Correlations with the SWCF-VR Equation - American Chemical Society

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Capturing Thermodynamic Behavior of Ionic Liquid Systems: Correlations with the SWCF-VR Equation Changchun He,† Jinlong Li,‡ Changjun Peng,*,† Honglai Liu,† and Ying Hu† †

State Key Laboratory of Chemical Engineering and Department of Chemistry and ‡Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China S Supporting Information *

ABSTRACT: An equation of state for square-well chain fluids with variable well-width range (SWCF-VR EoS) [Li et al. Fluid Phase Equilib. 2009, 276, 57] was applied to ionic liquid (IL) systems. ILs were treated as the square-well chain with hydrogen bonding. The corresponding association parameters were given according to our previous work [He et al. Fluid Phase Equilib. 2011, 302, 139]. The nonassociation parameters were obtained by correlating the experimental liquid densities. Excellent agreements were observed between experimental and theoretical results for pure ILs, and the molecular parameters were linearly correlated with the molecular masses of the [Cnmim][NTf2] members (n = 2, 3, ..., 8, 10). It is found that the other thermodynamic properties such as the vapor pressure and the enthalpy of vaporization, etc., can be reasonably predicted by using the obtained molecular parameters. The phase behavior of the binary systems containing ILs was well-represented with a simple mixing rule. For the vapor−liquid equilibria (VLE) of a system of volatile fluid + IL at low pressures, a temperature-independent binary interaction parameter was adopted. Satisfactory results were achieved for both the self- and cross-associating systems. The influence of temperature on the binary interaction parameters was taken into account in the correlation for the gas−liquid equilibria (GLE) of CO2 + IL mixtures and liquid−liquid equilibria (LLE) of IL-containing systems. For CO2 + IL mixtures, the multipolar interactions between like and unlike molecules, and the cross-association between CO2 and IL molecules were neglected to reduce the computational complexity, and the correlated results agree well with the experimental ones over a wide range of temperatures and pressures. The LLE of alkanol + IL systems were acceptably reproduced with moderate deviations between the experimental and calculated mass fractions. In the water-rich phase of water + IL with LLE, the neglect of electrostatic interaction caused correlated results to deviate from experimental ones greatly.

1. INTRODUCTION Recently, ionic liquids (ILs), in which either the cations or anions or both of them have higher molecular mass than traditional ions, have been given much attention due to their unique physicochemical properties.1−7 For instance, despite consisting of cations and anions, ILs are usually in liquid state at room temperature; compared with conventional solvents, they are almost nonvolatile. In addition, they also have wide liquid range, excellent thermal and chemical stability, high solubilities of many solutes, good electroconductibility, nonflammability, diversity of cations and anions, and function adjustability. These properties lead to some potential advantages in application to many industrial processes such as organic chemical reaction,8−11 separation engineering,12−14 energy and resource utilization,15 environmental conservation,16,17 materials processing,18,19 and aviation and astronavigation,20 as well as biomedical R&D.21 The characterization, modeling, and prediction of the physiochemical properties and phase equilibria are the key issues for the application of ILs. A reliable thermodynamic model will be crucial. A growing number of experimental data on pVT and phase equilibria of ILs also have provided an impetus for the extension of thermodynamic models to this field. Up to the present, a series of models22−41 have satisfactorily been used to reproduce the properties. In earlier pioneering works, the researchers neglected some complicated interactions in IL systems such as hydrogen © 2012 American Chemical Society

bonding, ionic association, dipolar interaction, electrostatic interaction, and so on. The simple cubic equations of state (EoSs) were the first choice. Shariati et al.22 correlated the phase behavior of [C2mim][PF6] + CHF3 with the Peng− Robinson (PR) EoS. Yokozeki and Shiflett23−26 applied the modified Redlich−Kwong (RK) EoS to the binary IL systems. With the use of four binary cross-interaction parameters, this modified EoS can successfully mimic the vapor−liquid equilibria (VLE), liquid−liquid equilibria (LLE), and vapor−liquid− liquid equilibria (VLLE) for binary mixtures containing ILs. Subsequently, other types of EoSs were also applied to this field. Wang et al.27,28 correlated the densities of ILs, solubilities of CO2 in ILs and VLE of binary systems containing imidazolium-based ILs with the square-well chain fluid (SWCF) EoS.29 They28 regarded an imidazolium-based IL as a “diblock compound” composed of an alkyl group and imidazolium ring anion pair. Similar to the group contribution method, this approach gives a useful example for modeling IL structure. Xu et al.30 modeled the densities of ILs, solubilities of CO2 in ILs, VLE, LLE, and VLLE of some binary systems containing ILs, and LLE of ternary systems containing ILs by using a new lattice-fluid (LF) Received: Revised: Accepted: Published: 3137

September 29, 2011 January 19, 2012 January 20, 2012 January 20, 2012 dx.doi.org/10.1021/ie202237e | Ind. Eng. Chem. Res. 2012, 51, 3137−3148

Industrial & Engineering Chemistry Research

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parameters, three for binary parameters characterizing the interactions between CO2 and the three groups and two for crossassociation parameters. Li et al.42 treated ILs in aqueous solutions as partially ionized weak electrolytes and added the hydrogenbonding, electrostatic interaction, and ionic association between the cations and anions to the SWCF-VR EoS. The elaborate model can accurately describe the liquid densities, osmotic coefficients, and vapor pressures of IL aqueous solutions. The SWCF-VR EoS was established by our group based on SWCF EoS.43 Compared with the SWCF EoS,29 there are two improvements in the new EoS. One is the derivation of an expression for the square-well dispersion term based on the second-order perturbed theory44 and the PY2 approximation of the integral equation;45 and the other is the separation of the chain formation term into the hard-sphere chain formation term and the correction term from the effect of square-well potential. Analogous to the SWCF EoS, the nearest neighbor connectivity and the effect of next-to-nearest neighbor indirectconnectivity are considered through the nearest neighbor and next-to-nearest neighbor cavity correlation functions (CCFs). The SWCF-VR EoS has been used for the calculation of vapor−liquid equilibria (VLE) for nonassociating small molecules,46,47 polymers,46,47 refrigerants,48 ILs,31 and associating small molecules.49 However, despite the application of the SWCF-VR EoS to ILs, Li et al.31 ignored the hydrogen-bonding and variable well-width in IL molecules. We therefore employed the EoS to reproduce the phase behavior of IL systems by introducing the hydrogen bonding and variable well width. The rest of this paper is outlined as follows. Section 2 briefly describes the EoS. The correlations of the experimental densities and predictions of some caloric properties for pure ILs are presented in sections 3.1 and 3.2. The phase behavior for the binary systems containing ILs is shown in the rest of this section. Finally, we conclude this work in section 4.

EoS. It is worth noting that besides the temperature-dependent binary interaction parameter, they introduced another adjustable binary parameter to describe the effective segment number in calculating LLE. Their comprehensive and satisfactory work has laid a solid foundation for the further improvement. Li et al.31 adopted an EoS for square-well chain molecules with variable range (SWCF-VR) to compute the pVT and phase equilibria properties with a given reduced-well width (λ = 1.5). Breure32 et al. developed a group contribution EoS for the prediction of the phase equilibria in CO2 + ILs. It is a promising method for representing the phase behavior of the mixtures containing ILs. In addition, the classical activity coefficient models also exhibit superiority in representing the liquid properties of IL mixtures at low pressures.33 To describe IL systems more accurately, the complicated interactions mentioned above should be taken into account partially or totally. By accounting the cross association between CO2 and IL molecules, quadrupolar interaction of CO2 molecules, dipolar interaction of IL molecules, and multipolar interaction between CO2 and IL molecules, Kroon and co-workers34 successfully reproduced the solubilities of CO2 in [Cnmim][BF4] (n = 2, 4, 6) and [Cnmim][PF6] (n = 4, 6, 8) at elevated pressures using the EoS developed from the truncatedperturbed-chain polar statistical associating fluid theory (tPCPSAFT). In their work, an IL was viewed as a mixture of anion and cation, the molecular parameters of ILs were calculated from those for the ions based on the literature data. The results show the binary interaction parameter for CO2 + IL increases linearly with temperature and decreases asymptotically with the molecular mass for the two families. Subsequently, they reestimated the tPC-PSAFT parameters for ILs by fitting to the latest experimental liquid densities over a wider temperature range and restricting the model to predict the extremely low vapor pressures and recorrelated the solubilties of CO2 by using the new set of parameters,35 and better agreement between the calculated and experimental data was obtained. Moreover, the solubilties of O2, CO, and CHF3 in [C4mim][PF6] were also calculated with the new tPC-PSAFT parameters. Andreu and Vega36 checked the ability of the soft-SAFT EoS to capture the solubilities of CO2 in [Cnmim][BF4] (n = 2, 4, 6, 8) and [Cnmim][PF6] (n = 4, 6, 8). They modeled CO2 and IL as a Lennard-Jones chain with explicit quadrupolar interaction and with an associating site, respectively. The association parameters for ILs were fixed as those for 1-alkonols optimized in their previous work with the assumption that there is no effect of the length of chain on the strength of association bonds.37 The correlated results with a temperature-independent binary interaction parameter agree well with the experimental solubilities of CO2 in ILs. Moreover, the solubility behavior of CO2, H2, and Xe in [Cnmim][NTf2] was described with the softSAFT EoS by treating the ILs as Lennard-Jones chains with three associating sites.38 Afterward, the VLE and LLE of the mixtures containing [Cnmim][NTf2] were predicted with the recalculated molecular parameters from the new available experimental data.39 Some physical properties of pure [Cnmim][NTf2] were also investigated.39 Ji and Adidharma40 employed heterosegmented SAFT model to reproduce the densities of ILs by dividing them into the alkyls, cation heads, and anions. The electrostatic and polar interactions between cations and anions were represented by the cross interaction between the two kinds of ions. In the subsequent work, they examined the performance of the model to reproduce the solubilities of CO2 in ILs.41 Satisfactory results were achieved with five adjustable

2. EQUATION OF STATE In the SWCF-VR EoS for a K-component associating mixture, each molecule i is divided into ri tangent square-well spheres with the same diameter σi. According to our previous work,43 the residual Helmholtz function of the mixture can be written as ⎛ βΔAHS ‐ mono βAr,SWCF βΔASW ‐ mono ⎞ ⎟⎟ = r ⎜⎜̃ + rN rN N0 ̃ 0 ̃ 0 ⎝ ⎠ Δ(HS ‐ chain) βΔASW βΔAHS ‐ chain + + N0 N0 assoc βΔA + N0

(1)

K where N0 = Σi=1 Ni0 is the total number of molecules in a mixture K irrelevant to association and r̃ = Σi=1 xi0ri is the mean segment number. Ni0, xi0, and ri denote the number of molecules, the mole fraction, and the number of segments of component i, respectively. β = 1/kT, k is the Boltzmann constant, and T is the temperature. The superscripts “r,SWCF”, “HS-mono”, “SWmono”, “HS-chain” and “assoc” represent the residual property, the hard-sphere repulsion, the square-well dispersion interaction, the formation of chain hard-sphere and the association interaction between segments, respectively. “Δ(HS-chain)” describes the effect of square-well dispersion on the formation of molecule chain.

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dx.doi.org/10.1021/ie202237e | Ind. Eng. Chem. Res. 2012, 51, 3137−3148


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The expressions of the first four terms on the right of eq 1 are given in our previous work.42 For hydrogen-bonding association contribution, Liu et al.50 proposed a simplified sticky-point model and the association contribution to residual Helmholtz function, which can be expressed as βΔAassoc = N0

K

∑ i=1

⎤ ⎡ 1 xi 0⎢ln Xi + (1 − Xi)⎥ ⎦ ⎣ 2

λ ij =

κ ij = δεij =

(2)

Xi = (1 +

∑ ρj0XjΔij)

(3)

where ρj0 is the molecule number density of component j. The parameter Δij is the strength of bond between associating segments and can be calculated by π ‐ (2e)σ 3 Δij = κ ijτij−1ySHS ij i Sj (4) 3 Here κij and σij are the fraction of the segment surface participating in association and the collision diameter of a hard-sphere segment, respectively. The parameter τij is a sticky strength parameter computed by

τij−1 = eβδεij − 1

κ ii + κ jj (13)

2 δεiiδεjj

(14)

3. RESULTS AND DISCUSSION 3.1. pVT Properties of Pure ILs. Before applying an EoS to mixtures, an essential step is to obtain the model parameters of the corresponding pure compounds. As shown by some results from molecular simulation52 and experiments,53 there is a complex network of hydrogen-bonding in an imidazolium-based IL, and for the water-free IL systems, both the electrostatic interaction and ionic association are negligible. We therefore modeled ILs as square-well chains only with hydrogen-bonding (one associating site) as shown in Figure 1a, and all other

−1

j=1

(12)

Here, the subscripts “ii” and “jj” denote the pure fluids i and j, respectively, kij is a binary adjustable parameter introduced to improve the correlation precision for mixtures.

where Xi is the mole fraction of component i not bonded and ρ0 denotes the total molecule number density. Xi can be solved from the following equation K

λ iiσii + λjjσjj σii + σjj

(5)

Here δε is the association energy parameter. In eq 4, ySiSjHS‑(2e) denotes an effective cavity correlation function between associating segment Si and Sj and can be written as ‐ (2e) = ln ySHS S i j

(3 + α2)B2 E2 η /F2 − (1 + β2)E23/F2 2 2(1 − η) +

Figure 1. Sketches of several theory-based models used to represent ILs. (a) SWCF-VR; (b) tPC-PSAFT; (c) soft-SAFT with one associating site; and (d) soft-SAFT with three associating sites. The interaction potential ε between two segments is represented by the filled colors: (blue) −εLJ ≤ ε < 0 (−εLJ is the minimal potential in the Lennard-Jones model); (white) ε = 0; (red) ε > 0; and (black) ε → +∞.

(1 + β2)E23/F2 2 2(1 − η)2

E 3 − (1 + δ2) 22 ln(1 − η) F2

interactions such as electrostatic interaction, ionic association, and multipolar interaction are empirically suggested to be embedded in and are not included explicitly. Figure 1 also presents the sketches of the tPC-PSAFT (Figure 1b) and soft-SAFT (Figure 1c and d) models; they regarded ILs as ion pairs with dipole moment and Lennard-Jones chains with one or three associating sites, respectively. The SWCF-VR EoS in this work have six temperature-independent molecular parameters, namely number of segments (r), collision diameter of a hard-sphere segment (σ), reduced-well width (λ), well depth (ε/k), segment surface fraction participating in association (κ), and association energy (δε/k). The former four are nonassociation parameters, and the latter two are association parameters. Andreu and Vega36,38,39 assumed that the association parameters of the softSAFT EoS for ILs were equal to those of 1-alkanols,37 which were set to constant values by considering that the length of chain should not affect the strength of association bonds. To reduce the number of the adjustable parameters and simplify the computational procedure, in this work, we also fixed the two association parameters as κ = 0.001 and δε/k = 3600 K, which are close to the mean values of the optimized association parameters of the SWCF-VR EoS for alkanols.49 The nonassociation parameters were then estimated by fitting to the

(6)

where η = π/6∑i=1K ρi0riσi3 is the reduced segment density of mixture. The parameters α2, β2, and δ2 are model constants given by α 2 = − a2 + b2 − 3c 2 ,

β2 = − a2 − b2 + c 2 ,

δ2 = c2 (7)

a2 = 0.45696,

b2 = 2.10386,

c 2 = 1.75503

(8)

The parameters B2, E2, and F2 in eq 6 are defined as B2 =

σi + σj 2

,

E2 =

σi 2 + σj 2

,

2

F2 =

σi 3 + σj3 2 (9)

51

The standard Lorentz−Berthelot rule the cross parameters, viz σii + σjj σij = 2 εij = (1 − k ij) εiiεjj

was used to estimate

(10) (11) 3139



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Good agreement between the experimental and calculated results is observed. Furthermore, the molecular parameters for the [Cnmim][NTf2] (n = 2, 3,..., 8, 10) family are linearly correlated with the molecular masses. As shown in Table 1 and

experimental liquid densities with the objective function expressed as 100 OF1 = NP

2 NP ⎛ exp cal ⎞ ρ − ρ i i ∑ ⎜⎜ exp ⎟⎟ ρi ⎠ i=1 ⎝

Table 1. Parameters for Linear Expressions between Optimized Molecular Parameters MP and molecular masses MW of the [Cnmim][NTf2] family (n = 2, 3, ..., 8, 10)

(15)

where ρ is the density of ionic liquid and NP is the number of data points. The superscripts “exp” and “cal” denote the experimental and calculated data. The molecular parameters of 43 pure ILs were obtained, which are given in Table S1 of the Support Information. The overall average deviation of densities is 0.06%, which is a little improved in comparison with the results of Li et al.31 (0.07%, 23 ILs) and Xu et al.30 (0.12%, 46 ILs). For most of the ILs, the optimized square-well widths are in the range of 1.7 ≤ λ ≤ 1.9. In contrast, the square-well widths for most of conventional fluids vary in the range of 1.6 ≤ λ ≤ 1.8.46,49 Moreover, the square-well depths calculated by Li et al.31 (λ = 1.5) are greater than those in this work, which is largely caused by the dependence between square-well depths and widths. Besides, the separation of the hydrogen-bonding from the total interactions also contributes to the decrease of the square-well depths. Figures 2 and 3

a

parameter

aa

b

R2

r rNAσ3 rε/k rλ

0.0150406 1.10292 3.20243 0.0283478

0.0379725 −196.615 346.660 −0.125010

0.9913 0.9988 0.9866 0.9877

MP = aMW + b.

Figure 4, all of the squares of correlation coefficients (R2) are greater than 0.985, indicating the reliability of the linear relation which is capable to predict the pVT properties of other members of this family. [C1mim][NTf2] is selected as a demonstration. The predicted densities are reported in Table S1 of the Support Information and Figure 2. A density AAD of 0.20%, slightly greater than those of the correlated members, is an acceptable value. The linear relationship is indistinctive for other families such as [Cnmim][BF4] and [Cnmim][PF6], probably because of the inconsistent ranges of temperatures and pressures in the experimental data. 3.2. Vapor Pressures, Vaporization Enthalpies, and Normal Boiling Points of Pure ILs. Although ILs are regarded as nonvolatile fluids, it is substantiated that ILs exhibit extremely low vapor pressure.6,60,61 Zaitsau et al.60,61 measured the vapor pressures of [Cnmim][NTf2] (n = 2, 4, 6, 8) by the integral effusion Knudsen method in a temperature range from 440 to 500 K. The vapor pressures predicted by the SWCF-VR and soft-SAFT EoSs39 are plotted in Figure 5. It is found that the results produced by the SWCF-VR EoS have the same order of magnitude with experimental ones and the soft-SAFT EoS obviously overestimated the vapor pressures. In addition, the vaporization enthalpies and normal boiling points (NBPs) of the ILs can also be predicted by our EoS. In Zaitsau et al.’s work,61 these properties of the ILs were also derived from the above experimental vapor pressures by using a simple equation. Table 2 gives a comparison between the experiment-based61 and EoS-predicted results. The trend of predictions with the SWCF-VR EoS is similar to the that of the experiment-based one. However, larger vaporization enthalpies are obtained by the SWCF-VR EoS. For the NBPs, the experiment-based data show a nonmonotonic trend with a maximum in [C4mim][NTf2], while both the predicted values by the SWCF-VR EoS and group contribution method62 increase monotonically with the alkyl chain. 3.3. VLE for Binary Mixtures Containing ILs. In a binary system of a volatile fluid and an IL, the IL concentration in the vapor phase is negligible at low temperatures. As is known, there is the strong interaction between a volatile fluid and IL only in the high dense phase, such as liquid and supercritical phases, while the vapor and gas phases far away from the supercritical point do not exhibit the strong interaction due to the low density. In this work, the VLE temperatures of some binary systems are up to 353 K, but far away from the supercritical points of the binary system. Without association, the vapor is close to ideal gas. In addition, the saturated vapor pressures of

Figure 2. Experimental54−58 (symbols) and calculated (lines) densities of [Cnmim][NTf2] (n = 1, 2, ..., 8, 10) at 0.1 MPa.

Figure 3. Experimental59 (symbols) and calculated (lines) densities of [C4mim][BF4].

compare the experimental and calculated densities of [Cnmim][NTf2] (n = 1, 2, ..., 8, 10) at 0.1 MPa and [C4mim][BF4] over a wide temperature and pressure range, respectively. 3140



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Figure 4. Linear dependence of molecular parameters on molecular masses for [Cnmim][NTf2] family (n = 1, 2, ..., 8, 10). (a) r vs MW; (b) rNAσ3 vs MW; (c) rε/k vs MW; and(d) rλ vs MW.

only the pure volatile fluid exists in vapor phase in calculating the VLE of the system. An adjustable binary interaction parameter is introduced for the characterization of the interaction between the unlike segments. Using the molecular parameters of the pure compounds, we calculated the VLE of the mixture using the SWCFVR EoS by minimizing the following objective function, 100 OF2 = NP Figure 5. Experimental60,61 (symbols) and predicted (lines) saturated pressures for [Cnmim][NTf2] (n = 2, 4, 6, 8, lines from top to bottom): (solid lines) SWCF-VR EoS; (dashed lines) soft-SAFT EoS.39

2

(17)

Here p is the vapor pressure. The molecular parameters for volatile fluid were optimized in our previous work46,49 except for cyclohexene and n-nonane; and the parameters of the two hydrocarbons were estimated with the saturated vapor pressures and liquid densities and listed in Table 3. The reproduced results for the vapor pressures of volatile fluid + IL systems are reported in Table S2 of the Support Information. For comparison, the deviations from the experimental data of the SWCF28 and LF30 EoSs are also included in the table. Note that three adjustable binary interaction parameters were used in the SWCF EoS. For the lattice-fluid EoS developed by Xu et al., the interaction energies ε were treated as functions of temperature, and for the SWCF-VR EoS, all of the molecular parameters are temperature-independent. Due to the correction of interaction energies in the mixing rule, the SWCF-VR EoS shows bigger deviations than the lattice fluid EoS with one

the ILs involved in the binary systems are extremely low in 353 K. It can be concluded from some experimental data,60,61 such as 0.0062 Pa at 441.70 K for [C2mim][NTf2]. According to the estimations by the following equation, S x /p yIL ≈ pIL IL

NP ⎛ pexp − pcal ⎞ ∑ ⎜⎜ i exp i ⎟⎟ pi ⎠ i=1 ⎝

(16)

the concentrations of ILs in the vapor phases are negligible even at 353 K. Here yIL and xIL are the molar fractions of the S IL in vapor and liquid phases, and pIL and p are the saturated vapor pressure of the pure IL and the vapor pressure of the binary system, respectively. Thus, it is reasonably assumed that

Table 2. Experimental and Predicted Vaporization Enthalpies and NBPs for [Cnmim][NTf2] (n = 2, 4, 6, 8)

a

IL

T (K)

ΔvapHexp (kJ·mol−1)

ΔvapHcal 61 (kJ·mol−1)

% ADΔvapHa

Tbexp (K)

Tbcal (K)61

% ADTba

TbGC (K)62

% ADTba

[C2mim][NTf2] [C4mim][NTf2] [C6mim][NTf2] [C8mim][NTf2]

463.0 477.6 461.8 475.2

118.8 118.3 123.4 132.3

130.6 141.8 152.3 153.3

9.93 19.86 23.42 15.87

907 933 885 857

914.71 980.15 1020.47 1207.80

0.85 5.05 15.31 40.93

806.1 851.8 897.6 943.4

11.12 8.70 1.42 10.08

% ADDΔvapH = |(ΔvapHexp − ΔvapHcal)/ΔvapHexp| × 100, % ADTb = |(Tbexp − Tbcal)/Tbexp| × 100. 3141



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Table 3. Molecular Parameters and Correlated Results for Cyclohexene and n-Nonane optimized parameters

% AAD

fluid

MW (g·mol−1)

T (K)

r

NAσ3 (cm3·mol−1)

ε/k (K)

λ

pS

VL

ref

Cyclohexene n-Nonane

82.144 128.255

254−546 308−585

2.28860 3.66394

32.9608 36.6615

219.683 211.602

1.72574 1.68919

1.83 0.56

1.08 0.66

63 63

temperature-independent binary parameter for the binary mixtures containing IL at several temperatures. Namely, the binary parameter in our model has more significant temperature dependency. Despite the slightly bigger deviations, we can see that the SWCF-VR EoS can reliably describe the VLE of a selfor cross-associating volatile fluid + IL system with only one temperature-independent binary interaction parameter. The successful correlations are shown graphically by two examples in Figures 6 and 7, namely benzene + [Cnmim][NTf2] (n = 1, 2, 4)

Figure 8. Experimental33 (symbols) and predicted (lines) mixing enthalpies of ethanol + [C6mim][NTf2] at 353.15 K. k12 = 0.102931.

CO2 + IL is a valuable work. Some researchers correlated the experimental solubilities of CO2 in ILs with various EoSs. Table 4 lists the consideration of the interactions between molecules from CO2 + IL in the SWCF-VR and several other EoSs. For the accurate description of the phase behavior of the system, the quadrupolar interaction between CO2 molecules was involved in the tPC-PSAFT34,35 and soft-SAFT EoSs,36,38 and the contributions from the dipolar interaction between IL molecules and multipolar interaction between CO2 and IL molecules were added into the tPC-PSAFT EoS. In addition, the tPCPSAFT and soft-SAFT EoSs considered the contributions of the cross-association between CO2 and IL molecules and selfassociation between IL molecules, respectively. It is worth noting that although the transient dimers were found in supercritical CO2 by Raman scattering,65 the hydrogen-bonding between the CO2 molecules were not considered in the listed EoSs. We think the assumption is reasonable because CO2 is a hydrogen-free compound and the dimers are not caused by the hydrogenbonding. Moreover, the latest research found that the solubility of a special IL in CO2 was up to about a mass fraction of 0.07,66 but the CO2 phase of the observed systems in this work can still be viewed as an IL-free phase. In our model, we neglected the multipolar interactions in the systems owing to the complicated expressions for the Helmholtz free energy. The cross-association between CO2 and IL were also taken into no account to further simplify our model. In other words, the SWCF-VR EoS modeled CO2 + IL as a self-associating system. Different from the temperature-independent binary parameters for the systems in Section 3.3, especially at high CO2 fractions, the binary interaction parameters for the SWCF-VR EoS were set to functions of temperature to represent the gas−liquid equilibria (GLE) more precisely. The temperature dependency of binary parameters was used for CO2 + IL. The relationship between binary interaction parameter k12 and temperature T is expressed as a quadratic function,

Figure 6. Experimental33 (symbols) and calculated (lines) VLE of benzene + [Cnmim][NTf2] (n = 1, 2, 4) at 353.15 K. k12 = −0.046272, −0.039150, −0.035765.

Figure 7. Experimental64 (symbols) and calculated (lines) VLE of alkanol + [C4mim][OcSO4] at 313.15 K. k12 = 0.094266, 0.052669, 0.018782.

at 353.15 K and alkanol (methanol, ethanol, and 1-propanol) + [C4mim][OcSO4] at 313.15 K. Using the information about crossinteraction, one can predict other thermodynamic properties of a mixture. Figure 8 gives a comparison between the experimental33 and predicted mixing enthalpies of ethanol + [C6mim][NTf2] at 353.15 K. The results are satisfactory. 3.4. Solubilities of CO2 in ILs. Due to the great influence of CO2 on human life, modeling the phase behavior of

k12 = a(T /273.15)2 + b(T /273.15) + c

(18)

Here, a, b, and c are the adjustable coefficients and are listed in Table 5. In the tPC-PSAFT EoS,34,35 the binary interaction parameter k12 is a linear function of temperature T with a positive slope. Meanwhile, in the soft-SAFT EoS, two 3142



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Table 4. Consideration of Interactions between CO2 + IL Molecules in Several EoSs this work

tPC-PSAFT34,35

soft-SAFT36,38

lattice-fluid30

no no no yes no no σij = 0.5(σi + σj) εij = (εiεj)0.5(1 − kij)

yes yes yes no yes yes σij = 0.5(σi + σj) εij = (εiεj)0.5(1 − kij)

no yes no yes no no σij = 0.5(σi + σj)(1 − lij) εij = (εiεj)0.5(1 − kij)

no no no no no no εij = (εiεj)0.5(1 − kij)

interaction CO2

dipolar interaction quadrupolar interaction dipolar interaction association cross-association multipolar interaction mixing rule

IL cross

Table 5. Parameters for Quadratic Expressions between Binary Interaction Parameters k12 and Temperatures T in CO2 + IL Mixtures with GLE

a

IL

aa

b

c

[C4mim][BF4] [C6mim][BF4] [C8mim][BF4] [C2mim][PF6] [C4mim][PF6] [C6mim][PF6] [C2mim][NTf2] [C4mim][NTf2] [C6mim][NTf2] [C8mim][NTf2]

0.183909 0.181287 0.107965 0.0663123 0.154344 −0.0128192 0.264524 0.209621 0.169685 0.231257

−0.406565 −0.369971 −0.254978 −0.142640 −0.352799 0.0436217 −0.582636 −0.473613 −0.372514 −0.530738

0.319653 0.307460 0.258740 0.158263 0.308445 0.0762289 0.418001 0.376786 0.325150 0.433583

k12 = a(T/273.15)2 + b(T/273.15) + c.

temperature-independent binary interaction parameters were used to characterize the interaction, as shown in Table 4. The molecular parameters for CO2 were obtained from ref 46. The objective function for correlation is identical with that of VLE for binary mixtures containing ILs, i.e. eq 17. Figures 9−11

Figure 10. Experimental70−72 (symbols) and calculated (lines) bubblepoint pressures of CO2 + [Cnmim][PF6]. (a) n = 2; (b) n = 4; and (c) n = 6. The term k12 is referred to in Table 5.

Figure 9. Experimental67−69 (symbols) and calculated (lines) bubblepoint pressures of CO2 + [Cnmim][BF4]. (a) n = 4; (b) n = 6; and (c) n = 8. The term k12 is referred to in Table 5.

Figure 11. Experimental73 (symbols) and calculated (lines) bubblepoint pressures of CO2 + [Cnmim][NTf2]. (a) n = 2; (b) n = 4; (c) n = 6; and (d) n = 8. The term k12 is referred to in Table 5.

show the comparison between the experimental and correlated solubilities of CO2 in [Cnmim][BF4] (n = 4, 6, 8), [Cnmim][PF6] (n = 2, 4, 6), and [Cnmim][NTf2] (n = 2, 4, 6, 8) at several temperatures. As observed from the figures, the correlated results for these systems agree well with the experimental

ones over a wide pressure range (0−100 MPa). This is very inspiring because of the successful representation of the abrupt slope changes in these systems using a simple model. In the range of correlated temperatures, the obtained binary interaction 3143



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the wide applications to some processes such as liquid−liquid extraction. We selected four cross-associating systems, namely methanol + [C2mim][NTf2], ethanol + [C2mim][NTf2], 1-propanol + [C2mim][NTf2], and water + [C6mim][BF4] at 0.1 MPa. The objective function for correlations is expressed as

parameters for the systems with the SWCF-VR EoS are about 0.08 < k12 < 0.13, less than the initial work34 (0.15 < k12 < 0.23) and greater than the subsequent work35 (0.01 < k12 < 0.08) by using the tPC-PSAFT EoS. Moreover, for the [Cnmim][BF4] and [Cnmim][NTf2] families, there is a relationship between the coefficients a and b in eq 17, i.e. 2a < b < 2.5a (a > 0 and b < 0). Consequently, the binary interaction parameters k12 exhibit minimums in the temperature range from 273 to 341 K. It is perhaps caused by the total effect of temperature on the various interactions between CO2 and IL molecules. Our EoS takes into no account the temperature dependency of welldepth, which is introduced in the SAFT-based EoSs. The temperature dependency of well-depth in the SAFT-based EoSs can be given by ε = ε*(1 + e/kT )

100 OF3 = NP1 +

(19)

(a > 0, b > 0)

L1,exp L ,cal − w IL1 )2 ∑ (wIL

i=1 NP2

100 NP2

L 2,exp L ,cal − w IL2 )2 ∑ (wIL (22)

i=1

where wIL denotes the mass fraction of IL and the subscripts 1 and 2 represent the two liquid phases. The molecular parameters for the alkanols and water were regressed in ref 49. The binary interaction parameters are quadratic functions of temperatures as well as those for the above CO2 + IL systems. Figure 13 makes a graphic comparison between the experi-

Here, ε* is the a temperature-independent energy parameter and e/k is a constant related to the Pitzer’s acentric factor and critical temperature. In the application of the tPC-PSAFT EoS to CO2 + IL mixtures, the binary interaction parameters increases linearly with temperature and can be written as k12 = a + bT

NP1

(20)

where a and b are the slope and intercept of the straight line for k12 vs T. Thus, the cross well depth is computed by ε12 =

ε10ε02 (1 + e/kT )(a + bT )

0 (a + be/k + ae/kT + bT ) = ε12 0 (a + be/k + 2 abe/k ) ≥ ε12

(21)

Namely, if the temperature dependency of well depth is not considered in tPC-PSAFT EoS, there will also be a minimum for the binary parameter. Note that the cross association and cross multipolar interactions were considered in the tPC-PSAFT EoS. As a result of neglecting the cross association in our EoS, cross multipolar interactions, etc., these cross interactions are integrated into the cross energy parameter ε12. Thus, these interactions between CO2 and IL molecules can influence the binary parameter. A plot of the binary parameters k12 versus temperatures T for CO2 + [Cnmim][NTf2] (n = 2, 4, 6, 8) is presented in Figure 12.

Figure 13. Experimental74 (symbols) and calculated (lines) results of LLE coexistence curves for alkanol + [C2mim][NTf2]: (solid lines) SWCF-VR EoS; (dashed lines) lattice-fluid EoS.30 k12 is referred to in Table 6.

mental74 LLE points and correlated LLE coexistence curves for three alkanol + [C2mim][NTf2] systems. The coefficients used to describe the temperature-dependent binary interaction parameters are presented in Table 6. Although there are moderate Table 6. Parameters for Quadratic Expressions between Binary Interaction Parameters k12 and Temperatures T in the Binary Systems with LLE system 1-propanol 1-butanol 1-pentanol water a

[C2mim][NTf2] [C2mim][NTf2] [C2mim][NTf2] [C6mim][BF4]

aa

b

c

−1.251061 −0.581133 −3.009690 −0.135645

2.535712 1.219597 7.250926 0.241255

−1.228371 −0.588129 −4.326381 −0.082082

k12 = a(T/273.15)2 + b(T/273.15) + c.

deviations between the experimental and calculated mass fractions of [C2mim][NTf2], the correlations with SWCF-VR EoS are satisfactory due to the simple modeling. Xu and coworkers30 also represented the LLE of these systems with a new lattice-fluid EoS. To modify the effective chain lengths of two components, they introduced another adjustable binary parameter apart from the temperature-dependent binary interaction parameter k12. The new parameter leads to the adjustable

Figure 12. Temperature dependency of binary parameters k12 for CO2 + [Cnmim][NTf2] (n = 2, 4, 6, 8): (triangles) n = 2; (squares) n = 4; (diamonds) n = 6; and (circles) n = 8.

3.5. LLE of Binary Mixtures Containing ILs. Reproducing the LLE of a mixture with an EoS is a challenging work. Nevertheless, it is important to investigate the LLE because of 3144



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between CO2 and IL molecules, the SWCF-VR EoS successfully reproduced the phase behavior of CO2 + IL over a wide temperature and pressure range. For the LLE of alkanol + [C2mim][NTf2], the deviations between the experimental and calculated mass fractions of the IL are acceptable due to the simple modeling. The efforts show that the SWCF-VR EoS can give a reliable representation for the thermodynamic behavior of IL systems. Without taking into account the electrostatic interaction between ions, the LLE coexistence curve was poorly calculated in the water-rich phase of water + [C6mim][BF4]. The electrostatic interaction in water + IL with LLE requires to be integrated into our model to correlate the phase behavior of these systems accurately.

molecule volumes varying with the composition of a mixture and obviously improves the calculated results. In this work, only the binary interaction parameter k12 was adopted. Similar to CO2 + IL, k12 for the three systems have minimums at 277 K for methanol, 287 K for ethanol, and 329 K for 1-propanol. Figure 14 gives the experimental75 and correlated results for

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ASSOCIATED CONTENT

S Supporting Information *

Table S1: the optimized molecular parameters and correlated results for pure ILs. Table S2: the correlated results of VLE for binary mixtures containing ILs. This material is available free of charge via the Internet at http://pubs.acs.org.

■

Figure 14. Experimental75 (symbols) and calculated (lines) results of LLE coexistence curves for water + [C6mim][BF4]: (solid lines) SWCF-VR EoS. k12 is referred to in Table 6.

AUTHOR INFORMATION

Corresponding Author

*Phone: 86-21-6425 2767. Fax: 86-21-6425 2767. E-mail: [email protected].

water + [C6mim][BF4]. Different from the IL-rich phase, the water-rich phase has big deviations. The calculated contents of [C6mim][BF4] in the water-rich phase are a great deal lower than the experimental ones. The reason is that the IL can be ionized in the water-rich phase more easily and our model treats the IL as a square-well chain with hydrogen-bonding and does not takes into account the electrostatic interaction. A more reliable model should be developed for phase behavior of water + IL systems. Inspiringly, Li et al.42 has done the complicated work by incorporating the electrostatic interaction and ionic association into the SWCF-VR EoS and successfully captured the VLE for water + IL. It is natural that the modified model can be extended to the LLE for water + IL.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support for this work was provided by the National Natural Science Foundation of China (No. 20876041, 21136004), National Basic Research Program of China (2009CB219902), and the 111 Project (Grant B08021) of China.

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4. CONCLUSIONS This work extended the SWCF-VR EoS to the IL systems. An IL molecule was modeled as an associating square-well chain with given association parameters. The nonassociation molecular parameters for pure ILs were determined by fitting to the experimental liquid densities. The pVT properties of 43 ILs were well captured with an overall average absolute density deviation of 0.06%. For the members of the [Cnmim][NTf2] family, there is a linear relationship between the molecular parameters and molecular masses. To examine the robustness of the model, we predicted other thermodynamic properties including the vapor pressure and the enthalpy of vaporization for several [Cnmim][NTf2] ILs. The extremely low vapor pressures predicted by our model have the same order of magnitude with the experimental ones and good agreement was observed between the experimental and predicted vaporization enthalpies. Furthermore, by using the molecular parameters for the pure ILs, the model was applied to the VLE, GLE, and LLE of the binary mixtures containing ILs. The VLE of a volatile fluid + IL system can be satisfactorily represented with only one temperature-independent binary interaction parameter. In calculating the GLE and LLE, the binary interaction parameters were set to quadratic functions of temperature to describe the cross-interaction more accurately. Without the addition of the multipolar interactions in CO2 + ILs and cross-association 3145

NOMENCLATURE A = Helmholtz function Δ(HS‑chain) ΔA SW = contribution to residual Helmholtz function from the effect of square-well dispersion on hard-sphere chain formation a, b, c = coefficients of function in eqs 18, 20, and 21 a2, b2, c2 = model constants in eq 10 B2, E2, F2 = parameters defined by eq 11 e/k = constant related to the Pitzer’s acentric factor K = number of component(s) k = Boltzmann constant kij = binary interaction parameter between components i and j k12 = binary interaction parameter between components 1 and 2 MP = molecular parameter of SWCF-VR EoS MW = molecular mass N = molecule number NP = number of data points NA = Avogadro’s number p = pressure R2 = square of correlation coefficient r = number of segment r̃ = mean segment number of a mixture T = temperature V = molar volume wf = weighting factor dx.doi.org/10.1021/ie202237e | Ind. Eng. Chem. Res. 2012, 51, 3137−3148


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Xi = mole fraction of component i NOT bonded xi = liquid mole fraction of component i yi = vapor mole fraction of component i ySi SjHS‑(2e) = effective cavity correlation function between associating segment Si and Sj Greek Letters

α2, β2, δ2 = model constants β = 1/kT Δij = strength of bond between associating segment i and j δε = association energy ε = square-well depth η = reduced segment density κ = fraction of segment surface participating in association λ = reduced-well width μ = chemical potential ρ = molecule number density σ = collision diameter of hard-sphere segment τ = sticky parameter

Superscripts

cal = calculated value exp = experimental value HS-chain = contribution from hard-sphere chain formation and association interaction HS-mono = contribution from hard-sphere repulsion of monomer L = liquid phase r,SWCF = residual property of square-well chain fluid S = saturated value SW-mono = contribution from square-well dispersion interaction between segments V = vapor phase Subscripts

0 = property without association i,j = pure component indexes ij = interaction of component i and j IL = property of IL Abbreviations

[C2mim][BF4] = 1-ethyl-3-methylimidazolium tetrafluoroborate [C4mim][BF4] = 1-butyl-3-methylimidazolium tetrafluoroborate [C6mim][BF4] = 1-hexyl-3-methylimidazolium tetrafluoroborate [C8mim][BF4] = 1-octyl-3-methylimidazolium tetrafluoroborate [C2mim][PF6] = 1-ethyl-3-methylimidazolium hexafluorophosphate [C4mim][PF6] = 1-butyl-3-methylimidazolium hexafluorophosphate [C6mim][PF6] = 1-hexyl-3-methylimidazolium hexafluorophosphate [C8mim][PF6] = 1-octyl-3-methylimidazolium hexafluorophosphate [C4mmim][PF6] = 1-butyl-2,3-dimethylimidazolium hexafluorophosphate [C4mim][I] = 1-butyl-3-methylimidazolium iodide [C6mim][I] = 1-hexyl-3-methylimidazolium iodide [C8mim][I] = 1-octyl-3-methylimidazolium iodide [C1 mim][NTf2] = 1-methyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [C2mim][NTf2] = 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

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[C3mim][NTf2] = 1-propyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [C4mim][NTf2] = 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [C5mim][NTf2] = 1-pentyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [C6mim][NTf2] = 1-hexyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [C7mim][NTf2] = 1-heptyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [C8mim][NTf2] = 1-octyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [C10mim][NTf2] = 1-decyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [C2mmim][NTf2] = 2,3-dimethyl-1-ethylimidazolium bis(trifluoromethylsulfonyl)imide [C2eim][NTf2] = 1,3-diethylimidazolium bis(trifluoromethylsulfonyl)imide [C3mmim][NTf2] = 2,3-dimethyl-1-propylimidazolium bis(trifluoromethylsulfonyl)imide [C 3 mpy][NTf 2 ] = 3-methyl-1-propylpyridinium bis(trifluoromethylsulfonyl)imide [C3mpyr][NTf2] = 1-propyl-1-methylpyrrolidinium bis(trifluoromethylsulfonyl)imide [C4mpyr][NTf2] = 1-butyl-1-methylpyrrolidinium bis(trifluoromethylsulfonyl)imide [P(14)666][NTf2] = trihexyl(tetradecyl)phosphonium bis(trifluoromethanesulfonyl)imide [N(4)111][NTf 2 ] = trimethyl(butyl)ammonium bis(trifluoromethylsulfonyl)imide [N(6)111][NTf 2 ] = trimethyl(hexyl)ammonium bis(trifluoromethylsulfonyl)imide [N(6)222][NTf 2 ] = triethyl(hexyl)ammonium bis(trifluoromethylsulfonyl)imide [C1mim][MeSO4] = 1-methyl-3-methylimidazolium methylsulfate [C2mim][MeSO4] = 1-ethyl-3-methylimidazolium methylsulfate [C4mim][MeSO4] = 1-butyl-3-methylimidazolium methylsulfate [C2mim][EtSO4] = 1-ethyl-3-methylimidazolium ethylsulfate [C4mim][OcSO4] = 1-butyl-3-methylimidazolium octylsulfate [C2mim][CF3SO3] = 1-ethyl-3-methylimidazolium trifluoromethanesulfonate [C4mim][CF3SO3] = 1-butyl-3-methylimidazolium trifluoromethanesulfonate [C4mim][C(CN)3] = 1-butyl-3-methylimidazolium tricyanomethane [C4mim][dca] = 1-butyl-3-methylimidazolium dicynamide [C 4 mim][CTf 3 ] = 1-butyl-3-methylimidazolium tris(trifluoromethylsulfonyl)methide [P(14)666][Cl] = trihexyl(tetradecyl)phosphonium chloride [P(14)666][Ac] = trihexyl(tetradecyl)phosphonium acetate

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