PC-SAFT - ACS Publications - American Chemical Society

Article pubs.acs.org/JPCA

Implementation of Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT), Generalized (G)SAFT+Cubic, and Cubic-Plus-Association (CPA) for Modeling Thermophysical Properties of Selected 1‑Alkyl-3-methylimidazolium Ionic Liquids in a Wide Pressure Range Ilya Polishuk* Department of Chemical Engineering and Biotechnology, Ariel University, 40700, Ariel, Israel ABSTRACT: This study is the first comparative investigation of predicting the isochoric and the isobaric heat capacities, the isothermal and the isentropic compressibilities, the isobaric thermal expansibilities, the thermal pressure coefficients, and the sound velocities of ionic liquids by statistical associating fluid theory (SAFT) equation of state (EoS) models and cubic-plus-association (CPA). It is demonstrated that, taking into account the high uncertainty of the literature data (excluding sound velocities), the generalized for heavy compounds version of SAFT+Cubic (GSAFT+Cubic) appears as a robust estimator of the auxiliary thermodynamic properties under consideration. In the case of the ionic liquids the performance of PC-SAFT seems to be less accurate in comparison to ordinary compounds. In particular, PC-SAFT substantially overestimates heat capacities and underestimates the temperature and pressure dependencies of sound velocities and compressibilities. An undesired phenomenon of predicting high fictitious critical temperatures of ionic liquids by PC-SAFT should be noticed as well. CPA is the less accurate estimator of the liquid phase properties, but it is advantageous in modeling vapor pressures and vaporization enthalpies of ionic liquids. At the same time, the preliminary results indicate that the inaccuracies in predicting the deep vacuum vapor pressures of ionic liquids do not influence modeling of phase equilibria in their mixtures at much higher pressures.

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INTRODUCTION The unique physical properties of ionic liquids present a major interest for modern science and technology. Thus, the number of references dealing with the implementation of equation of state (EoS) models for estimating phase equilibria and densities of these substances is rapidly increasing in the recent years.1,2 Unfortunately, much less attention thus far has been paid to modeling auxiliary thermodynamic properties of ionic liquids, such as the isochoric and the isobaric heat capacities, the isothermal and the isentropic compressibilities, the isobaric thermal expansibility, the thermal pressure coefficient, and the sound velocity. A major obstacle for implementing the PVT equations of state for estimating these properties is the fact that the latter models typically treat only the intermolecular interactions and therefore cannot assess the values of the ideal gas heat capacities. Ge et al.3 have proposed to solve this problem for ionic liquids by implementing the empirical Joback’s group contribution method4,5 Unfortunately, the reliability of this method should be considered as doubtful unless the truthful and widely agreed values of the critical and boiling temperatures of ionic liquids6 will be evaluated. Consequently, the reliable values of CoP are currently available for only a limited number of ionic liquids. In particular, Paulechka et al.7 and Blokhin et al.8 © 2013 American Chemical Society

have implemented the statistical thermodynamic methods for estimating the ideal gas state properties of four members of the 1-butyl-3-methylimidazolium bis[trifluoromethanesulfonyl]imide series, namely, [C2mim][NTf2], [C4mim][NTf2], [C6mim][NTf2], and [C8mim][NTf2]. In addition, Paulechka et al.9 have evaluated the CoP values of 1-butyl-3-methylimidazolium hexafluorophosphate ([C4mim][PF6]). At the same time, the measured in wide pressure range sound velocities and the estimated auxiliary properties of [C4mim][NTf2], [C6mim][NTf2], [C4mim][PF6], [C3mim][NTf2], and [C5mim][NTf2] are available in literature.10−12 Although the CoP values of the two latter compounds thus far have not been reported, they can be easily extrapolated from the results available for other members of the [Cxmim][NTf2] homologues series. A reliable estimation of the auxiliary properties of complex compounds such as the ionic liquids doubtlessly presents a severe test for EoS models. In the current study this test is posed to three equations, namely, the perturbed-chain statistical association fluid theory (PC-SAFT),13 the generalized for heavy compounds version of SAFT+Cubic (GSAFT+Cubic),14 and Received: October 12, 2012 Revised: January 31, 2013 Published: February 7, 2013 2223

dx.doi.org/10.1021/jp310115p | J. Phys. Chem. A 2013, 117, 2223−2232

The Journal of Physical Chemistry A

Article

Table 1. Fictitious Pure Compound Critical Temperatures Predicted by PC-SAFT2 compound

[C2mim][NTf2]

[C4mim][NTf2]

[C5mim][NTf2]

[C6mim][NTf2]

Tc (K) compound

215.738 [C7mim][NTf2]

[C3mim][NTf2] 220.467 [C8mim][NTf2]

225.427 [C10mim][NTf2]

229.537 [C12mim][NTf2]

233.871 [C14mim][NTf2]

Tc (K)

237.282

241.294

247.870

253.737

259.002

Table 2. Input Data and the Values of GSAFT + Cubic Parameters compound [C2mim][NTf2] [C3mim][NTf2] [C4mim][NTf2] [C5mim][NTf2] [C6mim][NTf2] [C8mim][NTf2] [C10mim][NTf2] [C4mim][PF6]

ρexperimental (g/L), conditions, data source

predicted Tc (K)

m

ε/k (K)

σ (Å)

a (bar·mol2/L2)

1076.47

5.36171

297.758

4.35070

206.364

1056.72

5.58169

294.668

4.38290

222.148

1045.32

5.79798

299.874

4.41440

239.398

1035.72

6.02269

302.903

4.43909

257.554

1033.98

6.22411

318.416

4.47489

277.626

1012.34

6.69502

308.911

4.50415

316.609

1005.85

7.03473

320.194

4.56666

356.987

1016.22

4.57995

343.364

4.26702

133.736

1485.34 (23.59 bar, 333.16 K, ref 29) 1551.00 (999.11 bar, 333.16 K, ref 29) 1475.7 (1 bar, 298.15 K, ref 12) 1515.1 (595.9 bar, 298.15 K, ref 12) 1437.04 (1 bar, 298.15 K, ref 10) 1475.94 (591 bar, 298.15 K, ref 10) 1404.45 (1 bar, 298.15 K, ref 12) 1443.56 (595.9 bar, 298.15 K, ref 12) 1338.24 (1 bar, 333.15 K, ref 10) 1380.65 (595.9 bar, 333.15 K, ref 10) 1278 (1 bar, 343.15 K, ref 30) 1303 (300 bar, 343.15 K, ref 30) 1228.6 (1 bar, 353.15 K, ref 31) 1257.9 (350 bar, 353.15 K, ref 31) 1357.72 (1 bar, 308.15 K, ref 32) 1368.99 (200 bar, 308.15 K, ref 32)

the cubic-plus-association EoS (CPA).15 The details of these approaches have been given in the original references. In what follows the aspects of their implementation in this study are discussed.

the undesired phenomena take place at the sufficiently low temperatures. Therefore Paduszyński and Domańska’s2 version has been selected to represent the PC-SAFT EoS. Following the patterns of the latter study, the PC-SAFT parameters for [C4mim][PF6] have been evaluated as: m = 8.1968, ε/k = 297.56 K, σ = 3.4257 Å, εAB/k = 2278.41 K and κAB = 0.0154 (assuming five positive sites A and five negative sites B per ion pair). This set of parameters yields the fictitious critical temperature at 218.114 K. It should be pointed out that the values of the association parameters are same for all of the considered ionic liquids. Thus the current version of PC-SAFT has three adjustable parameters. The second model examined in this study, GSAFT+Cubic,14 is characterized by a weaker theoretical basis in comparison with PC-SAFT. Having simpler algebraic expression and smaller polynomial order, SAFT+Cubic27,28 is free of the numerical pitfall of the fictitious pure compound critical phenomena. GSAFT+Cubic is the generalized version of the latter model, implying intercorrelation of parameters. It has been originally developed for heavy hydrocarbons. However it has been assumed14 that this approach can be applied for modeling other heavy compounds, including ionic liquids. Although GSAFT+Cubic treats ionic liquids as inert substances, the influence association, polar, and ion interactions are considered by this model indirectly. For calculating its parameters, GSAFT+Cubic requires an input of two experimental density data points at arbitrary temperatures and pressures and involves evaluation of four parameters, which are however not fitted but obtained by solving algebraic equations.14 Consequently, in spite of the fact that GSAFT +Cubic has more substance-dependent parameters than the Paduszyński and Domańska’s2 version of PC-SAFT, it requires less experimental data for their assessment. Table 2 lists the experimental points used and the resulting parameters of GSAFT+Cubic. As seen, the values of ε/k are similar for both GSAFT+Cubic and PC-SAFT. At the same time, GSAFT+Cubic exhibits bigger values of σ and smaller values of m. Remarkable, the ratio

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THEORY PC-SAFT is commonly considered as one of the most successful versions of the statistical association fluid theory.16 Nevertheless, this model is not free of the undesired numerical pitfalls characteristic for several SAFT models.17−22 In particular, Privat et al.17 have demonstrated that PC-SAFT predicts the additional fictitious liquid−liquid phase equilibria in pure compounds. In the cases of ordinary substances such as n-alkanes these critical points take place at the very low temperatures, usually far below the melting points and, therefore, are not supposed to interrupt the actual fluid phases. However in the cases of ionic liquids the situation can be different. For example, with the parameters evaluated by by Ji et al.23 for [C4mim][BF4], PC-SAFT in its simplest version (without any specific interactions other than repulsive and dispersive forces) yields an additional fictitious critical point at 393.742 K and 10016.5 bar. Although with the current universal constants matrix of PC-SAFT13 these phenomena cannot be completely eliminated, their negative impact might be reduced. A particularly successful parametrization yielding relatively low fictitious critical temperatures of ionic liquids has been performed by Paduszyński and Domańska2 (see also refs 24−26). Table 1 lists their values. As seen, the model still cannot be implemented for estimating properties of the supercooled ionic liquids, and the fictitious critical temperatures rise with the increase of the 1-alkyl-3-methylimidazolium chain length. Consequently, in the cases of the heavier members of the [Cxmim][NTf2] series, the numerical pitfall already presents a real threat for the robustness of modeling. Nevertheless, at least in the cases of the lighter members of the series for which the thermophysical data are available, 2224



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Table 3. Absolute Average Deviations property

compound

AAD% of GSAFT+Cubic

AAD% of PC-SAFT

AAD% of CPA

ρ

[C2mim][NTf2] [C3mim][NTf2] [C4mim][NTf2] [C5mim][NTf2] [C6mim][NTf2] [C8mim][NTf2] [C10mim][NTf2] [C4mim][PF6] [C3mim][NTf2] [C4mim][NTf2] [C5mim][NTf2] [C4mim][PF6] [C3mim][NTf2] [C4mim][NTf2] [C5mim][NTf2] [C4mim][PF6] [C2mim][NTf2] [C3mim][NTf2] [C4mim][NTf2] [C5mim][NTf2] [C6mim][NTf2] [C4mim][PF6] [C2mim][NTf2] [C3mim][NTf2] [C4mim][NTf2] [C5mim][NTf2] [C6mim][NTf2] [C4mim][PF6] [C2mim][NTf2] [C3mim][NTf2] [C4mim][NTf2] [C5mim][NTf2] [C4mim][PF6] [C4mim][NTf2] [C4mim][PF6] [C4mim][NTf2] [C4mim][PF6]

0.170 0.096 0.077 0.101 0.061 0.207 0.147 0.047 0.690 1.737 1.488 2.029 2.050 2.602 3.672 3.436 3.108 2.744 1.767 2.620 2.237 0.731 12.03 11.10 16.04 6.277 6.969 11.03 10.24 10.32 17.67 8.483 10.82 4.780 0.726 1.716 2.557

0.193 0.050 0.045 0.069 0.105 0.090 0.120 0.096 4.362 3.739 3.537 1.590 4.370 7.353 4.287 2.927 6.920 3.133 9.715 3.619 3.988 3.748 8.436 11.59 22.02 10.60 12.63 22.45 13.08 14.29 31.16 10.60 21.37 42.71 46.97 40.71 46.76

2.012

W

κS

κT

γ

α

CP CV

0.420

20.98

34.32

68.32 39.89

35.54 49.36

79.69 71.23

15.79 6.548

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T range (K)

P range (bar)

no. of points

reference

283.19−373.18 298.15−333.15 298.15−328.2 298.15−333.15 298.15−333.15 293.15−393.15 293.15−393.15 298.15−323.15 298.15−338.15 283.15−323.15 288.15−338.15 283.16−323.25 298.15−333.15 298.15−323.15 298.15−333.15 298.15−323.15 283.19−373.18 298.15−333.15 298.15−323.15 298.15−333.15 298.15−333.15 298.15−323.15 283.19−373.18 298.15−333.15 298.15−323.15 298.15−333.15 298.15−333.15 298.15−323.15 283.19−373.18 298.15−333.15 298.15−323.15 298.15−333.15 298.15−323.15 298.15−323.15 298.15−323.15 298.15−323.15 298.15−323.15

7.24−1000.87 1−595.9 1−591 1−595.9 1−595.9 1−300 1−350 1−1000 1−2000 1−1510 1−1500 1−1509.25 1−595.9 1−1500 1−595.9 1−1000 7.24−1000.87 1−595.9 1−1500 1−595.9 1−595.9 1−1000 7.24−1000.87 1−595.9 1−1500 1−595.9 1−595.9 1−1000 7.24−1000.87 1−595.9 1−1500 1−595.9 1−1000 1−1500 1−1000 1−1500 1−1000

84 165 168 165 163 96 80 144 113 39 114 62 165 204 165 144 84 165 204 165 163 144 84 165 204 165 163 144 84 165 204 165 144 204 144 204 144

29 12 10 12 10 30 31 11 12 10 12 11 12 10 12 11 29 12 10 12 10 11 29 12 10 12 10 11 29 12 10 12 11 10 11 10 11

RESULTS The absolute average deviations of the models under consideration are listed in Table 3, and the selected results are presented graphically. Figure 1 depicts the representative examples of modeling densities. As seen, the experimental data from different sources (even from the same laboratory10,34) exhibit remarkable mutual deviations, which hinders the detailed evaluation of the model accuracy. Since the density data have been considered during evaluation of the model parameters, this modeling should be recognized as correlative rather predictive. Both SAFT equations under consideration yield comparable accurate modeling of the data. Unlike that, CPA exhibits larger deviations, whose absolute values are nevertheless still relatively small.1 It should be pointed out that, unlike both SAFT equations that estimate the critical temperatures of ionic liquids in proximity to the results of molecular simulations2,49 (see Table 2), CPA significantly overestimates these data (Tc = 2380.65 K for [C2mim][NTf2] and 1532.39 K for [C4mim][NTf2]). This result is one of the possible reasons for the

of m*σ of GSAFT+Cubic and PC-SAFT is nearly same for the [NTf2] ionic liquids (∼0.77). It should however be emphasized that in the cases of both models the parameter values are empirical in nature rather than having solid molecular background. The third EoS to be considered is CPA.15 In spite of its simplicity, thus far CPA has been successfully implemented for modeling phase equilibria in large variety of complex systems.16 Therefore evaluating the accuracy of CPA in estimation thermophysical properties presents doubtless interest. Recently1 the parameters of CPA have been published for two ionic liquids under consideration, namely, [C2mim][NTf2] and [C4mim][NTf2]. CPA has five substance-dependent adjustable parameters, which is more than PC-SAFT and GSAFT+Cubic. Maia et al.1 have proposed several sets of these parameters for each ionic liquid. The set 6 for [C2mim][NTf2] and the set 8 for [C4mim][NTf2] are slightly more accurate that the other ones in predicting auxiliary thermodynamic properties, and therefore, they have been adopted in this study. All of the calculations have been performed in the Mathematica 7 software, and the pertinent routines can be obtained from the author by request. 2225



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Figure 1. Densities. Experimental data: [C2mim][NTf2],29,33 [C4mim][NTf2],10 [C6mim][NTf2],10,34 [C10mim][NTf2].31 Modeling: dashed lines, PC-SAFT; solid lines, GSAFT+Cubic; dotted−dashed lines, CPA.

Figure 4 demonstrates that in the cases of ionic liquids the performance of GSAFT+Cubic does not substantially differ from ordinary compounds.14 As seen, the model yields overall reliable predictions of the sound velocity data in wide pressure range. The results of PC-SAFT are characterized by underestimation of both temperature and pressure dependencies of sound velocities. In addition, it can be seen that CPA is a particularly poor estimator of the data under consideration. Similar patterns of behavior of the models appear also in the case of both isothermal and isentropic compressibilities (Figures 5 and 6). As seen, CPA substantially underestimates these data, PC-SAFT underestimates their temperature and pressure dependence, and GSAFT+Cubic predicts them more reliably. In spite of the fact that the isobaric thermal expansibilities and the thermal pressure coefficients are supposed to exhibit analogous behavior for the similar in nature ionic liquids considered here, it is not a case of the literature data. In particular, in some cases the isotherms intersect already at the relatively low pressures, and in other ones they are nearly parallel to each other. These results indicate a questionable reliability of the data under consideration, which substantially hinders the comparative evaluation of the EoS models. Nevertheless, it still can be concluded (see Table 3) that GSAFT+Cubic predicts the isobaric thermal expansibilities and the thermal pressure coefficients better than PC-SAFT. In addition, it can be seen that CPA is once again characterized by the particularly poor accuracy. In the following discussion let us consider the accuracy of the equations under consideration in modeling vapor pressures and vaporization enthalpies, while taking a representative example

inaccurate representation of the temperature dependence of the density data. In addition, the small values of the infinity pressure densities characteristic for the cubic equations seriously affect the accuracy in modeling the pressure dependence of densities. Unlike the densities, the data on the auxiliary properties in this study have been predicted. Figure 2 depict the experimental data on the isobaric heat capacities at atmospheric pressure and the results yielded by the models under consideration. Taking into account the significant mutual deviations between the experimental data sources and the large uncertainty characteristic for the caloric properties, it still can be concluded that GSAFT+Cubic is a mainly successful estimator of the isobaric heat capacities in a wide temperature range. Unfortunately, it is not a case of PC-SAFT and CPA. In particular, while CPA tends to underestimate the data, PC-SAFT substantially overpredicts them. Moreover, the proximity of the unrealistic behavior exhibited by the model can be clearly seen in the case of [C8mim][NTf2]. The isobaric and the isochoric heat capacities evaluated from sound velocities and densities in wide pressure range are presented on Figure 3. Since this indirect method is sensitive to small uncertainties of the measured data, its results should be considered with the appropriate precaution. Nevertheless it is evident GSAFT +Cubic has a clear superiority over PC-SAFT and CPA. Remarkable, in spite of its flaw theoretical basis, CPA deviates from the data under consideration less than PC-SAFT. Sound velocity is a directly and highly accurately measured auxiliary property. Therefore it presents particular interest for examining the robustness and reliability of EoS models. 2226



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Figure 2. Isobaric heat capacities at atmospheric pressure. Experimental data: [C2mim][NTf2]: ▼,3 ●,35 □;36 [C4mim][NTf2]: ▼,3 ●,8 △,37 □;38 [C6mim][NTf2]: ▼,3 ○,39 ●,40 △41 (100 bar), ◊,42 □;43 [C6mim][NTf2]: ▼,3 ●,35 △41 (100 bar), □;43 [C4mim][PF6]: ▲,37 □,38 △,44 ●;45 Predictions: dashed line, PC-SAFT; solid line, GSAFT+Cubic; dotted−dashed line, CPA.

experimental data, its predictions are close to the molecular simulation data at the lower temperatures. In addition, it can be seen that PC-SAFT most probably substantially underestimates the experimentally unavailable low-temperature vapor pressure data. An important question that should be asked in this occasion concerns the possible impact of the inaccuracies in predicting the deep vacuum vapor pressures of ionic liquids on modeling phase equilibria in their mixtures at the much higher pressures. To address this question, let us briefly consider two representative cases: the high pressure equilibria in the system CO2 (1) + [C4mim][NTf2] (2) and the liquid−liquid equilibrium (LLE) in

of [C4mim][NTf2]. Figure 7 shows that CPA has a clear advantage in predicting the vaporization enthalpies, while GSAFT+Cubic seems to be more accurate than PC-SAFT. Concerning the vapor pressure data it should be pointed out that they have been considered by the parameter fitting of both PC-SAFT and CPA. Unlike that, the results of GSAFT+Cubic are entirely predictive. Since the experimental data are available only in the relatively narrow temperature range of ∼460−520 K, it is helpful to consider their recent Monte Carlo molecular simulation49 at the lower temperatures. Figure 7 demonstrates that CPA has a doubtless superiority over both SAFT models. Although GSAFT+Cubic is the worst estimator of the available 2227



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Figure 3. Isobaric and isochoric heat capacities of [C4mim][NTf2] and [C4mim][PF6] in the extended pressure range. Literature data:10,11 black right-filled circle, 298.15 K; blue left-filled circle, 313.15 K; red top-filled circle, 323.15 K. Predictions: dashed lines, PC-SAFT; solid lines, GSAFT +Cubic; dotted−dashed lines, CPA.

Figure 4. Sound velocities. Experimental data: [C3mim][NTf2],12 [C4mim][NTf2],10 [C5mim][NTf2],12 [C4mim][PF6].11 Predictions: dashed lines, PC-SAFT; solid lines, GSAFT+Cubic; dotted−dashed lines, CPA. 2228



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Figure 5. Isothermal compressibilities. Literature data: [C2mim][NTf2],29 [C4mim][NTf2],10 [C5mim][NTf2],12 [C4mim][PF6].11 Predictions: dashed lines, PC-SAFT; solid lines, GSAFT+Cubic; dotted−dashed lines, CPA.

Figure 6. Isentropic compressibilities. Literature data: [C3mim][NTf2],12 [C4mim][NTf2],10 [C5mim][NTf2],12 [C4mim][PF6].11 Predictions: dashed lines, PC-SAFT; solid lines, GSAFT+Cubic; dotted−dashed lines, CPA. 2229



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Figure 8. High-pressure phase equilibria in the system CO2 (1) + [C4mim][NTf2] (2). Points, experimental data.52 Predictions: dashed lines, PC-SAFT; solid lines, GSAFT+Cubic; dotted−dashed lines, CPA.

Figure 7. Vapor pressure and vaporization enthalpy of [C4mim][NTf2]. Points, experimental data;46−48,50 dotted line, molecular simulation.49 Modeling: dashed line, PC-SAFT; solid line, GSAFT +Cubic; dotted−dashed line, CPA. Figure 9. LLE in the system [C4mim][NTf2] (1) + H2O (2) at atmospheric pressure. Points, experimental data.53 Predictions: dashed WS lines, PC-SAFT (kLB 12 = k12 = 0); dotted−dashed lines, PC-SAFTUNIFAC; solid lines, GSAFT+Cubic.

the system [C4mim][NTf2] (1) + H2O (2) at the atmospheric pressure. It should be pointed out that the accuracy of modeling mixtures, besides the EoS itself, is defined by selection of the mixing and combining rules, the number of binary adjustable parameters, quality of fitting, and other issues going beyond the scope of the present study. Therefore the current analysis is restricted to the simplest case of the entirely predictive application of the models with zero values of the binary adjustable parameters. Figure 8 depicts the high pressure equilibria in the system CO2 (1) + [C4mim][NTf2] (2). The models under consideration are attached by their original mixing rules.1,13,14 As seen, all of them fail in precise prediction of the data, while only GSAFT+Cubic describes the topology of the phase behavior correctly and CPA has a clear superiority below 50 bar. Figure 9 illustrates the LLE in the system [C4mim][NTf2] (1) + H2O (2) at the atmospheric pressure. The SAFT+Cubic parameters for water have been fitted to the densities and sound velocities with zero critical volume translation factor (m = 1, ε/k = 120.19 K, σ = 2.9424 Å, a = 5.8013 bar·mol2/L2, c = 0.12 L/mol, εAB/k = 1700 K and κAB = 0.011, bonding association scheme 4C16). As seen, although the model does not yield precise predictions of the data, its results exhibit an overall advantage over PC-SAFT attached by the more sophisticated mixing rules (Lorentz−Berthelot + WS Wolbach and Sandler,54 kLB 12 = k12 = 0, and the predictive 2 ∞ methodology that utilizes γ values calculated with modified UNIFAC, PC-SAFT-UNIFAC). In addition, inspection of the

results of CPA1 indicates that with the zero binary adjustable parameter this model yields the substantially less accurate predictions of the LLE data under consideration.

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CONCLUSIONS Traditionally, the main objective of the EoS models such as the cubic equations and the various versions of SAFT is the correlation and prediction of phase equilibria in mixtures. However, in such cases the accuracy of modeling is defined not only by the EoS themselves, but also by the mixing schemes. Unlike that, the success of predicting the thermophysical properties is defined by the intrinsic robustness and reliability of the equations. Therefore consideration of these properties present the most evident test for evaluating thermodynamic models. A question asked in this study was: can the EoS developed for ordinary organic compounds be implemented for predicting thermophysical properties of ionic liquids without making special considerations of the complex nature of these substances? In the case of GSAFT+Cubic an answer to this question seems to be positive. Its accuracy in predicting the isochoric and the isobaric heat capacities, the isothermal and the isentropic compressibilities, the isobaric thermal expansibilities, the thermal pressure coefficients, and the sound velocities of 2230



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in Chemical Engineering, Massachusetts Institute of Technology, Cambridge, 1984. (5) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids; McGraw-Hill: New York, 2001. (6) Esperança, J. M. S. S.; Canongia Lopes, J. N.; Tariq, M.; Santos, L. M. N. B. F.; Magee, J. W.; Rebelo, L. P. N. Volatility of Aprotic Ionic LiquidsA Review. J. Chem. Eng. Data 2010, 55, 3−12. (7) Paulechka, Y. U.; Kabo, G. J.; Emel’yanenko, V. N. Structure, Conformations, Vibrations, and Ideal-Gas Properties of 1-Alkyl-3methylimidazolium bis(trifluoromethylsulfonyl)imide Ionic Pairs and Constituent Ions. J. Phys. Chem. B 2008, 112, 15708−15717. (8) Blokhin, A. V.; Paulechka, Y. U.; Strechan, A. A.; Kabo, G. J. Physicochemical Properties, Structure, and Conformations of 1-Butyl3-methylimidazolium Bis(trifluoromethanesulfonyl)imide [C4mim]NTf2 Ionic Liquid. J. Phys. Chem. B 2008, 112, 4357−4364. (9) Paulechka, Y. U.; Kabo, G. J.; Blokhin, A. V.; Vydrov, O. A.; Magee, J. W.; Frenkel, M. Thermodynamic Properties of 1-Butyl-3methylimidazolium Hexafluorophosphate in the Ideal Gas State. J. Chem. Eng. Data 2003, 48, 457−462. (10) de Azevedo, R. G.; Esperança, J. M. S. S.; Szydlowski, J.; Visak, Z. P.; Pires, P. F.; Guedes, H. J. R.; Rebelo, L. P. N. Thermophysical and Thermodynamic Properties of Ionic Liquids over an Extended Pressure Range: [bmim][NTf2] and [hmim][NTf2]. J. Chem. Thermodyn. 2005, 37, 888−899. (11) Esperança, J. M. S. S.; Guedes, H. J. R.; Blesic, M.; Rebelo, L. P. N. Densities and Derived Thermodynamic Properties of Ionic Liquids. 3. Phosphonium-Based Ionic Liquids over an Extended Pressure Range. J. Chem. Eng. Data 2006, 51, 237−242. (12) Esperança, J. M. S. S.; Visak, Z. P.; Plechkova, N. V.; Seddon, K. R.; Guedes, H. J. R.; Rebelo, L. P. N. Density, Speed of Sound, and Derived Thermodynamic Properties of Ionic Liquids over an Extended Pressure Range. 4. [C3mim][NTf2] and [C5mim][NTf2]. J. Chem. Eng. Data 2006, 51, 2009−2015. (13) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244−1260. (14) Polishuk, I. Generalization of SAFT + Cubic Equation of State for Predicting and Correlating Thermodynamic Properties of Heavy Organic Substances. J. Supercrit. Fluids 2012, 67, 94−107. (15) Kontogeorgis, G. M.; Voutsas, E. C.; Yakoumis, I. V.; Tassios, D. P. An Equation of State for Associating Fluids. Ind. Eng. Chem. Res. 1996, 35, 4310−4318. (16) Kontogeorgis, G. M.; Folas, G. K. Thermodynamic Models for Industrial Applications. From Classical and Advanced Mixing Rules to Association Theories; John Wiley & Sons: New York, 2010. (17) Privat, R.; Gani, R.; Jaubert, J.-N. Are Safe Results Obtained When the PC-SAFT Equation of State is Applied to Ordinary Pure Chemicals? Fluid Phase Equilib. 2010, 295, 76−92. (18) Privat, R.; Conte, E.; Jaubert, J.-N.; Gani, R. Are Safe Results Obtained When SAFT Equations are Applied to Ordinary Chemicals? Part 2: Study of Solid-Liquid Equilibria in Binary Systems. Fluid Phase Equilib. 2012, 318, 61−76. (19) Yelash, L.; Müller, M.; Paul, W.; Binder, K. A Global Investigation of Phase Equilibria Using the Perturbed-Chain Statistical-Associating-Fluid-Theory Approach. J. Chem. Phys. 2005, 123, 14908/1−14908/15. (20) Yelash, L.; Müller, M.; Paul, W.; Binder, K. Artificial Multiple Criticality and Phase Equilibria: an Investigation of the PC-SAFT Approach. Phys. Chem. Chem. Phys. 2005, 7, 3728−3732. (21) Polishuk, I. About the Numerical Pitfalls Characteristic for SAFT EOS Models. Fluid Phase Equilib. 2010, 298, 67−74. (22) Polishuk, I.; Mulero, A. The Numerical Challenges of SAFT EoS Models. Rev. Chem. Eng. 2011, 27, 241−251. (23) Ji, X.; Held, C.; Sadowski, G. Modeling Imidazolium-Based Ionic Liquids with ePC-SAFT. Fluid Phase Equilib. 2012, 335, 64−73. (24) Paduszyński, K.; Chiyen, J.; Ramjugernath, D.; Letcher, T.; Domańska, U. Liquid-Liquid Phase Equilibrium of (PiperidiniumBased Ionic Liquid + an Alcohol) Binary Systems and Modelling with NRHB and PCP-SAFT. Fluid Phase Equilib. 2011, 305, 43−52.

ionic liquids should be considered as satisfactorily accurate (considering the uncertainty of the literature data on some properties), and it is comparable to the accuracy achieved in the cases ordinary compounds, such as the heavy n-alkanes and the heavy aromatic.14 Being originally evaluated for heavy organic substances with very small vapor pressures, GSAFT+Cubic neglects phase equilibria in pure compounds. Therefore this model yields inaccurate predictions of vapor pressures not only for ionic liquids, but for ordinary organic substances as well. This drawback can be easily corrected by fitting the canceled yet parameter c. However this practice is incompatible with the simple algebraic scheme of calculating the model parameters. In addition, it seems that the inaccuracies in predicting the deep vacuum vapor pressures do not influence modeling of phase equilibria in mixtures at the much higher pressures. Unlike GSAFT+Cubic, the performance of PC-SAFT in predicting auxiliary thermodynamic properties of ionic liquids deteriorates in comparison to ordinary substances. In particular, PC-SAFT substantially overestimates heat capacities and underestimates the temperature and pressure dependencies of sound velocities and compressibilities. Additional analysis is required for detecting the reasons for this worsening and its possible relation to the numerical pitfall affecting the model. In the case of the cubic EoS based CPA the expectations were not be initially high while considering its performance for thermophysical properties of ordinary substances.51 Nevertheless, it should be pointed out that the predictions of CPA deviate from the literature data of heat capacities less than those of PC-SAFT. At the same time, the success of CPA in modeling vapor pressures and vaporization enthalpies makes this simple EoS a viable competitor for more complex SAFT equations.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +972-3-9066346. Fax: +972-3-9066323. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Acknowledgments are made to Dr. Kamil Paduszyński (Department of Physical Chemistry, Faculty of Chemistry, Warsaw University of Technology) and Mrs. Filipa M. Maia (Center for Energy Resources Engineering, Department of Chemical and Biochemical Engineering, Technical University of Denmark) for their valuable assistance in the appropriate implementation of PC-SAFT and CPA EoS models.

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REFERENCES

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