Extension of a Group Contribution Method To Estimate the Critical

Article pubs.acs.org/IECR

Extension of a Group Contribution Method To Estimate the Critical Properties of Ionic Liquids of High Molecular Mass José O. Valderrama,*,†,‡ Luis A. Forero,‡,§ and Roberto E. Rojas∥ †

Faculty of Engineering, Department of Mechanical Engineering, and ∥Faculty of Sciences, Department of Chemistry, University of La Serena, Casilla 554, La Serena, Chile ‡ Center for Technological Information (CIT), Monseñor Subercaseaux 667, La Serena, Chile § Faculty of Chemical Engineering, Universidad Pontificia Bolivariana, A.A. 56006, Medellín, Colombia S Supporting Information *

ABSTRACT: The group contribution method proposed by Valderrama and Robles in 2007 and updated by Valderrama and coworkers in 2012 to estimate the critical properties of ionic liquids is extended to evaluate these properties for ionic liquids of higher molecular mass. The modifications are done to follow the behavior that these properties have for other type of substances, such as the asymptotic tendency of the normal boiling temperature as the molecular mass M increases. The magnitudes of the modifications are found so that the pressure test and the density test previously defined by the authors are fulfilled for most substances. The proposed extension does not change anything of the original method already in use which is valid for ILs with M < 500 g/mol. A total of 316 ionic liquids with M > 500 g/mol are considered in this work. Of these, 310 passed the pressure test. Also, 111 of these ILs have experimental density values and 103 pass the density test, with absolute average deviation of 4.1 %. A spreadsheet for calculating the critical properties and performing the tests is provided as Supporting Information. The spreadsheet file includes at present the properties for 1630 ionic liquids of molecular mass going from 77 to 1730.

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INTRODUCTION In two previous papers the authors proposed and applied a group contribution method to determine the critical properties (TC, PC, VC), the normal boiling temperature (Tb), and the acentric factor (ω) of ionic liquids1,2 The proposed method unified the best characteristics of two of the most used techniques of Lydersen3 and of Joback and Reid4 and was originally named the Modified Lydersen−Joback−Reid method.1 Valderrama and Robles1 estimated the critical properties of 50 ionic liquids using the Modified Lydersen−Joback−Reid method and proposed three new groups and the values of the contributions for TC, PC, VC, and Tb. Since there are not experimental critical properties to evaluate the accuracy of the estimates, these calculated values were tested for accuracy and consistency by determining the density of the ionic liquids, for which experimental data were available. For this, an independent equation not employed in determining the critical properties was applied. Later, the authors presented an additional test for determining the consistency of the estimated critical temperature, the normal boiling temperature, the critical pressure and the acentric factor. The new testing method included the calculation of the saturation pressure at the normal boiling temperature using an equation of state and an accurate model to represent the temperature function of the attractive term in the equation of state.2 As several authors have indicated, most ionic liquids start to decompose, in many cases, at temperatures approaching the normal boiling point.5 Therefore, critical properties cannot be measured. The estimated properties are useful approximations that must be interpreted as ″these would be the values of TC, PC, VC, Tb and ω, if the properties were possible to be measured.″6 © 2015 American Chemical Society

Since experimental data do not exist and there is no reasonable and generally accepted theory yet available to calculate these properties for ionic liquids, the group contribution method presented by Valderrama and Robles1 and later extended and revised by Valderrama et al.2 is considered to give reasonable estimates. The proposed Valderrama−Robles group contribution method has been used in various applications by other authors. The estimated critical properties have been used to correlate and estimate different properties of ionic liquids such as heat capacity and liquid density.7,8 Wu et al.9−11 developed group contribution methods to estimate the surface tension and the thermal conductivity of ionic liquids in which the estimated critical temperature was involved. Other authors have used equations of state in various applications. The critical properties required by the equations of state were evaluated using the Valderrama−Robles method.12−18 However, none of these studies included applications to high molecular ionic liquids (M > 500 g/mol).

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GROUP CONTRIBUTION METHODS Group contribution methods have been matter of several studies through the years and some limitations have been pointed out in the literature.19 The method of Valderrama and Robles is an extension of the popular and traditional methods of Lydersen and of Joback and Reid and has the same inherent Received: Revised: Accepted: Published: 3480

January 20, 2015 March 13, 2015 March 18, 2015 March 30, 2015 DOI: 10.1021/acs.iecr.5b00260 Ind. Eng. Chem. Res. 2015, 54, 3480−3487

Article

Industrial & Engineering Chemistry Research

Figure 1. Properties of the n-alkanes as the number of carbons (NC) increases: (a) normal boiling temperature; (b) critical temperature; (c) critical pressure; (d) critical volume. (●) Values predicted by the Lydersen−Joback−Reid method; (○) values recommended in literature.20,21

with M = 1731 g/mol). For this high molecular mass ionic liquids the method of Valderrama and Robles predicts very high values for the normal boiling temperature and for the critical temperature (Tb = 1607 K and Tc = 3325 K), ignoring the most probable shape of the curve (similar to the alkanes) in which as the molecular mass increases, the curve asymptotically increases to a limiting value. In the case of n-alkanes, some authors have suggested that the limit for Tb is 1000 K and for Tc is 1200 K.21

limitations of that method. The main one is its wrong prediction of the critical properties for very high molecular mass compounds. For a family of compound, say the n-alkanes, the model used for Tc and Tb are increasing functions of the molecular mass. At very high molecular mass the methods predict even negative values. This happens because the original method is being extended to situations that are far away from the limits in which experimental data were used to estimate the model contributions and model parameters. Also, the model parameters were fitted using data for a group of substances with certain characteristics, for instance for low molecular mass substances. In fact, Joback and Reid4 determined the values of the contributions using data of substances of molecular mass up to about 300 g/mol. What is known is that, as the molecular mass increases, the normal boiling temperature and the critical temperature tend to an asymptotic value and do not go to infinity or do not become negative, as predicted by the models. Figure 1 shows this behavior for the normal boiling temperature. The values used in the figure are taken from the DIPPR databank for alkanes with carbon number from 1 to 36.20 Values for high molecular mass alkanes are approximate extrapolations suggested by some authors.21 Also, the values predicted with the modified Lydersen−Joback−Reid method are included in Figure 1. Some ionic liquids synthesized until now have high molecular mass up to the order of 1700 g/mol (for instance 1-butyl-3methylimidazolium tetrakis[(4-perfluorohexyl)phenyl]borate

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THE EXTENSION The extension of the method to be applied to high molecular mass ionic liquids considers the modification of the properties calculated with the original method so the corrected values follow the asymptotic behavior found for high molecular mass hydrocarbons. The modifications are intended to be applied for ionic liquids with M > 500 g/mol. For the normal boiling temperature, the original method proposed is Tb(K) = 198.2 +

∑ nΔTb

(1)

For M > 500 g/mol, Tb is now defined as Tb*(K) = Tb(K) − δTb

(2)

The term δTb is defined in such a way that the asymptotic behavior assumed for the hydrocarbons is observed. For the critical temperature, the original method is 3481

DOI: 10.1021/acs.iecr.5b00260 Ind. Eng. Chem. Res. 2015, 54, 3480−3487

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Industrial & Engineering Chemistry Research Tc(K) =

Tb A + B ∑ nΔTc − (∑ nΔTc)2

(3)

For this property the extended method proposes: Tc*(K) = fc

Tb* A + B ∑ nΔTc − (∑ nΔTc)2

(4)

The term fc is a factor that includes the structural index “mass connectivity index” (λ) previously defined by the authors.22 The factor fc was determined imposing that the ratio Tb/Tc assumes a value similar to that of other high molecular mass substances (an average value of 0.82). For the critical volume, the original method proposed Vc(cm 3/mol) = 6.75 +

∑ nΔVc

(5)

The extension for M > 500 g/mol proposes V c*(cm 3/mol) = Vc − δVc

Figure 2. Normal boiling point of the [Cxmim] [PF6] series as the number of carbons (NC) in the alkyl group of the cation increases: (●) values predicted by the Valderrama−Robles original method; (○) values calculated with the propose corrections.

(6)

The term δVc is defined in such a way that the average deviation in the prediction of density is minimized. For the critical pressure, the original method proposed Pc(bar) =

M [C + ∑ nΔPc]2

The Term f T for the Critical Temperature. The term f T is proposed by observing and analyzing the relation between Tb and Tc. The proposed decrease in Tb directly affects Tc as seen by the models presented in Table 1 (Tc is equal to Tb divided by a sum of group contributions). However, it is also observed that for high molecular mass organic substances, the value of the ratio Tb/Tc varies between defined ranges, about 0.78 to 0.88 (average 0.82). Figure 3 shows this for the series n-alkanes, n-alkanols, and aliphatic carboxylic acids. For some ionic liquids of high molecular mass the value of Tb/Tc calculated by the original method is too low and does not follow the observed tendency of high molecular mass organic substances. For instance for 1-ethyl-3-methylimidazolium tris[(trifluoromethyl)sulfonyl]methide (with M = 522.4 g/ mol) the value of Tb/Tc is 0.63. For N-octadecyl-isoquinolinium tetrakis [3,5-bis(trifluoromethyl) phenyl]borate (with M = 1245.9 g/mol) the value of Tb/Tc is 0.58. The function f T is incorporated so the value of Tb/Tc is changed according to the observed behavior of high molecular mass hydrocarbons, shown in Figure 3. The Term δVc for the Critical Volume. For the term δVc two facts are considered: (i) the value of the critical volume cannot increase to infinity as the molecular mass increases; and (ii) the observed behavior for n-alkanes presented in Figure 1 indicates that the critical volume also increases in an asymptotic form, although the decreasing from the straight line behavior is smaller than in the case of critical temperature and normal boiling temperature. The Term f P for the Critical Pressure. The tendency of the critical pressure for high molecular mass hydrocarbons presented in Figure 1 indicates that this property already goes asymptotically to a defined low limit as the molecular mass increases (to around 3 to 5 bar, depending on the method). After analyzing the effect of changing the critical pressure it is observed that variations of Pc do not cause important changes in the pressure test so the function f P is very close to one. Iteration Procedure. To evaluate the optimum modified values for the normal boiling temperature, for the critical temperature, and the critical volume (δTb, fc, δVc, and f p), several simple models are tested. For δTb a quadratic expression is chosen. For f T, a simple Padé expression23 is tested while a

(7)

The extension for M > 500 g/mol proposes: Pc*(bar) = fP ·Pc

(8)

The term f P is defined by analyzing its effect after all the other changes have been done. That is made by minimizing the average absolute deviation in the calculated vapor pressure at the normal boiling temperature.

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RESULTS AND DISCUSSION The magnitude and the expressions for calculating the additional terms included in the new models (δTb, fc, δVc, and f P) are determined by minimizing deviations in the predicted pressure (for all substances) and density (for those substances for which experimental density values were available). The form of the expressions that define δTb, fc, δVc, and f P are established following information available in the literature and knowledge about how the critical properties and the normal boiling temperature change and behave for high molecular mass substances.20,21 This is of course known for simple substances such as the hydrocarbons, and it is the hypothesis of our work that for ionic liquids a similar behavior can be assumed. This hypothesis is proven later in this paper by the results of the two tests that check the accuracy and appropriateness of the estimated properties. The Term δTb for the Normal Boiling Temperature. The term δTb is proposed as an extension of the behavior of this property for hydrocarbons and the analysis of Tb for a series of ionic liquids, taken as reference. The 1-alkyl-3methylimidazolium hexafluorophosphate ([Cxmim] [PF6]) chain is considered, and estimated Tb values for this series are calculated using the method of Valderrama and Robles.1 The estimated asymptotic behavior for the ionic liquid chain is considered to be similar to that of the n-alkanes. Considering the more complex structure of ionic liquids than n-alkanes the value of δTb for ionic liquids should be lower than that for nalkanes. Figure 2 shows the expected behavior for this property for high molecular mass [Cxmim] [PF6]. 3482

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Table 1. Modified Group Contribution Method.2 In the Equations, M is in g/mol, Tb and Tc in K, Pc in bar, and Vc in (cm3/mol) model equation M < 500 g/mol normal boiling temp

critical temp

M > 500 g/mol Tb* (K) = Tb (K) − δTb

Tb (K) = 198.2 + ∑nΔTb

Tc(K ) =

δTb = 0.000388M2 − 0.39M + 113

Tb

Tc*(K ) = fT ·

2

A + B ∑ nΔTc − C(∑ nΔTc)

A = 0.5703; B = 1.0121

Pc(bar) = critical pressure

Pc*(K ) = fP ·Pc f p = 1.05λ + 0.0011

C = 0.2573 critical volume

∑ nΔV

V c*(cm3/mol) = Vc − δVc

c

δVc = 15.13λ + 3.328

D = 6.75

ω= acentric factor

⎡P ⎤ (Tb − 43)(Tc − 43) log⎢ c ⎥ (Tc − Tb)(0.7Tc − 43) ⎣ Pb ⎦ −

ω=

⎡P ⎤ (Tc − 43) ⎡ Pc ⎤ log⎢ ⎥ + log⎢ c ⎥ − 1 (Tc − Tb) ⎣ Pb ⎦ ⎣ Pb ⎦

⎡ P* ⎤ (Tc* − 43) ⎡ Pc* ⎤ log⎢ ⎥ + log⎢ c ⎥ − 1 (Tc* − Tb*) ⎣ Pb ⎦ ⎣ Pb ⎦

Pb = 1.01325 bar

the parameters for the linear expression for δVc (c1 and c2) are determined by simultaneously fitting the deviation between predicted density and calculated density as proposed by Valderrama and Robles (2007) and keeping Tb/Tc close to the average value of 0.82 which is an average value for high molecular weight organic substances. To achieve this, the objective function includes the difference |0.82 − Tb/Tc| which is to be minimized. Data for 111 ionic liquids with M > 500 g/mol for which density data are available are considered to find the optimum parameters in the expressions for δTb, δVc, and f T. After finding these parameters the expression for f P is determined by minimizing the deviation between the predicted vapor pressure at the normal boiling point and the reference value of 1.01325 bar. That can be made because the critical pressure has no effect on the density test and on the Tb/Tc ratio but it affects the pressure test. The final parameters and expressions are presented in Table 1. The objective function F1 to determine δTb, δVc, and f T is defined as

Figure 3. Reduced normal boiling point of some organic homologous series. (×) alkanes; (●) aliphatic carboxylic acids; (▲) n-alkanols. Values taken from the DIPPR compilation.20

111

F1 =

plain linear relation is tried for δVc. For the critical pressure it is observed that the molecular mass has no much effect for M > 500 g/mol and the factor f P is assumed to be linear with the mass connectivity index (and varies around the value 1.0). Therefore, the expressions that define these new terms are

(10)

δVc = c1 + c 2λ

(11)

fP = d1 + d 2λ

(12)

∑ i=1

|ρ lit − ρcalc | ρ lit

+

calc |0.82 − Tbr | 0.82

(13)

The objective function F2 to determine f P is defined as 111

F2 =

∑ i=1

(9)

1 − b1λ b2 + b3λ

fT =

⎡ P* ⎤ (Tb* − 43)(Tc* − 43) log⎢ c ⎥ * * * (Tc − Tb )(0.7Tc − 43) ⎣ Pb ⎦ −

Pb = 1.01325 bar

δTb = a1 + a 2M + a3M2

A + B ∑ nΔTc − C(∑ nΔTc)2

A = 0.5703; B = 1.0121 1 − 0.05794λ fT = 1.025 − 0.05704λ

M [C + ∑ nΔPc]2

Vc(cm3/mol) = 6.75 +

Tb*

|P calc − 1.01325| 1.01325

(14)

The terms added to the original models for each of the properties produce variable changes in the value of the critical properties and normal boiling temperature. These variations have been obtained by analyzing the behavior of the properties for high molecular mass hydrocarbons and on the optimization (minimization) of the deviations between calculated and experimental density, the analysis of the Tb/Tc ratio, and on the theoretical approach to the normal vapor pressure. For Tb the maximum value of δTb is 600 K. This variation plus the additional factor f T produces a maximum change in the critical

The parameters of the quadratic expression for δTb (a1, a2, and a3), those for the Padé expression for f T (b1, b2, and b3), and 3483

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Industrial & Engineering Chemistry Research Table 2. Groups Considered in the Valderrama−Robles Method for Ionic Liquids1 ΔTc

Mi −CH3 −CH2− >CH− >C< CH2 CH− C< C CH C− −OH −O− >CO −CHO −COOH −COO− −HCOO− =O (any other) −NH2 −NH3 −NH− >N− =N− −CN −NO2 −F −Cl −Br −I −P −B −S− OSO −CH2− >CH− CH− >C< C< −O− −OH (phenols) >CO −NH− >N− =N−

[>C−]−

−COO− [−O]−

[>N< ]+ [>N]+

15.035 14.027 13.019 12.011 14.027 13.019 12.011 12.011 13.019 12.011 17.008 16.000 28.011 29.019 45.018 44.010 45.018 16.000 16.023 17.031 15.015 14.007 14.007 26.018 46.006 18.999 35.453 79.904 126.905 30.974 10.811 32.066 64.065

0.0275 0.0159 0.0002 −0.0206 0.017 0.0182 −0.0003 −0.0029 0.0078 0.0078 0.0723 0.0051 0.0247 0.0294 0.0853 0.0377 0.036 0.0273 0.0364 0.0364 0.0119 −0.0028 0.0172 0.0506 0.0448 0.0228 0.0188 0.0124 0.0148 −0.0084 0.0352 0.0006 −0.0563

14.027 13.019 13.019 12.011 12.011 16.000 17.008 28.011 15.015 14.007 14.007

0.0116 0.0081 0.0114 −0.018 0.0051 0.0138 0.0291 0.0343 0.0244 0.0063 −0.0011

ΔPc without rings 0.3031 0.2165 0.114 0.0539 0.2493 0.1866 0.0832 0.0934 0.1429 0.1429 0.1343 0.13 0.2341 0.3128 0.4537 0.4139 0.4752 0.2042 0.1692 0.1692 0.0322 0.0304 0.1541 0.3697 0.4529 0.2912 0.3738 0.5799 0.9174 0.1776 0.0348 0.6901 −0.0606 with rings 0.1982 0.1773 0.1693 0.0139 0.0955 0.1371 0.0493 0.2751 0.0724 0.0538 0.0559

ΔVc

ΔTb

66.81 57.11 45.7 21.78 60.37 49.92 34.9 33.85 43.97 43.97 30.4 15.61 69.76 77.46 88.6 84.76 97.77 44.03 49.1 49.1 78.96 26.7 45.54 89.32 123.62 31.47 62.08 76.6 100.79 67.01 22.45 184.67 112.19

23.58 22.88 21.74 18.25 18.18 24.96 24.14 26.15

−10.50 73.23 73.23 50.17 11.74 74.60 125.66 152.54 −0.03 38.13 66.86 93.84 34.86 −24.56 117.52 147.24

51.64 30.56 42.55 17.62 31.28 17.41 −17.44 59.32 27.61 25.17 42.15

27.15 21.78 26.73 21.32 31.01 31.22 76.34 94.97 52.82 68.16 57.55

92.88 22.42 94.97 72.24 169.06 81.10

ature and the acentric factor, including the density and vapor pressure tests, is provided as Supporting Information so the numbers presented in this paper can be checked and the properties for any other ionic liquid can be estimated. The Excel file is easy to use and details about its functioning are provided in the Excel file itself. As previously explained by the authors,1,2 the calculation of the density is done as a global test of the consistency of the estimated critical properties. Errors are within deviations found between experimental data reported by different authors for some ionic liquids. Overall deviations and some other statistical values are similar as those found in previous papers by the authors, a fact that reinforces the statement given above that

temperature of 2117 K. For the critical volume the maximum value of δVc is 250 cm3/mol. The pressure is the less affected property being the maximum variation of 1.1 bar. In terms of percentages with respect to the values proposed by the original method the changes are 37% for Tb, 64% for Tc, 7.0% for Vc and 7.0% for Pc. For better clarity and to avoid confusion with previous works Table 1 summarizes the group contribution method and Table 2 provides the values of the contributions (the same as in the original method). In Table 2 the groups considered in the original method, the equivalences between groups, and the value of the contributions are detailed. The spreadsheet for calculating the critical properties, the normal boiling temper3484

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Industrial & Engineering Chemistry Research the changes introduced in this paper do not affect the overall estimation of the proposed method, but correctness must prevail.

Table 3. Deviations in Correlating the Density and the Normal Vapor Pressure (P = 1.0325 bar) for 1630 Ionic Liquids with Molecular Mass Lower than 500 g/mol (1314 ILs) and Higher than 500 g/mol (316 ILs)

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TEST RESULTS In the Figures 4 and 5 the results for the density and vapor pressure tests are presented. In both cases only those ionic

%|ΔY| of ILs no. of ILs

%ΔY

111 316

3.6

density test pressure test

556

−1.1

1314

−0.8

667

−1.1

1630

max %|ΔY|

M > 500 g/mol −0.01 4.1 30.8

density test pressure test

density test pressure test

%|ΔY|

0.04

5

49

M < 500 g/mol 4.8 19.9 2.3

11

overall (any M) 4.76 30.8 2.8

49

500 g/mol are included. Table 3 summarizes the results obtained for both the density and the pressure tests. In the previous paper in which the pressure test was proposed2 results were reported for 1130 ionic liquids and now the amount increased to 1630. That quantity corresponds to the 1130 ILs with M < 500 g/mol previously reported, 184 new ILs with M < 500 g/mol, and 316 new ILs with M > 500 g/mol. For ILs with M > 500 g/mol, the density test is done for the 111 ionic liquids for which experimental density data are available. As previously established and explained by the authors the limiting deviation accepted for passing the test for an individual ionic liquid is 10%, expressed as absolute deviation.1,2,6 The absolute average deviation for these 111 ionic liquids is 4.1% for the density and only 8 ionic liquids do not pass the test. Of these eight ionic liquids, six present deviations below 20%. The vapor pressure test was applied for the 316 ionic liquids with M > 500 g/mol. The absolute average deviation according to the pressure test is 5.0%. These results are similar to those found in our previous paper, for ionic liquids with M < 500 g/mol. In that case2 a total of 1130 ILs were considered, the average absolute deviation in the vapor pressure test was 7.8%, 29 ILs did not pass the test, and 19 ILs gave deviations below 20%.

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CONCLUSIONS According to the results, the following conclusions are drawn: (i) a new simple method has been proposed to estimate the critical properties of ionic liquids with molecular mass higher than 500 g/mol; (ii) the method can be applied to any ionic liquid that contain any of the 39 groups defined by the method; (iii) the method requires only the molecular mass, the mass connectivity index, and the structure of the molecule; (iv) the average absolute deviation in the density test is 4.8% and in the pressure test 2.8%. If the maximum accepted tolerance is 10%, the number of ionic liquids that do not pass the density test are 109 (out of 667), and 7 (out of 1630) do not pass the pressure test. If tolerance for accepting the test is increased to 20% only 3485

DOI: 10.1021/acs.iecr.5b00260 Ind. Eng. Chem. Res. 2015, 54, 3480−3487

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Industrial & Engineering Chemistry Research Abbreviations

two ionic liquids do not pass the density test and two do not pass the pressure test.

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IL = ionic liquid log = base 10 logarithm

ASSOCIATED CONTENT

Greek Letters

* Supporting Information S

Easy-to-use spreadsheet and detailed results for the 1666 ionic liquids. This material is available free of charge via the Internet at http://pubs.acs.org. The spreadsheet for calculating the mass connectivity index is available in the web associated with the paper by Valderrama and Rojas (2010); http://goo.gl/WSzJ35.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: 56-51-204260. Fax: 56-51-551158. E-mail: jvalderr@ userena.cl. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful for the support of the Direction of Research of the University of La Serena and of the Center for Technological Information of La Serena-Chile. J.O.V. thanks the National (Chilean) Council for Scientific and Technological Research (CONICYT) for its Research Grant FONDECYT 1120162. L.A.F. thanks the faculty of Chemical Engineering of the University Pontificia Bolivariana, Medellin (Colombia).

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ρL = liquid density ρcalc = liquid density calculated with the proposed method ρlit = liquid density from the literature %Δ ρ = percent relative deviation between calculated and literature values for the liquid density ΔM = difference in molecular mass: M − M(cal) ΔTb = contribution to the normal boiling temperature the modified Lydersen−Joback−Reid method ΔTc, ΔPc, ΔVc = contribution to the critical properties in the modified Lydersen−Joback−Reid method Ω = temperature function in density equation (Table 1) λ = mass connectivity index ω = acentric factor δTb = factor to correct the normal boiling temperature for ionic liquids with M > 500 g/mol δVc = factor to correct the critical volume of ionic liquids with M > 500 g/mol

REFERENCES

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NOTATION A, B, C, D = coefficients in the Modified Lydersen−Joback− Reid method M = molecular mass no. = number of ionic liquids N = number of times that a group appears in a molecule (number frequency, in Table 1) NC = number of carbons in a hydrocarbon molecule or in the alkyl group of the IL cation Pb = normal boiling pressure (1.01325 bar) Pc = Critical pressure for ionic liquids with M < 500 g/mol (original method) Pc* = critical pressure for ionic liquids with M > 500 g/mol Pcalc = vapor pressure calculated at the normal boiling temperature R = ideal gas constant (83.144 bar cm3/mol K) T = temperature Tb = normal boiling temperature for ionic liquids with M < 500 g/mol (original method) Tb* = normal boiling temperature for ionic liquids with M > 500 g/mol Tc = critical temperature for ionic liquids with M < 500 g/ mol (original method) Tc* = critical temperature for ionic liquids with M > 500 g/ mol Vc = critical volume for ionic liquids with M < 500 g/mol (original method) Vc* = critical volume for ionic liquids with M > 500 g/mol Zc = critical compressibility factor f T = factor to correct the critical temperature of ionic liquids with M > 500 g/mol f P = factor to correct the critical pressure of ionic liquids with M > 500 g/mol 3486

DOI: 10.1021/acs.iecr.5b00260 Ind. Eng. Chem. Res. 2015, 54, 3480−3487

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DOI: 10.1021/acs.iecr.5b00260 Ind. Eng. Chem. Res. 2015, 54, 3480−3487