Density Prediction of Ionic Liquids at Different Temperatures and

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Ind. Eng. Chem. Res. 2010, 49, 4420–4425

Density Prediction of Ionic Liquids at Different Temperatures and Pressures Using a Group Contribution Equation of State Based on Electrolyte Perturbation Theory Junfeng Wang,†,‡ Zhibao Li,‡ Chunxi Li,*,† and Zihao Wang† State Key Laboratory of Chemical Resource Engineering, Beijing UniVersity of Chemical Technology, Beijing 100029, P.R. China, and Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, P.R. China

Based on the electrolyte perturbation theory, a group contribution equation of state that embodies hardsphere repulsion, dispersive attraction, and ionic electrostatic interaction energy was established to calculate the density of ionic liquids (ILs). According to this method, each ionic liquid is divided into several groups representing cation, anion, and alkyl substituents. The performance of the model was examined by describing the densities of a large number of imidazolium-based ILs over a wide range of temperatures (293.15-414.15 K) and pressures (0.1-70.43 MPa). A total number of 202 data points of density for 12 ILs and 2 molecular liquids (i.e., 1-methylimidazole and 1-ethylimidazole) were used to fit the group parameters, namely, the soft-core diameter σ and the dispersive energy ε. The resulting group parameters were used to predict 961 data points of density for 29 ILs at varying temperatures and pressures. The model was found to estimate well the densities of ionic liquids with an overall average relative deviation (ARD) of 0.41% for correlation and an ARD of 0.63% for prediction, which demonstrates the applicability of the model and the rationality of the soft-core diameter and dispersive energy parameters. 1. Introduction Ionic liquids are salts that have organic cations and inorganic/ organic anions. Most of them are liquids at room temperature. ILs have recently attracted considerable attention because of their unique properties, such as negligible vapor pressure, good electric conductivity, high thermal and chemical stabilities, wide liquid-state range, nonflammability, and good solvent capacity. A main advantage of ILs is that their physical and chemical properties can be readily adjusted by suitable cations, anions, and substituents in order to design an appropriate ionic liquid for a specific purpose. Liquid density is an important physical property for any material of interest that is needed for equipment sizing and also critically needed for the estimation of the lattice energy of an ionic species that can then be utilized to calculate the heat of formation. To date, a large amount of density data on ILs have been measured,1-19 but the practical application of ILs to industry processes is still limited by the scarcity of experimental data. Because experimental measurements are usually timeconsuming and expensive, it is desirable to develop predictive models for calculating the density data of ILs. Several thermodynamic models have been proposed for calculating the densities of ionic liquids. For example, the group additivity method proposed by Ye and Shreeve,8 in which the volume of an IL is assumed to be a linear sum of the volumes of cation and anion, can be used to estimate the density with good accuracy, but its application is restricted to 298.15 K and atmospheric pressure. In addition, the COSMO-RS model based on quantum chemistry was used to predict the density of ILs by Klamt.20 This model is a fully predictive one that uses only structural information of the molecules, but its prediction * To whom correspondence should be addressed. E-mail: Licx@ mail.buct.edu.cn. Tel.: 86-10-64410308. † Beijing University of Chemical Technology. ‡ Chinese Academy of Sciences.

accuracy at present is only qualitative.21 The objective of this work was therefore to develop an accurate model for the prediction of density of ILs over a wide range of temperatures and pressures. In our previous work, we developed an equation of state on the basis of hard-sphere perturbation theory by accounting for the dispersion interaction with Cotterman et al.’s equation of state for a Lennard-Jones fluid and electrostatic interaction with the mean spherical approximation (MSA) approach.22 The results indicated that the density enhancement of an IL with respect to its corresponding physical admixture can be satisfactorily explained in terms of the electrostatic interaction involved, which suggests that ILs can be treated as completely ionized electrolytes for the simplification of modeling. As an extension of that study, in this work, the ILs were divided into different groups representing the cation, anion, and alkyl substituents, and the group parameters were determined by the density data of some ionic liquids. In this work, each group has only two parameters, namely, the soft-core diameters σ and dispersive energy ε/k, and these parameters were further used to predict densities of new imidazolium-based ILs in a wide range of temperatures and pressures. 2. Theoretical Basis a. Model. The Helmholtz free energy for a pure IL system is composed of three parts, namely, the hard-sphere repulsion, the dispersive attraction, and the ionic electrostatic interaction, as described in detail in our previous work.22 Accordingly, the system pressure, P, can be expressed as a sum of these contributions P ) Phs + Pdis + Pelec

(1)

The first two terms on the right-hand side of eq 1 are available in the literature and can be expressed as23,24

10.1021/ie901590h  2010 American Chemical Society Published on Web 03/26/2010

Phs )

[

3ξ1ξ2 (3 - ξ3)ξ2 ξ0 6kT + + 2 π (1 - ξ3) (1 - ξ3) (1 - ξ3)3

3

()

4 ξ3 Pdis 1 ) mA1m FkT τ T˜ m)1

∑

ξn )

π 6

∑Fd

m

+

()

4 ξ3 1 mA2m 2 τ T˜ m)1

∑

(n ) 0, 1, 2, 3)

n

i i

]

Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010

(2)

m

(3)

(4)

i

T˜ ) kT/ε

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(5)

where di is the hard-sphere diameter of group i, which is temperature-dependent and related to the temperature-independent soft-core diameter of Lennard-Jones fluid σi by the equation

Figure 1. Segmentation of 1-methylimidazole to imidazolium cation.

with available parameters for the methyl group and can be used to calculate the density data of ionic liquids. For the ionic liquids studied, the parameters of the cation in the ionic liquids, σ and ε for [RMIM]+, can be calculated using the equations ε[RMIM]+ ) ε[MIM] + εR

(11)

3

di 1 + 0.2977T˜ ) σi 1 + 0.33163T˜ + 1.0477 × 10-3T˜2

(6)

The coefficients A1m and A2m in eq 3 were taken from ref 24 and are provided in the Supporting Information. The electrostatic contribution to the pressure of ILs, Pelec, as derived from integral equation theory under the mean spherical approximation (MSA),25 is given by the equation

[

Pelec ) -kT

( )]

1 aPn Γ3 + 3π 8 ∆

2

(7)

We assume that the pure ionic liquid is completely ionized into its ionic species. The density of an IL at specified temperature can be calculated by using eqs 1-7, together with the two ionic parameters, namely, the soft-core diameter σi and the dispersive parameter εi of ion i. b. Estimation of Parameters. To determine the group parameters, the following objective function, OBF, was minimized using the least-squares method OBF ) min

1 n

n

|Fexp - Fcal i i |

i)1

Fexp i

∑

(8)

To represent the properties of ionic liquids with the group contribution equation of state, the two parameters (σ and ε) of groups representing cations, anions, and alkyl groups are required. The parameters of the methyl group (CH3) were obtained from the literature,26 from which the two parameters of the imidazolium cation ([IM]+) and the ethyl group (C2H5) were determined by fitting the experimental densities of N-methylimidazole (MIM) and N-ethylimidazole (EIM) (taken from our previous work22) in the temperature range of 293.15-323.15 K with eqs 1-6, together with the combining rules in eqs 9 and 10 ε[IM]+ ) εRIM - εR

(9)

3

σ[IM]+ ) √σRIM3 - σR3

(10)

where R represents methyl, ethyl, and other alkyl groups. As shown in Figure 1, 1-methylimidazole is divided into one methyl radical (•CH3) and one imidazole radical (IM•), which further forms an imidazolium cation ([IM]+) as one electron is lost. For heterocyclic groups, such as IM• and [IM]+, losing or gaining an electron has a negligible effect on the soft-core diameter. This implies that the parameter values of imidazolium cation can be approximated as those of neutral imidazole group, which were determined from density data of 1-methylimidazole

σ[RMIM]+ ) √σ[MIM]3 + σR3

(12)

The parameters of bis(trifluoromethylsulfonyl)imides (Tf2N) anion were obtained by regressing the density data for 1,3-dimethylimidazolium Tf2N ([MMIM][Tf2N]) from the literature.1 The density data for 1-butyl-3-methylimidazolium Tf2N ([BMIM][Tf2N]), 1-hexyl-3-methylimidazolium Tf2N ([HMIM][Tf2N]), and 1-octyl-3-methylimidazolium Tf2N ([OMIM][Tf2N]) were used to determine the group parameters of butyl (C4H9), hexyl (C6H13), and octyl(C8H17) groups with the available parameters for [Tf2N] anion, CH3, and MIM. It was found that the soft-core diameters σ and energy parameters ε/k of CH3, C2H5, C4H9, C6H13, and C8H17 groups directly connected to the cationic imidazolium ring follow a binomial distribution with the number of carbon atoms in the alkyl groups, nC. The binomial functions obtained are as follows σnC ) -0.0176nC2 + 0.5055nC + 3.3186

R2 ) 0.999 (13)

(ε/k)nC ) 3.2978nC2 + 1.467nC + 109.95

R2 ) 0.983 (14)

The binomial relativity coefficient between the σ parameters of the alkyl groups and the number of carbon atoms in the alkyl groups is 0.999, indicating a high relativity. Therefore, the σ parameters of propyl (C3H7), amyl (C5H11), heptyl (C7H15), nonyl (C9H19), and decyl (C10H21) groups can be calculated by eq 13. The schematic correlation between the σ parameters of the alkyl groups and the number of carbon atoms in the alkyl groups is provided in the Supporting Information. Although the relativity between the ε/k parameter and the number of carbon atoms in the alkyl groups is less correlative, the ε/k value has small effect on the calculated density. The ε/k parameters of C3H7, C5H11, C7H15, C9H19, and C10H21 can also be calculated by the eq 14, as listed in the Table 1. Equations 13 and 14 provide a basis for estimating parameters of longer alkyls appearing in an imidazolium-based ionic liquid. The parameters of other anions, namely, hexafluorophosphate (PF6), tetrafluoroborate (BF4), chloride (Cl), trifluoromethyl sulfonate (CF3SO3), dicyanamide [N(CN)2], tricyanomethyl [C(CN)3], hydrogen sulfate (HSO4), and trifluoroacetate (CF3CO2) were obtained by correlating the experimental density data of [BMIM][PF6], [BMIM][BF4], [HMIM][Cl], [BMIM][CF3SO3], [BMIM][N(CN)2], [BMIM][C(CN)3], [BMIM][HSO4], [BMIM][CF3CO2], and [MMIM][Tf2N], respectively. The experimental data used for parameter fitting and the groups

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Table 1. Group Parameters Determined (ε/k, σ) group a

CH3 C2H5 C3H7b C4H9 C5H11b C6H13 C7H15b C8H17 C9H19b C10H21b [Tf2N][PF6][BF4][Cl][N(CN)2][C(CN)3][CF3SO3][CF3CO2][HSO4][IM]+ a

ε/k (K-1)

σ (Å)

98 144.60 144.03 186.14 199.73 205.28 281.81 345.56 390.27 454.40 558.39 331.06 265.87 134.31 298.70 225.26 308.82 267.35 271.63 379.17

3.75 4.3184 4.7001 5.0616 5.4075 5.6985 5.9781 6.2251 6.4398 6.6341 5.8517 4.3010 3.8302 3.0349 3.9620 4.5783 4.6163 4.2881 3.6598 3.9103

Figure 2. Liquid densities of [MMIM][Tf2N], [BMIM][Tf2N], [HMIM][Tf2N], and [OMIM][Tf2N] at 0.1 MPa. ∆, experimental data; s, correlated results.

Taken from the literature.26 b Parameters estimated from eqs 13 and

14.

determined are summarized in Table 2. The resulting group parameters are listed in Table 1. 3. Results and Discussion As shown in Table 2, the experimental data for ionic liquids/ pure liquids can be well represented using the group contribution equation of state with an overall ARD of 0.41% and a maximum ARD of 1.09%. Figure 2 shows a comparison of the correlated and experimental liquid densities for [MMIM][Tf2N], [BMIM][Tf2N], [HMIM][Tf2N], and [OMIM][Tf2N] at normal pressure. As shown in Figure 2, the densities of ionic liquids always decrease with increasing temperature because of weakening molecular interactions. The calculated densities agree well with the experimental data. To show the overall performance of the model, a comparison of the correlation results of density with the experimental data for 12 ILs and 2 molecular liquids is shown in Figure 3. As can be seen, the densities of these ionic liquids and molecular liquids can be represented with high accuracy. The results demonstrate the applicability of the model proposed for ionic liquids. Based on the above satisfactory results, a total of 961 data points for 29 ILs based on imidazolium cations with PF6, BF4, Cl, N(CN)2, C(CN)3, CF3SO3, HSO4, CF3CO2, and Tf2N anions covering wide ranges of temperature, 293.15-414.15 K, and

Figure 3. Correlated versus experimental densities for 12 ionic liquids and 2 molecular liquids as noted in Table 2.

pressure, 0.1-70.43 MPa, were predicted with the available σ and ε/k parameters as listed in Table 1. The overall information on the experimental data summarized in Table 3 includes the ionic liquids used for prediction, the numbers of data points, the temperature and pressure ranges covered, and the overall prediction accuracy in terms of ARD. As shown in Table 3, the experimental densities of ionic liquids can be well represented using the group contribution equation of state with an overall average relative deviation of 0.63%. Among these predicted data points, 68.8% were within a relative deviation

Table 2. Groups Determined and Calculation Accuracy in ARD from Fitting the Densities for 12 ILs and 2 Molecular Liquids ionic liquid/pure liquid MIM EIM [MMIM][Tf2N] [BMIM][Tf2N] [HMIM][Tf2N] [OMIM][Tf2N] [BMIM][PF6] [BMIM][BF4] [HMIM]Cl [BMIM][N(CN)2] [BMIM][C(CN)3] [BMIM][CF3SO3] [BMIM][HSO4] [BMIM][CF3CO2] total

temperature range (K) 293.15-323.15 293.15-323.15 298.15-353.15 293.15-353.15 293.15-358.15 298.15-343.15 293.49-414.93 293.49-414.92 298.15-343.15 297.15-355.85 293.15-393.15 295.75-342.95 293.15 298

pressure range (MPa) 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1-30 0.1 0.1 0.1

no. of data points 7 7 12 13 14 10 10 10 10 6 96 5 1 1 202

ARD (%) 0.08 0.14 0.13 0.11 0.62 1.09 0.70 0.31 0.17 0.14 0.61 0.25 0.77 0.65 0.41

group determined +

[IM] C2H5 [Tf2N]C4H9 C6H13 C8H17 [PF6][BF4]Cl[N(CN)2][C(CN)3][CF3SO3][HSO4][CF3CO2]-

ref 22 22 1 1 2 2 3 3 4 5 6 5 7 8

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Table 3. Predictive Accuracy of Density in ARD for 29 Imidazolium-Based Ionic Liquids ionic liquid [EMIM][Tf2N] [C3MIM][Tf2N] [BMIM][Tf2N] [C5MIM][Tf2N] [C7MIM][Tf2N] [OMIM][Tf2N] [C9MIM][Tf2N] [C10MIM][Tf2N] [BMIM][PF6] [HMIM][PF6] [C7MIM][PF6] [OMIM][PF6] [C9MIM][PF6] [EMIM][BF4] [C3MIM][BF4] [BMIM][BF4] [HMIM][BF4] [OMIM][BF4] [MMIM]Cl [BMIM]Cl [OMIM]Cl [EMIM][N(CN)2] [OMIM][N(CN)2] [EMIM][CF3SO3] [BMIM][CF3SO3] [HMIM][CF3SO3] [OMIM][CF3SO3] [EMIM][CF3CO2] [EMIM][HSO4] total

temperature range (K) 298.15-353.15 293.15-393.15 293.49-414.92 293.15 293.49-414.92 293.15 293.15-393.15 293.15-393.15 293.15 293.15 273.15-298.15 278.15-318.15 293.15-393.15 293.15 273.15-363.15 293.15-393.15 293.15 298.15-323.15 298 293.15 293.15-323.15 293.15-393.15 288.15-323.15 288.15-323.15 293.15-393.15 293.15 298 298.15-343.15 298 298 298 293.15-393.15 293.15-393.15 298 298 298 298.15

pressure no. of range (MPa) data points ARD (%) ref 0.1 0.1-30 0.1 0.1 0.1 0.1 0.1-30 0.1-30 0.1 0.1 0.1 0.1 0.1-10 0.1 0.1 0.1-10 0.1 0.1 0.1 0.1 0.1 0.1-10 0.1 0.1 0.1-10 0.1 0.1 0.1 0.1 0.1 0.1 0.1-35 0.1-10 0.1 0.1 0.1 0.1

12 96 10 1 10 1 96 96 1 1 6 9 77 1 13 77 1 7 1 1 31 77 36 36 77 1 1 10 1 1 1 91 77 1 1 1 1 961

0.52 0.60 0.33 0.20 0.19 0.75 0.74 0.80 0.95 0.22 0.12 0.39 0.48 0.54 0.66 0.53 0.27 2.76 0.09 1.04 0.28 0.37 0.38 0.56 0.60 0.57 0.79 0.20 0.64 0.50 1.29 0.48 0.35 0.99 1.77 0.87 0.66 0.63

1 6 3 7 3 7 6 6 7 7 9 12 13 7 14 13 7 15 16 7 17 13 17 17 13 7 18 4 8 8 11 19 13 8 8 8 9

of 0.5%, 18% were within 0.5-1.0%, 10.2% were within 1.0-2.0%, and only 2.1% were above 2.0%. Figure 4 shows a comparison of the predicted and experimental liquid densities for 1-ethyl-3-methylimidazolium BF4 ([EMIM][BF4]), [BMIM][BF4], [HMIM][BF4], and [OMIM][BF4] at atmospheric pressure. It can be seen that the model agrees well with the experimental data for [BMIM][BF4], [HMIM][BF4], and [OMIM][BF4], whereas a large deviation is observed for [EMIM][BF4]. The high deviation for [EMIM][BF4] is likely due to the inaccuracy of the experimental

Figure 4. Predicted (lines) versus experimental (symbols) densities at 0.1 MPa. Legend for experimental data: ∆, [EMIM][BF4];152, [EMIM][BF4];16 0, [BMIM][BF4];17 9, [BMIM][BF4];13 b, [HMIM][BF4];17 ], [OMIM][BF4];17 [, [OMIM][BF4].13

Figure 5. Predicted (lines) versus experimental (symbols) densities at 0.1 MPa. Legend for experimental data: ∆, [EMIM][Tf2N];6 2, [EMIM][Tf2N];3 O, [EMIM][Tf2N];1 0, [BMIM][Tf2N];3 ×, [C7MIM][Tf2N];6 ], [OMIM][Tf2N].6

density values used,15 which might be related to the water content and other impurities of the IL samples or the experimental method employed, because excellent agreement was also obtained for [EMIM][BF4] at atmospheric pressure with a relative deviation of 0.09%.16 Similarly, the average predictive deviation for [EMIM][CF3SO3] from the literature is 1.29%,11 whereas the prediction deviation from the literature19 is only 0.48% for the same IL in even wider ranges of temperature (293.15-393.15 K) and pressure (0.1-35 MPa). Other systems with higher deviations are [OMIM][CF3SO3] and [C3MIM][BF4], with ARDs of 1.77% and 1.04%, respectively, for which we are unable to verify the accuracy of the experimental data. Because of the limited availability of experimental density data assessment, the rejection of doubtful data was not done in this work. Figure 5 shows a comparison of the predicted and experimental liquid densities for [EMIM][Tf2N], [BMIM][Tf2N], [C7MIM][Tf2N], and [OMIM][Tf2N] at normal pressure. As shown in Figure 5, the density prediction accuracy monotonically decreases with increasing size of the alkyl group for the same series of ionic liquids; that is, the prediction accuracy follows the order [EMIM][Tf2N] ≈ [BMIM][Tf2N] > [C7MIM][Tf2N] > [OMIM][Tf2N]. This can be ascribed to the increasing nonsphericity of the cations as the alkyl-chain group becomes longer, which deviates from the spherical assumption of the particles embedded in the model. It is also noted that the prediction results can mimic the experimental tendencies of the cation effect on the density of ionic liquids; that is, as the length of the alkyl chain of the cation increases, the density of ionic liquids with the same anion decreases significantly. The decreasing density of ionic liquids with increasing size of cation mainly results from the weakening ionic electrostatic interaction because the Coulomb energy is inversely proportional to the closest approach between ions, which is determined by the bulk of ions. To show the overall prediction performance of the model, a comparison of the predicted results with the experimental data for 29 ILs over wide ranges of temperature and pressure is shown in the Figure 6. It can be seen that the experimental densities are well predicted by the model, with an average absolute relative deviation (ARD) less than 0.63%. The results indicate that the model is applicable for all of the systems studied, as the deviation basically occurs evenly in all of the systems and no data points show a biased deviation from the diagonal in Figure 6.

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Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010 Fcal ) calculated density, g/cm3 Fexp ) experimental density, g/cm3 σi ) soft-sphere diameter of species i, Å

Supporting Information Available: Table listing the coefficientsA1m and A2m. Figure showing the correlation between σ parameters of alkyls and the number of carbon atoms in alkyl groups. This information is available free of charge via the Internet at http://pubs.acs.org. Literature Cited

Figure 6. Predicted versus experimental densities for 29 imidazolium-based ionic liquids as noted in Table 3.

4. Conclusions A group contribution equation of state based on the electrolyte perturbation theory, in which the IL is treated as a completely dissociated electrolyte, has been developed to represent the densities of ionic liquids. The structure of an ionic liquid is considered to be composed of a cation, an anion, and alkyl substituent groups. Each group has two parameters, namely, the soft-core diameter σ and the dispersive energy ε. The performance of the model was examined by describing the densities for imidazolium-based ILs containing PF6, BF4, Cl, N(CN)2, C(CN)3, CF3SO3, CF3CO2, HSO4, and Tf2N anions, over wide ranges of temperature, 293.15-414.15 K, and pressure, 0.1-70.43 MPa. The group parameters were obtained by correlating 202 density data points for 12 ionic liquids and 2 pure liquids with an average deviation of 0.41%, and the resulting group parameters were used for the prediction of 961 density data points for 29 ionic liquids with good accuracy. This justified the applicability of the group contribution equation of state and the reasonableness of group parameters. As parameters for new groups become available, the model proposed here can be extended to a larger number of ionic liquids. Acknowledgment The authors are grateful for the financial support from the National Science Foundation of China (20376004, 20776015) and the Research Fund for the Doctoral Program of Higher Education of China (20090010110001). Nomenclature di ) hard-sphere diameter of species i, Å n ) number of data points nC ) number of carbon atoms in alkyl groups P ) system pressure, kPa Pdis ) pressure of the dispersive interaction, kPa Pelec ) pressure of the electrostatic interaction, kPa Phs ) pressure of a hard sphere, kPa T ) absolute temperature, K T˜ ) dimensionless temperature Zi ) valence of species i ε ) dispersion energy, kJ/mol Γ ) inverse shielding length, Å-1 κ ) Boltzmann constant, 1.38066 × 10-23 J/K Fi ) number density of species i, m-3

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ReceiVed for reView May 25, 2009 ReVised manuscript receiVed March 7, 2010 Accepted March 11, 2010 IE901590H