CHEE 311 J.S. Parent 1
7. Liquid Phase Properties from VLE Data (11.1)
The fugacity of non-ideal liquid solutions is defined as:
(10.42)from which we derive the concept of an activity coefficient:
(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:
How do we calculate/measure these properties?
lii
li f̂lnRT)T()P,T(
lii
li
i fx
f̂
liiii
li fxlnRT)T()P,T(
CHEE 311 J.S. Parent 2
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
CHEE 311 J.S. Parent 3
Liquid Solution Fugacity from VLE Data
Our understanding of molecular dynamics does not permit us to predict non-ideal solution fugacity, fi
l . 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,
The vapour mixture fugacity for component i is given by,(10.47)
If we conduct VLE experiments at low pressure, but at the required temperature, we can use the perfect gas mixture model,
by assuming that iv = 1.
Pyˆf̂ ivi
vi
Pyf̂ ivi
vi
li f̂f̂
CHEE 311 J.S. Parent 4
Liquid Solution Fugacity from Low P VLE Data
Since our experimental measurements are taken at equilibrium,
according to the perfect gas mixture model
What we need is VLE data at various pressures (all relatively low)
Py
f̂f̂
i
vi
li
CHEE 311 J.S. Parent 5
Activity Coefficients from Low P VLE Data
With a knowledge of the liquid solution fugacity, we can derive activity coefficients. Actual fugacity
Ideal solution fugacityOur low pressure vapour fugacity simplifies fi
l to:
and if P is close to Pisat:
leaving us with
sati
sati
lsati
sati
li
P
RT
)PP(VexpPf
i
lii
li
i fx
f̂
lii
ii fx
Py
satii
ii Px
Py
CHEE 311 J.S. Parent 6
Activity Coefficients from Low P VLE Data
Our low pressure VLE data can now be processed to yield experimental activity coefficient data:
satii
ii Px
Py
CHEE 311 J.S. Parent 7
Activity Coefficients from Low P VLE Data
CHEE 311 J.S. Parent 8
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 J.S. Parent 9
Correlation of Liquid Phase Data
Recall our development of activity coefficients on the basis of the partial excess Gibbs energy :
where the partial molar Gibbs energy of the non-ideal model is provided by equation 10.42:
and the ideal solution chemical potential is:
Leaving us with the partial excess Gibbs energy:
(10.90)
idii
Ei GGG
li
li ii
f̂lnRT)T(G
liii
idi
idi fxlnRT)T(G
i
lii
li
lii
li
Ei
lnRT
fx
f̂lnRT
fxlnRTf̂lnRTG
CHEE 311 J.S. Parent 10
Correlation of Liquid Phase Data
The partial excess Gibbs energy is defined by:
In terms of the activity coefficient,
(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)
With this information, they can generate model equations that practicing engineers apply routinely.
nj,P,Ti
E
i n)RT/nG(
ln
i
ii
E
lnxRTG
nj,P,Ti
EEi
n)nG(
G
CHEE 311 J.S. Parent 11
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 = x1ln1 + x2ln2
CHEE 311 J.S. Parent 12
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
sat11
11 Px
Pylnln
sat22
22 Px
Pylnln
2211E lnxlnxRT/G
CHEE 311 J.S. Parent 13
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 J.S. Parent 14
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.
CHEE 311 J.S. Parent 15
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:
Equations of this form “fit” excess Gibbs energy data quite well. However, they are empirical and cannot be generalized for multi-component (3+) mixtures or temperature.
)Tttancons()xx(D)xx(CBxRTx
G 22121
21
E
CHEE 311 J.S. Parent 16
Symmetric Equation for Binary Mixtures
The simplest Redlich/Kister expansion results from C=D=…=0
To calculate activity coefficients, we express GE in terms of moles: n1 and n2.
And through differentiation,
we find:
BxRTx
G
21
E
221
21E
)nn(
nnBRTnG
2n,P,T1
E
1 n)RT/nG(
ln
212
221 BxlnandBxln
CHEE 311 J.S. Parent 17
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 multi-
component behaviour can be predicted from binary data.
CHEE 311 J.S. Parent 18
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.
sat11
11 Px
Pylnln
sat22
22 Px
Pylnln
2211E lnxlnxRT/G
CHEE 311 J.S. Parent 19
Margule’s Equations
While the simplest Redlich/Kister-type expansion is the Symmetric Equation, a more accurate model is the Margule’s expression:
(11.7a)
Note that as x1 goes to zero,
and from L’hopital’s rule we know:
therefore,
and similarly
21212121
E
xAxAxRTx
G
1
210xln
xRTxG
lim
E
1
12
0x21
E
AxRTx
G
1
112 lnA 221 lnA
CHEE 311 J.S. Parent 20
Margule’s Equations
If you have Margule’s parameters, the activity coefficients are easily derived from the excess Gibbs energy expression:
(11.7a)
to yield:
(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.
21212121
E
xAxAxRTx
G
]x)AA(2A[xln 1122112221
]x)AA(2A[xln 2211221212
CHEE 311 J.S. Parent 21
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:
(11.13)for the excess Gibbs energy and:
(11.14)
(11.15)
for the activity coefficients.
Note that: as x10, ln1 A’12
and as x2 0, ln2 A’21
2/121
/21
/21
/12
21
E
xAxA
AAxRTx
G
2
2/21
1/12/
121xA
xA1Aln
2
1/12
2/21/
212xA
xA1Aln
CHEE 311 J.S. Parent 22
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 A,B A,C B,C
tertiary mixture binary binary binary
CHEE 311 J.S. Parent 23
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:
(11.16)
where(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.
)xxln(x)xxln(xRTG
2112212211
E
RTa
expVV
RTa
expVV 21
2
121
12
1
212
CHEE 311 J.S. Parent 24
Wilson’s Equations for Binary Solution Activity
Activity coefficients are derived from the excess Gibbs energy using the definition of a partial molar property:
When applied to equation 11.16, we obtain:
(11.17)
(11.18)
2112
21
1221
12212211 xxxx
x)xxln(ln
2112
21
1221
12121122 xxxx
x)xxln(ln
jn,P,Ti
EEii n
nGGlnRT
CHEE 311 J.S. Parent 25
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:
(11.22)
and the activity coefficients become:
(11.23)
where ij = 1 for i=j. Summations are over all species.
i j
ijji
E
xlnxRTG
k
jkjj
kik
iijji x
xxln1ln
CHEE 311 J.S. Parent 26
Wilson’s Equations for 3-Component Mixtures
For three component systems, activity coefficients can be calculated from the following relationship:
Model coefficients are defined as (ij = 1 for i=j):
3322311
i33
2332211
i22
1331221
i113i32i21i1i
xxx
x
xxx
x
xxx
x)xxxln(1ln
RT
aexp
V
V ij
i
jij
CHEE 311 J.S. Parent 27
Comparison of Liquid Solution Models
Activity coefficients of 2-methyl-2-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.