7. Liquid Phase Properties from VLE Data (11.1)
The fugacity of non-ideal liquid solutions is defined as:
li (T,P)  i (T )  RT ln fˆil
(10.42)
from which we derive the concept of an activity coefficient:
fˆil
i 
xi fil
(10.89)
that is a measure of the departure of the component behaviour
from an ideal solution.
Using the activity coefficient, equation 10.42 becomes:
li (T,P)  i (T)  RT ln  i xifil
How do we calculate/measure these properties?
CHEE 311
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Liquid Phase Properties from VLE Data
Suppose we conduct VLE experiments on our system of interest.
 At a given temperature, we vary the system pressure by
changing the cell volume.
 Wait until equilibrium is established (usually hours)
 Measure the compositions of the liquid and vapour
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Liquid Solution Fugacity from VLE Data
Our understanding of molecular dynamics does not permit us to
predict non-ideal solution fugacity, fil . We must measure them by
experiment, often by studies of vapour-liquid equilibria.
Suppose we need liquid solution fugacity data for a binary mixture
of A+B at P,T. At equilibrium,
fˆil  fˆiv
The vapour mixture fugacity for component i is given by,
fˆiv  ˆ iv yiP
(10.47)
If we conduct VLE experiments at low pressure, but at the required
temperature, we can use the perfect gas mixture model,
fˆiv  yiP
by assuming that iv = 1.
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Liquid Solution Fugacity from Low P VLE Data
Since our experimental measurements are taken at equilibrium,
fˆil  fˆiv
according to the perfect gas
 yiP
mixture model
What we need is VLE data at various pressures (all relatively low)
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Activity Coefficients from Low P VLE Data
With a knowledge of the liquid solution fugacity, we can derive
activity coefficients.
ˆl Actual fugacity
fi
i 
xi fil Ideal solution fugacity
Our low pressure vapour fugacity simplifies fil to:
i 
and if P is close to Pisat:
y iP
x i fil
l
sat

V
(
P

P
)
i
i
l
sat sat
fi  i Pi exp 

RT


 Pisat
leaving us with
i 
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y iP
x i Pisat
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Activity Coefficients from Low P VLE Data
Our low pressure VLE data can now be processed to yield
experimental activity coefficient data:
i 
CHEE 311
y iP
x i Pisat
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Activity Coefficients from Low P VLE Data
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7. Correlation of Liquid Phase Data
The complexity of molecular interactions in non-ideal systems
makes prediction of liquid phase properties very difficult.
 Experimentation on the system of interest at the conditions
(P,T,composition) of interest is needed.
 Previously, we discussed the use of low-pressure VLE data
for the calculation of liquid phase activity coefficients.
As practicing engineers, you will rarely have the time to conduct
your own experiments.
 You must rely on correlations of data developed by other
researchers.
 These correlations are empirical models (with limited
fundamental basis) that reduce experimental data to a
mathematical equation.
In CHEE 311, we examine BOTH the development of empirical
models (thermodynamicists) and their applications (engineering
practice).
CHEE 311
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Correlation of Liquid Phase Data
Recall our development of activity coefficients on the basis of the
partial excess Gibbs energy :
E
id
Gi  Gi  Gi
where the partial molar Gibbs energy of the non-ideal model is
provided by equation 10.42:
Gi  li  i (T )  RT ln fˆil
and the ideal solutionidchemical potential is:
l
Gi  id


(
T
)

RT
ln
x
f
i
i i
i
Leaving us with the partial excess Gibbs energy:
E
Gi  RT ln fˆil  RT ln x i fil
fˆil
 RT ln l
x i fi
 RT ln  i
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(10.90)
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Correlation of Liquid Phase Data
The partial excess Gibbs energy is defined by:
E
Gi
(nGE )

ni T,P,nj
In terms of the activity coefficient,
(nGE / RT )
ln  i 
ni
T,P,nj
(10.94)
Therefore, if as practicing engineers we have GE as a function of
P,T, xn (usually in the form of a model equation) we can derive i.
Conversely, if thermodynamicists measure i, they can calculate GE
using the summability relationship for partial properties.
(10.97)
GE
  xi ln  i
With this information, they
RT cani generate model equations that
practicing engineers apply routinely.
CHEE 311
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Correlation of Liquid Phase Data
We can now process this our MEK/toluene data one step further to
give the excess Gibbs energy,
GE/RT = x1ln1 + x2ln2
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Correlation of Liquid Phase Data
Note that GE/(RTx1x2) is reasonably represented by a linear
function of x1 for this system. This is the foundation for correlating
experimental activity coefficient data
 yP 
ln  1  ln 1 sat 
 x1P1 
 y P 
ln  2  ln 2 sat 
 x 2P2 
GE / RT  x1 ln 1  x 2 ln  2
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Correlation of Liquid Phase Data
The chloroform/1,4-dioxane system
exhibits a negative deviation from
Raoult’s Law.
This low pressure VLE data can be
processed in the same manner as the
MEK/toluene system to yield both
activity coefficients and the excess
Gibbs energy of the overall system.
CHEE 311
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Correlation of Liquid Phase Data
Note that in this example, the activity
coefficients are less than one, and
the excess Gibbs energy is negative.
In spite of the obvious difference
from the MEK/toluene system
behaviour, the plot of GE/x1x2RT is
well approximated by a line.
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8.4 Models for the Excess Gibbs Energy
Models that represent the excess Gibbs energy have several
purposes:
 they reduce experimental data down to a few parameters
 they facilitate computerized calculation of liquid phase
properties by providing equations from tabulated data
 In some cases, we can use binary data (A-B, A-C, B-C) to
calculate the properties of multi-component mixtures (A,B,C)
A series of GE equations is derived from the Redlich/Kister
expansion:
GE
 B  C( x1  x 2 )  D( x1  x 2 )2
(cons tan t T )
RTx 1x 2
Equations of this form “fit” excess Gibbs energy data quite well.
However, they are empirical and cannot be generalized for multicomponent (3+) mixtures or temperature.
CHEE 311
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Symmetric Equation for Binary Mixtures
The simplest Redlich/Kister expansion results from C=D=…=0
GE
B
RTx 1x 2
To calculate activity coefficients, we express GE in terms of moles:
n1 and n2.
nGE
B n1n2

RT (n1  n2 )2
And through differentiation,
(nGE / RT )
ln 1 
n1
T,P,n2
we find:
CHEE 311
ln 1  Bx 22
and
J.S. Parent
ln  2  Bx12
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7. Excess Gibbs Energy Models
Practicing engineers find most of the liquid-phase information
needed for equilibrium calculations in the form of excess Gibbs
Energy models. These models:
 reduce vast quantities of experimental data into a few
empirical parameters,
 provide information an equation format that can be used in
thermodynamic simulation packages (Provision)
“Simple” empirical models
 Symmetric, Margule’s, vanLaar
 No fundamental basis but easy to use
 Parameters apply to a given temperature, and the models
usually cannot be extended beyond binary systems.
Local composition models
 Wilsons, NRTL, Uniquac
 Some fundamental basis
 Parameters are temperature dependent, and multicomponent behaviour can be predicted from binary data.
CHEE 311
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Excess Gibbs Energy Models
Our objectives are to learn how to fit Excess Gibbs Energy models
to experimental data, and to learn how to use these models to
calculate activity coefficients.
 yP 
ln  1  ln 1 sat 
 x1P1 
 y P 
ln  2  ln 2 sat 
 x 2P2 
GE / RT  x1 ln 1  x 2 ln  2
CHEE 311
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Margule’s Equations
While the simplest Redlich/Kister-type expansion is the Symmetric
Equation, a more accurate
model is the Margule’s expression:
GE
 A 21x1  A12 x 2
RTx 1x 2
(11.7a)
Note that as x1 goes to zero,
E
G
RTx1x 2
 A 12
x1 0
and from L’hopital’s rule Ewe know:
G


ln

lim
1
RTx
x
x10
1 2
therefore,
CHEE 311
A12  ln 1
and similarly
J.S. Parent
A 21  ln  2
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Margule’s Equations
If you have Margule’s parameters, the activity coefficients are
easily derived from the Eexcess Gibbs energy expression:
G
 A 21x1  A12 x 2
RTx 1x 2
(11.7a)
to yield:
ln 1  x 22 [ A12  2( A 21  A12 )x1]
ln  2  x12 [ A 21  2( A12  A 21)x 2 ]
(11.8ab)
These empirical equations are widely used to describe binary
solutions. A knowledge of A12 and A21 at the given T is all we
require to calculate activity coefficients for a given solution
composition.
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van Laar Equations
Another two-parameter excess Gibbs energy model is developed
from an expansion of (RTx1x2)/GE instead of GE/RTx1x2. The end
/
/
results are:
GE
A12
A 21
 /
/
RTx 1x 2 A 21x1  A12
x2
(11.13)
for the excess Gibbs energy and:
ln 1 
/ 
A12 1 
ln  2 
/ 
A 211 

for the activity coefficients. 
2
/
A12 x1 

/
A 21x 2 
2
/
A 21x 2 

/
A12 x1 
Note that:
as x10, ln1  A’12
and
as x2  0, ln2  A’21
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(11.14)
(11.15)
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Local Composition Models
Unfortunately, the previous approach cannot be extended to
systems of 3 or more components. For these cases, local
composition models are used to represent multi-component
systems.
 Wilson’s Theory
 Non-Random-Two-Liquid Theory (NRTL)
 Universal Quasichemical Theory (Uniquac)
While more complex, these models have two advantages:
 the model parameters are temperature dependent
 the activity coefficients of species in multi-component liquids
can be calculated from binary data.
A,B,C
tertiary mixture
CHEE 311
A,B
A,C
binary
binary
J.S. Parent
B,C
binary
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Wilson’s Equations for Binary Solution Activity
A versatile and reasonably accurate model of excess Gibbs Energy
was developed by Wilson in 1964. For a binary system, GE is
provided by: E
G
 x1 ln( x1  x 2 12 )  x 2 ln( x 2  x1 21)
(11.16)
RT
where
12 
V2
 a 
exp  12 
V1
 RT 
 21 
V1
 a 
exp  21 
V2
 RT 
(11.24)
Vi is the molar volume at T of the pure component i.
aij is determined from experimental data.
The notation varies greatly between publications. This includes,
 a12 = (12 - 11), a12 = (21 - 22) that you will encounter in
Holmes, M.J. and M.V. Winkle (1970) Ind. Eng. Chem. 62,
21-21.
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Wilson’s Equations for Binary Solution Activity
Activity coefficients are derived from the excess Gibbs energy
using the definition of a partial molar property:
RT ln  i 
E
Gi
nGE

ni T,P,n
j
When applied to equation 11.16, we obtain:


12
 21
ln 1   ln( x1  x 2 12 )  x 2 


x

x

x

x

 1
2 12
2
1 21 


12
 21
ln  2   ln( x 2  x1 21)  x1


 x1  x 2 12 x 2  x1 21 
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(11.17)
(11.18)
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Wilson’s Equations for Multi-Component Mixtures
The strength of Wilson’s approach resides in its ability to describe
multi-component (3+) mixtures using binary data.
 Experimental data of the mixture of interest (ie. acetone,
ethanol, benzene) is not required
 We only need data (or parameters) for acetone-ethanol,
acetone-benzene and ethanol-benzene mixtures
The excess Gibbs energy
is written:
GE
  x i ln  x j  ij
RT
i
j
(11.22)
and the activity coefficients become:
x 
ln  i  1  ln  x j ij   k ki
i
k  x j  kj
(11.23)
j
where ij = 1 for i=j. Summations are over all species.
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Wilson’s Equations for 3-Component Mixtures
For three component systems, activity coefficients can be
calculated from the following relationship:
x
ln  i  1  ln( x1  i1  x 2  i2  x 3  i3 ) 
1
1i
x1  x 2 12  x 3 13
x 2  2i

x1 21  x 2  x 3  23
x 3  3i

x1 31  x 2  32  x 3
Model coefficients are defined as (ij = 1 for i=j):
  aij 
ij  exp 

Vi
 RT 
Vj
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Comparison of Liquid Solution Models
Activity coefficients of 2-methyl2-butene + n-methylpyrollidone.
Comparison of experimental
values with those obtained from
several equations whose
parameters are found from the
infinite-dilution activity
coefficients.
(1) Experimental data.
(2) Margules equation.
(3) van Laar equation.
(4) Scatchard-Hamer equation.
(5) Wilson equation.
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