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):