y1 data set
(experimental values). Eq. (12.1) is rearranged to give
Addition gives
From eqn. (12.1), therefore
Values of 1 and 2 from all models with their parameters are
combined with experimental values of P1
sat and P2sat to calculate P and y1 by eqs. (12.11)
and (12.12) at various x1.
Then, P-x1-y1 diagram can be plotted to compare the experimental
data and calculated values.
1 1 1 1 2 2 2 2andsat saty P x P y P x P
1 1 1 2 2 2
sat satP x P x P (12.11)
1 1 1 1 1 11
1 1 1 2 2 2
sat sat
sat sat
x P x Py
P x P x P
(12.12)
i ii sat
i i i i
y P y P
x f x P
(12.1)
24 24
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Comparison of
experimental and
calculated data by
Margules eqn. These
clearly provide an
adequate
correlation of the
experimental data
points.
experimental
calculated
Fig. 12.5(a)
The Gibbs Duhem eqn imposes a constraint on activity
coefficients that may not be satisfied by a set of experimental
values derived from P-x1-y1 data.
The Gibbs Duhem eqn is implicit in eq. (11.96), and activity
coefficients derived from this equation necessarily obey the Gibbs
Duhem eqn.
These derived activity coefficients cannot possibly be
consistent with the experimental values unless the experimental
values also satisfy the Gibbs Duhem eqn.
26
Thermodynamic Consistency
1 21 21 1
ln ln0 const T,P
d dx x
dx dx
(12.7)
, ,
ln
j
E
i
iP T n
nG RT
n
(11.96)
If the experimental data are inconsistent with the Gibbs Duhem
eqn, they are necessarily incorrect as the result of systematic
error in the data.
Therefore simple test is develop for the consistency with
respect to the Gibbs Duhem eqn of a P-x1-y1 data set.
Application of the test for consistency is represented by Eq.
(12.13) which requires calculation of the residuals
The right side of this equation is exactly the quantity that eq.
(12.7), the Gibbs/Duhem equation, requires to be zero for
consistent data.
The residual on the left therefore provides a direct measure of
deviation from the Gibbs/Duhem equation.
The extent to which a data set departs from consistency is
measured by the degree to which these residuals fail to scatter
about zero.
27
* *
1 1 21 2
2 1 1
ln lnln
d dx x
dx dx
(12.13)
Asterisk * denote the experimental values
1 21 21 1
ln ln0 const T,P
d dx x
dx dx
(12.7)
28
Experimental
values
calculated from
eqn. (12.1) &
(12.6)
1 2i ii sati i i i
y P y P (i , , ....N)
x f x P (12.1)
1 1 2 2ln lnEG
x xRT
(12.6)
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P-x1-y1 data and experimental
values, ln 1*, ln 2
* and (G
E/x1x2RT )*
are shown as points on Figs. 12.7(a)
and 12.7(b) .
The data points of Fig. 12.7(b) for
(GE/x1x2RT )* show scatter. The
straight line drawn is represented by
This is eq. (12.9a) with A21 = 0.70
and A12 = 1.35.
Values of ln 1, ln 2 at the given
values of x1, derived from this eqn,
are calculated by eqs. (12.10) and
derived values of P and y1 at the
same values of x1 come from eqs.
(12.11) and (12.12).
This results are plotted as the solid
lines of Fig. 12.7(a) and 12.7(b).
They clearly do not represent a good
correlation of the data.
1 2
1 2
0.70 1.35EG
x xx x RT
21 2 12 21 12 1ln 2x A A A x (12.10a)
22 1 21 12 21 2ln 2x A A A x (12.10b)
1 1 1 2 2 2
sat satP x P x P (12.11) 1 1 11
1 1 1 2 2 2
sat
sat sat
x Py
x P x P
(12.12)
experimental
calculated
Barkers method
30
Application of test for consistency
represented by Eq. 12.13 requires
calculation of the residuals
which plotted vs. x1 in Fig. 12.8.
The residuals distribute
themselves about zero, as required by
the test, but the residual , do
not, which show the data fail to satisfy
the Gibbs/Duhem eqn.
1 2 and lnEG RT
EG RT
1 2ln
Smith, J.M., Van Ness, H.C., and Abbott, M.M. 2005. Introduction
to Chemical Engineering Thermodynamics. Seventh Edition. Mc
Graw-Hill.
31
REFERENCE
PREPARED BY: NORASMAH MOHAMMED MANSHOR FACULTY OF CHEMICAL
ENGINEERING, UiTM SHAH ALAM. [email protected]
03-55436333/019-2368303