Vapor-Liquid Equilibrium for a Ternary System
December 3, 2014
Group # 11 Group Leader: Chase Kairdolf Oral Reporter: Sami Marchand
Written Reporter: Tiffany Robinson
Instructor: Dr. Elizabeth Melvin
EXECUTIVE SUMMARY
Objective: The objective of this experiment is to determine a method to remove isopropanol
from a water/ethanol/isopropanol system by applying the Van Laar thermodynamic model and
knowledge of the physical properties of the mixture.
Experiment: An Othmer Still was used to obtain vapor-liquid equilibrium data for the ternary
system. Equilibrium was established by creating a closed-loop system which recirculates the
condensate and accomplishing equivalent compositions in both the liquid and vapor phases.
When the temperature remained constant for fifteen minutes in the vapor phase and circulation
was achieved, it ensured the system had reached equilibrium. Samples of the mixture were
injected into a gas chromatograph and an analysis of the sketched peaks was used to
determine the mass fraction of each component contained in the mixture.
Results and Discussion: The Van Laar equation for ternary systems, based on the van der
Waals equation of state, was used to model the system. Extending the Wohl expansion and
neglecting all third and higher order terms, the Van Laar model is then used for this ternary
system. To find the theoretical fitting parameters in the Van Laar equation for ternary systems,
an infinite dilution assumption for a binary mixture must be made. Experimentally, the activity
coefficients were calculated from a modified form of Raoult’s Law for non-ideal mixtures. Once
the activity coefficients were obtained, the parameters for the Van Laar model were calculated.
The theoretical and experimental activity coefficients were compared using a parity plot. The
results, as shown in Figure 1, display that the theoretical and experimental activity coefficients
correlate at a 95% confidence interval
because the interval does not include zero.
The experimental/theoretical activity
coefficients for water, ethanol, and
isopropanol were 1.069/1.038, 2.469/2.157,
and 3.560/2.202, respectively.
Conclusions:
The theoretical and experimental activity
coefficients correlate at a 95% confidence
interval. From these results, the Van Laar
model appears to be valid; however, the
Wilson equation is more commonly used for
ternary systems and is likely to be a better fit.
Figure 1: This plot displays the experimental activity coefficients of the water on the x-axis and the theoretical activity coefficients of water in the ternary system on the y-axis.
INTRODUCTION
Background: ROH industries, a company that makes volatile organic compounds and solubilizes
them with water at varying concentrations, recently had an upset involving a particular
concentration of water and ethanol. The final product was contaminated with substantial
amounts of isopropanol. Using the Van Laar thermodynamic model and knowledge of the
physical properties of the water/ethanol/isopropanol system, a strategy was developed for
removing the isopropanol from the finished product.
Theory: The Van Laar equation is an asymmetric model for Gibbs energy that is meant to fit
experimental data and quantify compositional dependence of activity coefficients.3 The
asymmetric models for Gibbs energy uses an activity coefficient, a ratio of the fugacities that
accounts for deviations from ideality in a mixture, to model the molecular behavior in a system.
For a given system, vapor-liquid equilibrium occurs when the fugacities in the vapor and liquid
phases are equal and is given by the following equation:
𝑓𝑖𝑣 = 𝑓𝑖
𝑙 Equation 13
In this equation, 𝑓𝑖𝑣 represents the fugacity of a species in the vapor phase, and 𝑓𝑖
𝑙 denotes the
liquid-phase fugacity. In addition to the asymmetric models for Gibbs energy, a modified form of
Raoult’s Law for non-ideal mixtures was also used in calculating activities from experimental
data. Raoult’s Law is as follows:
𝑦𝑖𝑃 = 𝑥𝑖𝛾𝑖𝑃𝑖𝑠𝑎𝑡 Equation 23
Where 𝑦𝑖 is the vapor mole fraction, 𝑃 is the pressure of the overall system, 𝑥𝑖 is the liquid mole
fraction, 𝛾𝑖 is the activity coefficient, and 𝑃𝑖𝑠𝑎𝑡 is the saturation pressure of the species.
From a theoretical standpoint, by applying the Wohl expansion and neglecting third and higher
order terms, the Van Laar equation was extended to model the ternary system. The following
equation represents the natural logarithm of the activity coefficient for one species in the ternary
mixture:
ln(𝛾1) ={𝑥2
2𝛼12(𝛽12𝛼12
)2+𝑥3
2𝛼13(𝛽13𝛼13
)2+𝑥2𝑥3
𝛽12𝛼12
𝛽13𝛼13
(𝛼12+𝛼13−𝛼23𝛼12𝛽12
)}
[𝑥1+𝑥2(𝛽12𝛼12
)+𝑥3(𝛽13𝛼13
)]2 Equation 35
Where 𝛼 and 𝛽 are constants (𝛼𝑗𝑖 = 𝛽𝑖𝑗 and 𝛽𝑗𝑖 = 𝛼𝑖𝑗), 𝑥 is the mole fraction for species one,
two, and three as denoted by the subscripts, and 𝛾 represents the activity of the species. Since
the activity coefficient is a ratio of fugacity, it is a dimensionless group. The expression for
species 2 is obtained by interchanging subscripts 1 and 2 in Equation 3, and for species 3 by
interchanging subscripts 1 and 3 in Equation 3.5 To determine the theoretical constants, 𝛼 and
𝛽, an infinite dilution assumption was made to reduce Equation 3.
EXPERIMENTAL
Equipment: An Othmer Still, as shown in Figure 2, was used to obtain vapor-liquid equilibrium
data for the ternary system in this experiment. The system consisted of approximately 800mL,
at random concentrations, of water, ethanol, and isopropanol. The Othmer Still was not made of
Pyrex glass, thus the system was subject to thermal shock and was closely monitored to avoid
running the risk of cracking the still. Insulation also surrounded the still to help keep it warm and
avoid condensation and reflux. When filling the still with the ternary mixture, approximately
800mL was used to avoid allowing back pressure which would result in the inability to force the
liquid up for recirculation. A case style thermocouple measured the temperature of the still in
units of Kelvin. In addition to the Othmer Still and thermocouple, a gas chromatograph was used
to determine the mass fraction of each component in the liquid and vapor phases.
Procedure: For each trial, 270mL of isopropanol, water, and ethanol were prepared and poured
into the still to form a total solution of
approximately 800mL. Upon heating the
solution, aluminum foil and Parafilm were
used at point 6 in Figure 2 to ensure vapor did
not escape from the still and give false
equilibrium concentrations. Equilibrium was
established by creating a closed-loop system
and accomplishing equivalent compositions in
both the liquid and vapor phases. When the
temperature remained constant for fifteen
minutes and circulation was achieved, it
ensured the system had reached equilibrium.
Liquid samples from point 3 and vapor
samples from point 4, shown in Figure 2, were
taken from the still once the system reached
equilibrium. The samples were injected into a
gas chromatograph and an analysis of the
sketched peaks was used to determine the
mass fraction of each component contained in
the mixture.
5
1
2
7 3
4
6
Figure 2: Schematic of Othmer Still. The labels are as follows: (1) condenser, (2) hot plate, (3) liquid sample port, (4) vapor sample port, (5) thermocouple, (6) inlet (7) ethanol, water, and isopropanol mixture.
RESULTS AND DISCUSSION
Theoretical interaction parameters, 𝛼 and 𝛽, were found in literature and applied to the Van Laar
model for ternary mixtures. Applying these parameters, theoretical activity coefficients were
determined. A calibration curve,
presented in Figure 3, was used to
correct the mass and mole fractions
given by the gas chromatograph
peaks. The experimental activity
coefficients, determined from
Equation 2, were plotted against the
theoretical activity coefficients in a
parity plot shown in Figure 4.
Denoted by dotted red lines in the
parity plots, it can be concluded the
experimental and theoretical activity
coefficients for the Van Laar model correlate at a 95% confidence interval for each component.
All experimental parity plots, shown in the appendix, follow the same curved trend as the
bivariate parity plots of supplemental data from the Dortmund Data Bank.1 The blue line
depicted in the parity plots represents the ideal values at which the theoretical point is equal to
the experimental. The error produced from the parity plots for isopropanol and ethanol, give rise
to the need for the application of the Wilson equation rather than the Van Laar model for ternary
systems. The Wilson equation works well for mixtures of polar and nonpolar species and is
readily extended to multicomponent
mixtures.3 Since the results from the Van
Laar model did not appear consistent for a
ternary mixture, the Wilson equation was
applied and an analysis was conducted. A
new parity plot, shown in Figure 5, was
created and shows the Wilson equation
provides a more accurate fit for the
experimental data. Figure 5 includes both
the experimental and supplemental data
modeled with the Wilson equation. Results
show, by the dotted red lines, that the
Figure 4: This plot displays the ethanol experimental activity coefficients (x-axis) and the theoretical activity coefficients (y-axis); where the blue line represents 𝑦 = 𝑥, the red dotted line shows the confidence intervals for the data, and the solid red line is the line of best fit for the points.
Figure 3: This plot depicts the responses of the gas chromatograph machine when injected with samples of water, ethanol, and isopropanol. (See appendix for development of calibration equations for each component).
activity coefficients correlate at a 95%
confidence interval. The experimental data
using the Van Laar model provide a root
mean square error of 0.195, 0.219, and
0.823 for water, ethanol, and isopropanol,
respectively. The supplemental and
experimental data using the Wilson
equation provide a root mean square error
of 0.129, 0.125, and 0.359 for water,
ethanol, and isopropanol, respectively. The
parity plots using the Van Laar equation
result in an average error that is 67% larger
than the results from the Wilson equation.
From the conclusions reached using the
thermodynamic models for ternary systems, a strategy for removing the isopropanol was
developed. Azeotropic behavior is important in the process to separate mixtures. The ternary
mixture of isopropanol, water, and ethanol does not form an azeotrope. However, water and
ethanol form an azeotrope at 351 degrees Kelvin and a molar composition of about 10% water
and 90% ethanol. Water and isopropanol also form an azeotrope at 354 degrees Kelvin and a
molar composition of roughly 32% water and 68% isopropanol. Depending on what temperature
the system reaches equilibrium, azeotropes may not form. Although ethanol and isopropanol do
not form an azeotrope, the boiling points only differ by approximately 4 degrees. Since the
boiling points are so close to one another, conventional stage distillation would not effectively
separate the isopropanol. Instead, a two-step extractive distillation process would remove the
isopropanol from the ternary mixture. In the first step, the ternary mixture would be fed to a
distillation column where the water would leave the column with the bottoms stream. Ethanol
and isopropanol will vaporize to the distillate stream because the boiling points are lower. Using
the experimental data, the relative volatility of ethanol to isopropanol was determined to be 1.06
and it was found that an organic solvent must be used to raise the volatility. The solvent must
have a higher boiling point than the ethanol and isopropanol so it leaves the column in the
bottoms stream with the isopropanol. In the event that the isopropanol needs to be recovered as
a pure product, liquid-liquid extraction can be used to remove the organic solvent.
Figure 5: This plot displays the isopropanol experimental activity coefficients (x-axis) and the theoretical Wilson activity coefficients (y-axis); where the red dotted lines are the confidence intervals, the solid red line is the line of best fit, the black points are supplemental data, and the gray points are experimental data.
CONCLUSIONS
The experimental data obtained for the vapor-liquid equilibrium experiment correlates with the
Van Laar thermodynamic model at a 95% confidence interval. Although the Van Laar model
affords a sufficient correlation, the Wilson equation provides results that are, on average, 67%
more conclusive. The Wilson equation is ultimately a better fit, and more robust model for
ternary mixtures. Finally, because the system does not form an azeotrope and the boiling points
of isopropanol and ethanol hardly differ, a two-stage extractive distillation process would be the
most effective strategy for removing the isopropanol from the ternary mixture. To combat issues
similar to this in the future, a ternary diagram for the water/ethanol/isopropanol system is
provided in the appendix.
REFERENCES
1 Dortmund Data Bank Software and Separation Technology. “Vapor-Liquid Equilibrium
Data,” Available via http://www.ddbst.com/en/EED/VLE/VLE%20Ethanol%3B2-Propanol%3-
BWater.php. Accessed 19 November 2014.
2 Holmes and Van Winkle. “Prediction of Ternary Vapor-Liquid Equilibria from Binary Data,”
Available via https://www.academia.edu/7976146/Prediction_of_Ternary_Vapor-
Liquid_Equilibria_from_Binary_Data. Accessed 30 November 2014.
3 Koretsky, Milo. “Engineering and Chemical Thermodynamics,” 2nd ed., John Wiley & Sons,
Inc. (2013).
4 Pan, Yi-Chuan. “Evaluation of the Interaction Effect in Ternary Systems,” Available via
https://archive.org/stream/evaluationofinte00pany/evaluationofinte00pany_djvu.txt.
Accessed 17 November 2014.
5 Sandler, Stanley. “Chemical, Biochemical, and Engineering Thermodynamics,” 4th ed., John
Wiley & Sons, Inc. (2006).
6 NIST Chemical Webbook. “VLE-Calc: Calculator of Vapor-Liquid and Liquid-Liquid Phase
Equilibria,” Available via http://vle-calc.com/phase_diagram.html. Accessed 17 November
2014.
7 Wake Forrest College Department of Chemistry. “Gas Chromatography,” Available via
http://www.wfu.edu/chemistry/courses/organic/GC/index.html. Accessed 16 November 2014.
8 Word Press Passion World. “Ethanol/Water Azeotrope,” Available via
http://imeldalee18.wordpress.com/2011/02/18/ethanolwater-azeotrope/. Accessed 30
November 2014.
APPENDIX
Parity plots for water, ethanol, and isopropanol, respectively:
Ternary Diagrams (values are compositions in mass %):
r
Propagation of Uncertainty:
𝛾𝑤𝑎𝑡𝑒𝑟 𝛾𝑒𝑡ℎ𝑎𝑛𝑜𝑙 𝛾𝑖𝑠𝑜𝑝𝑟𝑜𝑝𝑎𝑛𝑜𝑙
+/- 0.2862
+/- 0.1263
+/- 0.1383
+/- 24%
+/- 7.8%
+/- 7.04%
Bivariate plots for supplemental data:
Wilson equation for ternary system:3
ln(𝛾𝑘) = −𝑙𝑛(∑ 𝑥𝑗Λ𝑘𝑗𝑚𝑗=1 ) + 1 − ∑
𝑥𝑖Λ𝑖𝑘
∑ 𝑥𝑗Λ𝑖𝑗𝑚𝑗−1
𝑚𝑖=1
Azeotropic data for ethanol/water/isopropanol:6
COMPONENTS Pure compound
boiling point Azeotropic
temperature Azeotropic
composition
--- K K mol/mol
water ethanol
373.2 351.6
351.448 0.103356 0.896644
water
isopropanol
373.2
355.7 353.87
0.320688
0.679312
ethanol isopropanol
351.6 355.7
Zeotropic Zeotropic
water ethanol
isopropanol
373.2 351.6 355.7
Zeotropic Zeotropic
Raw data:
Vapor mass frac Liquid mass frac
Ya Yb Yc Xa Xb Xc
0.2019473 0.3936266 0.4044262 0.599072 0.222265 0.1786632
0.18285 0.4059 0.41123 0.3342778 0.3321373 0.333585
0.1784279 0.401939 0.4196333 0.3521483 0.3239294 0.3239222
0.1729678 0.4952528 0.3317795 0.3488386 0.400048 0.2511136
0.1785076 0.494509 0.3269835 0.3651091 0.390988 0.2439028
0.1925323 0.3909318 0.4165358 0.5149738 0.2246054 0.2604208
Vapor mole frac Liquid mole frac
Ya Yb Yc Xa Xb Xc
0.42328428 0.32262245 0.25409327 0.81006047 0.117523727 0.0724158
0.39336317 0.34145528 0.26518155 0.59253619 0.230218942 0.17724486
0.38672193 0.34065301 0.27262506 0.61146203 0.219943086 0.16859489
0.37111166 0.41551081 0.21337753 0.60088339 0.269459698 0.12965692
0.37989106 0.41152158 0.20858736 0.61766776 0.258649569 0.12368267
0.4094191 0.32507349 0.2655074 0.75635373 0.128995947 0.11465032
Experimental Activities:
ga gb gc
1.069538662 2.46470773 3.561454729
1.358815831 1.33164719 1.518578473
1.294526955 1.3905875 1.641303727
1.264142968 1.38447434 1.670400362
1.258884621 1.42849017 1.711775166
1.107962195 2.26256931 2.350546986
Theoretical Activities using Van Laar:
ga gb gc
1.04549742 2.156800439 2.201337392
1.22786723 1.483769119 1.501934535
1.20519785 1.525622287 1.545614428
1.21627006 1.511530879 1.512062752
1.19680012 1.549432932 1.551881307
1.07626381 1.9401464 1.985190927
Theoretical Activities using Wilson:
ga gb gc
1.10696945 2.11703138 1.66217349
1.35724297 1.35834394 1.66217349
1.33105761 1.39409792 1.71969132
1.33638779 1.38558593 1.709054
1.31387685 1.41985481 1.7625055
1.16215573 1.80721941 2.43099169
Gas Chromatograph Schematic:
Calibration Curves for Water, Ethanol, Isopropanol (blue line represents ideality):