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New Analytical Expression for thePrediction of Vapour Pressures of IonicLiquidsM.H. Joshipuraa

a Chemical Engineering Department, Institute of Technology,Nirma University, S G Highway, Tragad Patia, Ahmedabad, IndiaPublished online: 07 Nov 2014.

To cite this article: M.H. Joshipura (2014): New Analytical Expression for the Prediction of VapourPressures of Ionic Liquids, Indian Chemical Engineer, DOI: 10.1080/00194506.2014.975757

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New Analytical Expression for the Prediction ofVapour Pressures of Ionic Liquids

M.H. Joshipura*Chemical Engineering Department, Institute of Technology, Nirma University, S G Highway, TragadPatia, Ahmedabad, India

Abstract: Ionic liquids (IL), considered as green solvents, are useful in different industrialapplications. Vapour pressure is one of the key properties of IL for designing differentprocesses. Recently, the zero pressure fugacity approach was proposed to predict the vapourpressures of IL using cubic equations of state. In the present work, this approach was improvedby fitting the vapour pressure data of ten ILs using six different cohesion factor models. Newanalytical expressions are proposed in the present work for predicting vapour pressure of ILusing acentric factor and Mass Connectivity Index. It was shown that one of the proposedmodels (Model J) predicts vapour pressure with global %AAD of 7.3%. The model comparedwith the models based on COSMO-RS theory and PC SAFT-equation of state (EOS), and itwas found that the proposed models work well compared to others. It is shown that theproposed models predict vapour pressure with greater accuracy and consistency.

Keywords: Ionic liquid, Vapour pressure, PR EOS, Cohesion factor models.

1. IntroductionIonic liquids (ILs) are finding widespread applications in chemical and allied industries. Theyincrease the safety of the process because of their low vapour pressure, as well as thermal andchemical stability. Moreover, they offer high ionic conductivity and can be used for a widertemperature range. Vapour pressure is an important property of IL for screening them assolvents and to know their phase transition behaviour. [1] Since IL’s have very low vapourpressures, it is very difficult to determine and estimate their vapour pressures.

Data for vapour pressure of ILs are very limited in the literature. Till 2009 only up to 60data points were available [2]. Experimental data of vapour pressures were reported by

*Author for Correspondence. Email: milind.joshipura@nirmauni.ac.in

INDIAN CHEMICAL ENGINEER © 2014 Indian Institute of Chemical Engineers2014, pp. 1–10Print ISSN: 0019-4506, Online ISSN: 0975-007X, http://dx.doi.org/10.1080/00194506.2014.975757

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different researchers like Paulechka et al. [3], Zaitsau et al. [4] and Emelyanenko et al. [5].Rocha et al. [6] reported highly accurate vapour pressure data for the 1-alkyl-3-methylimi-dazolium bis(trifluoromethylsulfonyl)imide series [Cnmim][Ntf2] with n = 2, 3, 4, 5, 6, 7, 8, 10and 12 in the range of 450–500 K. They have used modified Knudsen effusion apparatus forobtaining the vapour pressure data. They compared their data with above-mentionedresearchers and proved that their data is of higher accuracy.

Various models for prediction of vapour pressure of ILs were proposed in the literature.Some of the models were just fitted to the vapour pressure data and, hence, are not general innature. Diedenhofen et al. [7] proposed an approach based on COSMO-RS theory. Theyproposed two approaches to predict vapour pressure of ILs. In the first approach, they dividedthe Gibbs free energy of vaporisation into three parts, namely, cation, anion and pair Gibbs freeenergy. These Gibbs free energies were calculated using COSMO-RS, and vapour pressurevalues were calculated through an equation containing these energies. In the second approach,enthalpy of vaporisation and vapour pressure both were calculated by COSMO-RS. Thepresented results showed that Approach II was better compared to Approach I. Attempt togeneralise Antoine equation constants were also made to predict vapour pressure of IL byValderrama and Sanga [8]. Valderrama and Sanga have not reported the deviation in vapourpressure prediction, but they have reported and estimated Antoine constants for 20 pure ILs.Paduszyński and Domańska [9] proposed a model based on molecular theory using PC-SAFT.Ten (10) site models proposed by them were found to be accurate in predicting the variousproperties including density, surface tension, as well as vapour liquid equilibrium for pure ILand binary systems. However, for pure ILs, the deviation in vapour pressure was as high as330%. The minimum deviation they reported was 24%. Very recently, Valdarama and Luis [2]used 129 data points for the nine ILs of Rocha et al. [6] and the five data points reported forEmelyanenko et al. for [C4mim][dca] [5] to find a suitable model to predict vapour pressure ofILs. In their study, they analysed seven models for vapour pressure prediction available in theliterature to predict the vapour pressure in the range of 10−1 to 10−5 bar, but all of the modelsfailed to give accurate results. Hence, they proposed an analytical expression for vapour pressureprediction of ILs based on the Peng Robinson (PR) [10] EOS using the zero pressure fugacityapproach. One may refer to the original papers [2, 11] for a detailed discussion on the calculationof vapour pressure through zero pressure fugacity. Through this approach, they could fit thevapour pressure data very accurately using compound specific models. However, for generalisedmodels the deviation was found to be as high as 23%. In the present work, the approach ofValderrama was taken further to improve the generalisation of the compound specificparameters. In the present work, two important aspects were studied.

. Prediction of vapour pressure of IL through zero pressure fugacity approach usingdifferent cohesion factor expressions. The earlier approach of Valderama and Luis hadnot considered the cohesion factor relation of the type proposed by Stryjek-Vera andGibbons-Laughton [12] and Polishuk [13].

. Development of new generalised expressions based on Mass Connectivity Index (MCI) forpredicting vapour pressure of ILs. This will make the model predictive in nature, and itcan predict the vapour pressure data of large number of ILs in contrast to the modelswhich fit the experimental vapour pressure data.

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Prediction of vapour pressure using a concept of zero pressure fugacity requires criticalproperties of the ILs. Critical properties for all the ILs were predicted using the modelsproposed by Valderrama et al., 2008 [14, 15]. Values of MCI, which were used forgeneralisation, were calculated by group contribution method proposed by Valderrama et al.,2010 [16]. Accurate prediction of vapour pressure using cubic equations of state requires aproper cohesion factor expression. In the present work six different cohesion factorexpressions listed in Table 1 were considered. Models for prediction of vapour pressurewere developed, and the models were first compared with the models proposed byValderrama [2]. The best model (Model J) was then compared with the models ofDiedenhofen et al. [7],Valderarram and Sanga [8] and Paduszyński and Domańska [9].

2. Results and DiscussionIn the present work, the experimental database was taken the same as that in the work ofValderrama and Luis [2]. These experimental database was first fitted with six cohesion factormodels listed in Table 1. The compound specific parameters were generated for all the ILsconsidered for the study. These parameters were than generalised to give predictive models.Methodology for both compound specific and generalised models are discussed separately infollowing sections.

2.1. Compound Specific Parameter ModelsUsing the cohesion factors listed in Table 1, vapour pressures for ILs were fitted to obtaincompound specific parameters by minimising the function given by Equation (1). The compoundspecific parameters are listed in Table 2.

f ðm; nÞ ¼ %AAD ¼ 100N

AbsPsatcal�Psat

exp

� �

Psatexp

0@

1A ð1Þ

Table 1. Cohesion factor expression studied in the present work

Cohesion function Model equation Reference

Soave (S) aðTÞ ¼ 1 þ m 1�ffiffiffiffiffiffiTTc

q� �h i2[17]

Joshipura (J) aðTÞ ¼ exp m1 1� TTc

n oh i[18]

Heyen (H) aðTÞ ¼ exp m2 1 � TTc

� �nn oh i[2]

Figueria (SV) aðTÞ ¼ 1 þ ½m þ nð1þ ffiffiffiffiffiTr

p Þð0:7� Tr Þð1�ffiffiffiffiffiTr

p �� �2[12]

Figueria (GL) aðTÞ ¼ 1 þ mðTr� 1Þ þ nð ffiffiffiffiffiffiffiffiffiffiffiTr�1

p Þ [12]

Polishuk (P) aðTÞ ¼ 11 þ mðT2=3

r �1Þ [13]

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Table 2. Compound specific parameters for cohesion factors studied in the present work

Optimised cohesion factor parameters

Cohesion function S J H SV GL P

Sr. No IL abbreviation m m m n m n m n m

1 [C2C1im][NtF2] 1.3614 1.3567 1.0150 1.8524 1.8778 −0.9759 −3.0215 1.4150 0.74142 [C3C1im][NtF2] 1.3206 1.3228 1.0018 1.7782 1.7653 −0.8493 −2.4989 0.7126 0.73023 [C4C1im][NtF2] 1.3024 1.3080 1.0599 1.5058 1.6394 −0.6412 −1.4880 −0.8663 0.73034 [C5C1im][NtF2] 1.2744 1.2855 1.1697 1.1856 1.4576 −0.3457 −0.0605 −3.0870 0.71685 [C6C1im][NtF2] 1.2494 1.2649 1.2663 0.9982 1.3805 −0.2464 0.2767 −3.5572 0.70896 [C7C1im][NtF2] 1.2346 1.2530 1.3509 0.8824 1.2544 −0.0369 1.3519 −5.2410 0.70437 [C8C1im][NtF2] 1.2289 1.2481 1.3147 0.9164 1.2662 −0.0695 1.1801 −4.9485 0.70248 [C10C1im][NtF2] 1.2004 1.2251 1.6657 0.6287 1.0563 0.2665 2.7679 −7.4089 0.69349 [C12C1im][NtF2] 1.1666 1.1980 1.9877 0.4750 0.9677 0.3606 2.2670 −6.5052 0.682910 [bmim][dca] 1.6516 1.5790 3.0191 0.4239 1.3828 0.6400 6.6821 −15.3608 0.8309

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where, N is the number of data points, m and n are the parameters for cohesion factor model,Psatcal is the calculated vapour pressure using parameters m and n, and Psat

exp is the experimentalvapour pressure. Calculation procedure for vapour pressure was exactly the same as providedby Valderrama and Luis [2]. The code for the same was developed in Microsoft Excel ®. Theprocedure also gives the %Absolute Average deviation for the IL under consideration. Table3 reports global %AAD values, which were obtained by summing %AAD of individual ILand dividing it with number of ILs, for the six cohesion factor models. Table 3 also reportsmaximum and minimum deviation values obtained for each cohesion factor. It can be seenthat the two parameter models fit the data very accurately with all the %AAD values beingless than 1.0%. However, single parameter models also gave satisfactory fit with only thePolishuk model having %AAD >5%. These compound specific models can only be used to fitthe experimental vapour pressure data. However, for ILs such experimental data is notavailable easily. Hence, there is a need of developing generalised models which can predict thevapour pressure. Keeping this in mind a generalised model for each of the cohesion factormodels were developed in the present work.

2.2. Generalised Parameter ModelsParameters for the three models [namely, Soave (S), Hyene 1 (J in the present work) andHyene-2 (H)] were correlated by Valderrama and Luis [2] through MCI only. The %AADvalues for vapour pressures obtained by these correlations are reported in Table 5. It can beseen that the single parameter Models S and J are very inaccurate. Valderrama and Luisproposed Model H for the prediction of vapour pressure. Global %AAD for Model H was8.54% with maximum %AAD value of 19.68% for C14H23F6N3O4S2. The %AAD values forfour ILs were more than 10% out of which two were having greater than 15% AAD.Although Model H predicted vapour pressures better than Models S and J, inconsistency ofthe model may lead to high deviation in vapour pressure predictions. To overcome thisshortcoming, different correlations were considered, but generalising the parameters in termsof both acentric factor and MCI yielded the best results.

Table 3. Global %AAD in vapour pressure for studied cohesion factor models

Sr. No. Cohesion factor %AAD (Global) Maximum %AAD Minimum %AAD

1 S 4.06 11.34 0.542 J 3.74 9.64 0.953 P 5.52 17.88 1.004 H 0.77 1.7 0.285 SV 0.77 1.55 0.256 GL 0.93 1.91 0.28

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For single-parameter models like S, J and P, following expression was found to be mostaccurate:

m¼m0þm1kþm2xþm3k2 ð2Þ

where, ω is acentric factor and λ is MCI. Coefficients mi (i = 0 to 3) of Equation (2) for single-parameter models were obtained by minimising the difference between calculated andoptimised m parameters. This was performed using Microsoft Excel Solver ®, and the valuesof the coefficients are reported in Table 4.

Models H and GL were correlated using method proposed by Valderrama and Luis [2].Model equations are given by Equations (3)–(6).

For Model H:

1 þ mn ¼ 0:25k2 � 1:7069k� 1:5237x þ 5:8266 ð3Þ

mnðn � ð1 þ mnÞÞ ¼ �0:1239k2 þ 1:1693k � 0:2155x � 3:8984 ð4ÞFor Model GL:

1 � mn

2¼ 0:7505k2 � 4:5151k � 3:5630x þ 10:6739 ð5Þ

n

4¼ 0:9923k2 � 5:1481k � 7:8919x þ 8:8267 ð6Þ

Due to mathematical complexity in SV model, parameters were fitted empirically and themodels are given by Equations (7) and (8).

m ¼ 0:3338k2 � 2:1892k � 1:8460x þ 5:6364 ð7Þ

n ¼ �0:6892k2 þ 4:0103k þ 3:7304x � 7:4507 ð8ÞVapour pressure for all the ILs were predicted using Equations (2)–(8). Results in terms of%AAD of vapour pressures are listed in Table 5. All the single-parameter models gave%AAD value less than 8% with Model J being better compared to other models. All threetwo parameter models gave higher %AAD than single-parameter models. Only ModelH gave %AAD value less than 10%. It can be seen that Model J has the least maximum

Table 4. Coefficients of models for Equation 2

Model m0 m1 m2 m3

S 2.2662 −0.5256 0.1886 0.0535J 2.1749 −0.4871 0.1203 0.0556P 0.9960 −0.1461 0.0630 0.0139

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%AAD (12.97%) and also least number of ILs having %AAD values more than 10% (03).Results indicate that Model J predicts vapour pressure with a greater consistency. Theproposed Model J performs better than the model proposed by Valderram and Luis [2].

To check the performance of the proposed model (Model J), it was compared with themodel proposed by Valderrama and Sanga [8] as well as with the vapour pressure predictedby COSMO method of Diedenhofen et al. [7]. Experimental data for four ILs reported byDiedenhofen et al. [7] were used as a reference. Since the database is not the one which wasused to develop Model J, it will also test the robustness of the model. However, inaccuraciesof the experimental data may affect the numerical values of %AAD. But qualitatively one cancompare the performances. Table 6 shows the comparison of %AAD in predicting vapourpressure of four ILs. While Valderrama and Sanga model completely failed to predict the

Table 5. The %AAD in vapour pressure by generalised models (comparison with Valderrama and Luis)

Valderrama and Luis [2] This work

IL/Model S J (H1) H (H2) S J H SV GL P

C8H11F6N3O4S2 26.97 24.03 6.40 11.53 9.64 26.69 11.99 16.12 13.93C9H13F6N3O4S2 22.39 21.43 12.91 10.60 10.27 1.86 27.93 13.04 11.25C10H15F6N3O4S2 5.02 4.09 2.57 4.90 3.75 5.37 16.41 12.30 6.08C11H17F6N3O4S2 5.41 3.46 1.97 4.16 2.41 2.79 29.30 25.85 5.00C12H19F6N3O4S2 7.64 5.22 2.59 6.48 5.63 2.16 32.29 31.93 7.07C13H21F6N3O4S2 17.61 14.32 4.92 0.63 1.22 6.42 26.82 47.45 1.53C14H23F6N3O4S2 33.20 28.44 19.68 14.75 12.97 21.97 13.20 66.88 12.11C16H27F6N3O4S2 17.83 15.35 13.57 16.08 9.53 22.56 1.16 46.43 12.24C18H31F6N3O4S2 25.51 23.27 15.55 1.60 11.20 1.67 4.70 10.12 2.84C10H15N5 18.09 17.08 5.25 6.18 6.38 1.04 1.38 4.20 1.29

Global %AAD 17.97 15.67 8.54 7.69 7.30 9.25 16.52 27.43 7.33Max %AAD 33.20 28.44 19.68 16.08 12.97 26.69 32.29 66.88 13.93

%AAD >15 7 5 2 1 0 3 5 6 0%AAD >10 7 7 4 4 3 3 7 9 4%AAD <5 0 2 4 4 3 5 3 1 3

Table 6. The %AAD for vapour pressure predictions for different models

IL/Model CRSMD VS This Work (Model J)

C2mimNTf2 75.7 98.9 18C4mimNTf2 80.1 99.7 16.6C6mimNTf3 90.3 99.9 22.3C8mimNTf3 95.6 100 12.5Global 85.43 99.63 17.35

CRSMD, COSMO RS Model [7]; VS, Valderrama and Sanga Model [8].

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vapour pressure data, the COSMO method of Diedenhofen et al. [7] also could not predictthe vapour pressure properly. One can observe that the COSMO method and the VS methodunderestimate the vapour pressure by quite a few units where Model J is predictingsatisfactorily well.

3. Summary and Concluding RemarksVapour pressure is one of the key properties of ILs. Experimental determination of the sameis difficult, and hence, the data are limited in the open literature. There are variousapproaches to predict vapour pressure of ILs. Ranging from generalised methods proposedby Valderrama and Sanga to the use of molecular-based equation of state proposed byPaduszyński and Domańska. Researchers have also used COSMO-based models to predictthe vapour pressure. The generalised models are highly simplified, but they lack in accuracy,whereas models based on molecular EOS or COSMO require very high computational skillsand even after that the estimation of the vapour pressure is poor. Cubic equations of state-based approach proposed by Valderamma is both simple and accurate and, hence, should bewell accepted. In the current work, an attempt was made to improve the analytical expressionproposed by Valderrama. Incorporating MCI in the generalised expression has improved theconsistency of the vapour pressure predictions. The approach was used with six differentcohesion factor models, and it was found that all studied cohesion factor models fit vapourpressure data accurately. It was shown that the Model J predicts vapour pressures withgreater accuracy and more consistently compared to other cohesion factor models studied.Since the available experimental data for vapour pressure of IL are very limited, the proposedgeneralised expression can be utilised to predict the vapour pressure of other ILs. One needsonly critical properties, acentric factor and MCI of the IL. The research in future shouldprovide the required experimental vapour pressure data to facilitate a critical study ofestimation methods.

4. Nomenclature%AAD Percentage absolute average deviation defined by Equation (1)m, n Parameters of cohesion factor expressionN Number of data pointsPsat Vapour pressure

5. Greek lettersω Acentric factorλ Mass connectivity index

6. Subscriptsexp Experimental valuecal Calculated value

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