Volume, surface and UNIQUAC interaction parameters for imidazolium based ionic liquids via Polarizable Continuum Model

Fluid Phase Equilibria 234 (2005) 64–76

Volume, surface and UNIQUAC interaction parameters for imidazolium based ionic liquids via Polarizable Continuum Model Tamal Banerjee, Manish K. Singh, Ranjan Kumar Sahoo, Ashok Khanna ∗ Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India Received 5 March 2005; received in revised form 24 May 2005; accepted 26 May 2005 Available online 5 July 2005

Abstract Ionic liquids (ILs) have shown great potential as solvent/media for reactions and separations. However, the physico-chemical characteristics of ILs are scarce and the limitless different combinations of cations and anions further complicate the matter. The ternary tie line data along with the binodal curve gives a fair indication regarding the feasibility of the ILs as solvents. Most of the ternary (with ILs) liquid–liquid equilibrium data available in literature has been correlated through the NRTL model as r and q are not available. The absence of the volume and surface parameters poses a hindrance in calculation of the binary interaction parameters for UNIQUAC and UNIFAC models. A novel method has been developed for deriving these volume and surface parameters from the Polarizable Continuum Model (PCM). PCM is widely used for studying solvation effects. Here, the solute is represented by a charge distribution in a molecular shaped cavity embedded in an infinite polarizable dielectric medium. GEnerating POLyhedra (GEPOL), which is based on the concept of solvent excluding surface, is used for calculating this cavity. This novel approach for volume and surface parameters has been verified initially for 71 compounds belonging to several homologous series (paraffins, isoparaffins, olefins, naphthenes, aromatics, alkynes, alcohols, ketones, aldehydes, acids, esters and amines) and 24 solvents. The predicted values of r and q for alcohols, ethers and oxygen containing systems showed significant deviations (2.5–20% for r and 5–25% for q) from the literature values. The values of r and q obtained by PCM method have been applied to 17 ternary systems and 1 quaternary system belonging to these deviant components. The PCM method gives a significantly better fit (average of 60% improvement in rmsd) for all the systems studied. This approach has been used to estimate the structural parameters for 25 dialkylimidazolium based ILs. Subsequently, these values have been used to estimate the UNIQUAC interaction parameters for seven IL based ternary systems, giving a 40% improvement in rmsd over NRTL. © 2005 Elsevier B.V. All rights reserved. Keywords: Polarizable Continuum Model; Volume and surface parameters; Solute cavity; Gaussian 98; Imidazolium based ionic liquids

1. Introduction Ionic liquids (ILs) have received increased attention for the last 10 years. They are used as potential solvents for separating liquids. Their unique physical and chemical properties, such as specific solvent abilities, negligible vapor pressures and broad liquid temperature ranges, have led to promising applications as recyclable and environmentally benign solvents. A lot of research has been carried out on its use as solvents for extraction purposes. But the thermodynamic data related to ionic liquids are scarce. Much ∗

Corresponding author. Tel.: +91 512 2597117; fax: +91 512 2590104. E-mail address: [email protected] (A. Khanna).

0378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.05.017

work has focused on their ability to serve as solvents for reactions and their potential for phase transfer catalysis. They are known for their good heat transfer properties and high conductivities. In addition, the physical properties are tunable by judicious selection of cation, anion and substituent. ILs contain organic cation which are large in comparison to the inorganic cation. The coulombic attractions for ILs are in fact of comparable magnitude with respect to the intermolecular attractions of organic solvents. The increasing anion size generally leads to lower melting points while the melting points decrease for more asymmetric, larger cations. Some recent reviews [1–4] give an overview of the potential of these salts for synthesis, catalysis and separation.

T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

In order to use ILs as solvents, it is important to characterize their fundamental thermodynamic properties. The only phase behavior data that are currently available include some infinite dilution activity coefficients of chemicals in IL [5–8], gas liquid and solid solubilities in ILs [9,10] and ternary liquid–liquid equilibria data [11–15]. The infinite dilution activity coefficient gives a fair indication of the potential of using IL as solvent for separation of aliphatic and aromatic compounds. Further, it gives an indication of the interaction between the hydrocarbons and the IL. The liquid–liquid equilibria data is very essential for the design of the extraction equipment and also to know the thermodynamic limit of separation. Because of the lack of pure component surface and area parameters for ILs, the UNIFAC [16] and UNIQUAC [17] model could not be used for correlating the experimental data. The NRTL model has been invariably used in all the ternary systems reported in literature. These equations contain pure component structural parameters r and q. These are directly related, by constant values, to the van der Waals volume and surface area of the molecule, respectively. The van der Waals surface area and volume are characteristic properties of a molecule and, in principle, can be derived directly from the molecular structure. The binary interaction parameters are then estimated by regression of the experimental data. The r and q parameters are generally computed using a group contribution based approach as suggested by Bondi [18,19]. The use of this method is straightforward except for a few limitations. The surface and volume are calculated from experimental data, namely from bond distances and the van der Waals radii taken from Sutton [20], by means of the purely geometric method discussed by Bondi [19]. The underlying assumptions are that appropriate values of bond lengths and van der Waals radii are known and that the model geometry being used represents the facts. But the van der Waals radii are only moderately known and the rules for their transferability are still not fully worked out. The two aspects of atom shapes, i.e. (a) the true anisometry of bonded atoms parallel and normal to their covalent bonds in specific bonding states and (b) the cusp or the pear shape of a given molecule, have not been considered in the geometric model. Neither of these deviations from sphericity has yet been established in sufficient detail for rigorous calculation of volume and area. The group contributions cited by Bondi were obtained by considering structure groups bonded directly to carbon atoms. The group contributions cannot be used for molecules in which the structure groups are not directly attached to the carbon atom. Many polar molecules like water and ammonia and nitrogen containing compounds fall in this category. Brelvi [21] had suggested simple correlations for r and q as functions of critical volume and of radius of gyration of the molecule. Critical volume was shown to be a better parameter for correlating the structure parameters. The profiles for both polar and non-polar molecules were presented. Brelvi’s work however is still not applicable to molecules for which critical volume or radius of gyration is not known. New molecules

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or compounds molecules that have not been synthesized yet fall in this category. The UNIQUAC structure parameters, r and q, are calculated as: r=

Vw , Vws

q=

Aw Aws

(1)

where Vw and Aw are the van der Waals volume and area of the molecule given by Bondi [19], and Vws and Aws are the van der Waals volume and area of a standard segment chosen arbitrarily by Abrams and Prausnitz [16]. This standard segment is defined as a sphere such that for a linear polymethylene molecule of infinite length the equation below is satisfied where z is the coordination number usually taken to be 10.
z (r − q) = r − 1 (2) 2 Empirical equations have also been used to compute the r and q values of the unknown compounds. One such correlation as used by Doma’nska and Mazurowska [22] is: ri = 0.029281VM ,

qi =

(z − 2)ri 2(1 − li ) + z z

(3)

Here, VM is the molar volume of the component at 298.15 K, z the coordination number and li is the bulk factor. But for many ionic liquids, the molar volumes at standard temperature are not known and the general tendency to calculate the molar volume is by adding the volumes of the various groups in the molecule. The focus of this work is to present a way based on quantum chemical calculation to estimate the molecular volume and surface areas of ionic liquids which is rigorous, and is not limited to any class of compounds, and which does not suffer from the limitations of Bondi’s and Brelvi’s methods.

2. Molecular surfaces and volumes The molecular surface area and volume are the most popular geometrical descriptors of a compound. The surface area is closely related through various physical and quantum mechanical models to the intermolecular dispersion energy and the free energy of cavity formation [27–31]. It is also used extensively for the identification of binding regions in structure based drug design [32], prediction of the hydrobhobicity of compounds [33] and the prediction of NMR [34]. The algorithms to calculate the surface area and volumes are broadly categorized into numerical and analytical formulae. One such algorithm divides the molecule into number of slices along the main principal axis of inertia of the molecule in the X direction with step X. For each slice, a set of circles corresponding to the interaction of van der Waals spheres and cutting plane is generated. The lengths of non-occluded arcs are calculated for each circle using step dl and the surface is

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T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

then calculated as:  SA = ln X

(4)

an analytical formula while ‘Vin ’ is calculated using a numerical scheme [36].

n

Another term commonly used is the ‘solvent-accessible surface area’ of the molecule. Connolly [35] has developed analytical equations for the calculation of the area proceeding from the model rolling on the molecular van der Waals surface. This has been described briefly in the next section. The solvent-accessible surface area is thus the algebraic sum of the convex area (Acx ), the saddle areas (As ) and the concave areas (Acv ). SA = Acx + As + Acv

(5)

The convex area is simply the area defined by the center of the rolling solvent sphere (see Fig. 1) when in contact with the van der Waals surface of one atom in the molecule. The saddle area and concave area correspond to the areas defined by the centre of the rolling solvent sphere when in contact with the van der Waals surface of two atoms and three and more atoms in the molecule, respectively. In the same way, molecular volumes and the solventexcluded volume [36] are calculated and are widely used as molecular descriptors. The analytical way to calculate the solvent-excluded volume is to divide the volume in two parts. The first part accounts for the intrinsic van der Waals volume of the molecule while the second calculates the interstitial volume consisting of the packing defects between atoms that are too small to admit a solvent molecule of a given radius. Thus,    VSEV = Vcx + Vs + Vcv + VP + Vac + Vin (6) The first three terms have already been discussed, the fourth term corresponds to the volume of the internal polyhedron (VP ), the last two terms account for the possible cusp pieces appearing in the case of separated atomic spheres with limited access of solvent molecule between them. ‘Vac ’ uses

Fig. 1. Solute surfaces.

3. Polarizable Continuum Model In the continuum models, the solvation phenomenon is usually formulated by defining a cavity in which a molecule will be inserted in the midst of the dielectric medium which represents the solvent. In these models, the solvent distribution outside the solute is usually assumed to be constant outside the solute cavity and its value is taken to reproduce the density of the solvent. The main aim of these models is to calculate the solvation free energy

Gsolv = Ges + GvdW + Gcav

(7)

Here, the first term accounts for the electrostatic contribution, the second for the van der Waals dispersion and repulsion energies and the third for the energy required to create the solute cavity. Various continuum models have been reviewed by Tomasi and Persico [23]. These models mainly differ in the way the solute is described, the solute charge distribution, the interaction with the dielectric, i.e. the electrostatic potential and how the solute cavity and surface is described. The Polarizable Continuum Model of Tomasi et al. [24] has been used in the present work. This approach represents the solvent as a polarizable continuum and places the solute in a cavity within the solvent. It is an apparent surface charge approach, where the reaction potential is described in terms of an apparent charge distribution spread on the cavity surface. A cavity of molecular shape is used. The tessellation of the spheres making up the surface was originally based on parallels and meridians and, later, on the inscribed pentakisdodecahedron. The quantum mechanical calculation is performed making use of two nested cycles. The internal cycle calculates the final charges on the surface elements from the molecular charge distribution. The external one determines an improved solute charge distribution. It is possible to model the cavity in three different ways (Fig. 1). All surfaces are derived from the union of van der Waals spheres centered on the solute atoms: (a) The simplest definition is that of the mere union of spheres. This is called the van der Waals surface (Fig. 1(a)) and is sufficient to describe the continuum for molecules of small size. (b) Lee and Richards [25] introduced the solvent-accessible surface (Fig. 1(b)). The accessible surface is traced out by the probe sphere center as it rolls over the solute. It is a kind of expanded van der Waals surface. (c) Later, Richards [26] introduced the re-entrant surface, which together with the contact surface (the part of the van der Waals surface that can be touched by a watersized probe sphere) forms the molecular surface or equivalently, solvent excluding surface. The molecular surface (Fig. 1(c)) is the surface traced by the inward-facing

T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

surface of the probe sphere. The re-entrant surface consists of the inward-facing part of the probe sphere when it is in contact with more than one atom. The solventexcluded volume is the volume enclosed by the molecular surface. It is the volume that the probe sphere is excluded from. The solvent-excluded volume is the sum of the van der Waals volume and the interstitial volume. As has been mentioned in earlier sections, the PCM calculations differ in the way cavity is formed. The shape and size of the cavity are critical factors in the elaboration of a method. An ideal cavity should reproduce the shape of the solute. The cavity shapes actually employed are the following ones: (1) regular shapes, namely spheres, ellipsoid and cylinders or (2) molecular shapes. A cavity is also characterized by its size, i.e. volume and surface area.

4. GEPOL algorithm The GEnerating POLyhedra (GEPOL) algorithm of Nilsson et al. [42] is the most common algorithm used for computing the cavity. It defines the surface in terms of a set of interlocking spheres, the original van der Waals ones, supplemented by other spheres also depending on the solute molecular radius. The tessellation (partitioning the spheres) is performed using a pentakisdodecahedron (a polyhedron derived from the dodecahedron by replacing each pentagonal face with five triangles) inscribed within each sphere. This solid has 60 faces, all of equal area. The pentakisdodecahedron belongs to a family of polyhedra with triangular faces. The surface of the sphere is correspondingly partitioned into 60 equivalent curvilinear triangles, almost equilateral. Higher order polytopes are used to describe the areas and the centers of irregular tesserae deriving from the intersection of two spheres. The hidden tesserae contained in the volume of the cavity are not considered. This new partition of a spherical surface is more suited to treat spheres of very different radii. The algorithm has well survived the critical examination of many users, and others are using it in solvation methods and other applications. This method has been found useful to compute the surface and volume of the cavity, with a better accuracy and at a lower cost than achieved by means of other algorithms. The GEPOL algorithm can calculate the surfaces for all kind of systems and organic compounds [37–42].

5. Computational details for PCM using GEPOL algorithm The work here uses cavities based on atomic spheres, using the GEPOL algorithm. The difference with respect to the preceding PCM versions is that the spheres of the heavy atoms are defined taking into account their hybridization state. The radii of the spheres are determined by fitting the solvation

67

free energy over a large set of molecules, taking into account non-electrostatic contributions. GEPOL has been adopted as the default option for PCM calculations in the Gaussian 98 [43] electronic structure program. It expedites geometry optimization and energy calculations with better results. The implementation of the PCM model in Gaussian 98 can be invoked using the SelfConsistent Reaction Field (SCRF) keyword in combination with PCM-specific modifiers. The procedure for the calculations using Gaussian 98 is outlined as follows: (1) The first step is to obtain the equilibrium geometry of the molecule in the ideal gas phase from molecular energy minimization using the Density Functional Theory (DFT) with B3LYP functional at 6-31++G (d, p) basis set level. (2) The second step is to carry out the PCM calculation using the optimized structure obtained in the previous step. This is done using the scrf keyword specifying the pcm option along with solvent = water and read options. Here, read was used to indicate a separate section of options providing calculation parameters for specifying the characteristics of the cavity. The extra options used were radii = bondi and alpha = 1.0. The first option indicates the use of Bondi’s atomic radii. Alpha is used to specify the scaling factor for the definition of the surfaces. The radius of each atomic sphere is determined by multiplying the van der Waals radius by scale. In the original PCM model, the cavity was defined in terms of spheres with radii R, proportional to the van der Waals radii: R = αRvdW

(8)

In continuum models, such as PCM, a scale factor is used to account for the fact that the dielectric of the first solvation layer around a molecule or a group is different from that in bulk solution [23]. Instead of directly using the van der Waals radii, the radii of the fused spheres are multiplied by the scale factors. In the present work, a scale of 1.0 was used because the major interest was to compute the cavity instead of the solvation free energies. The output file of PCM contains the overall surface (APCM ) and the overall volume (VPCM ) of the solute molecule under consideration. Since ionic liquids consist of a cation and an anion, the anion is placed randomly on top of the cation in all possible orientation. The structure having the least total energy is then used for PCM calculation. In the minimum energy configuration, the cation and anion are linked through the use of a dummy atom. Similar approach has also been adopted out by Meng et al. [61]. For the estimation of r and q, we have used a simple relation: 3

rpred

3

˚ )(1 × 10−8 cm) Nav (V PCM A = Vws

(9)

68

qpred =

T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

˚ 2 )(1 × 10−8 cm)2 Nav (APCM A Aws

(10)

where ‘Nav ’ is the Avogadro’s number. The standard segment volume Vws (15.17 cm3 /mol) and area Aws (2.5 × 109 cm2 /mol) was used the same values as used by Bondi. From these observations, we hypothesize that for any molecule for which r and q parameters are not known, Eqs. (9) and (10) can be used to calculate the unknown r and q parameters. In Eqs. (9) and (10), VPCM and APCM are the molecular volume and surface area computed using PCM calculation. In order to substantiate our finding, we have used the r and q values of our work for deviant containing components—primarily the oxygen containing compounds (particularly alcohols) and then estimated the goodness of fit using the UNIQUAC model. The UNIQUAC [16] model is given by: Φi z θi Φi  ln γi = ln + qi ln + li − xj l j + q i xi 2 Φi xi j       θj τij ,

(11) × 1 − ln  θj τji  − k θk τkj j

where qi xi θi = , qT

qT =



qk xk ,

k

z li = (rk − qk ) + 1 − rk , 2

j

Φi =

ri xi , rT

τij = exp

rT =



rk x k

k

−Aij T

z = 10 (coordination number)



= Aij , (12)

The goodness of fit is usually defined by root mean square deviation, which is defined as: 1/2  m  2 c  j j 2  (xik − xˆ ik )  (13) rmsd =  2mc k=1 i=1 j=1

Here, ‘m’ refers to the no tie lines, ‘c’ the number of components and ‘k’ is the number of phases.

6. Results and discussion 6.1. Validation of PCM approach for r and q of homologous series Various homologous series have been considered as shown in Table 1. The computed molecular volumes and areas for all members of each homologous series has been calculated using the GEPOL algorithm and the r and q values calculated using Eqs. (9) and (10). The predicted volume and surface parameters have then been compared in Table 1 with the surface and volume parameters reported in literature. The

absolute errors are defined as percent deviations and are given by: Er =

rliterature − rpredicted × 100 rliterature

(14)

Eq =

qliterature − qpredicted × 100 qliterature

(15)

The predictions are very good, with the overall percentage relative errors being 3.7% and 8.02% for r and q, respectively, for all the homologous series considered. Deviations are found to be high in the oxygen containing components/series. Deviation greater than 2.5% for r and 5% for q are marked in Table 1. This is largely due to the extent of hydrogen bonding and the lesser availability of surface area on the compound. The largest deviations occur in the alcohol series. Alcohols show a large deviation in predicted values of r and q: 9.7% and 18.44%, respectively. This is consistent with the literature since the area and volume of the alcohols are less available than calculated [46]. The predicted values of r and q for alcohols are smaller than those reported in literature. This procedure has been carried out for all the compounds in study. For aromatics, the predicted q values are larger by 4% than those reported in literature. Comparison for the computations has been shown in Figs. 2 and 3 for r and q, respectively. The predictions for r are better than those for q. On investigating all the homologous series it can be inferred that there is a general trend of a negative deviation for prediction of ‘r’ and a positive deviation for ‘q’ (Table 1). Thus, the calculated area is generally less than the actual area while for the volume it is vice versa. The alcohols and aromatics are the only anomaly in this trend. The aromatic compounds having a negative deviation in ‘q’ have a larger calculated area. This is due to the spherical shape of an aromatic compound owing to the presence of double bonds. In comparison, the straight chain compounds assume a cylindrical shape. The area per unit volume calculated for a spherical shape is larger than of a cylinder/ellipsoid which explains the higher values of ‘q’. For the branched alkanes because of the branches it leads to a reduced calculated area by PCM since a solvent molecule rotates along the solute surface which would be smaller than the actual surface area. So more the number of branches around a carbon atom on a chain, more would be the deviation from literature q values. This is consistent with the high values of errors in ‘q’ for 2,4-dimethylhexane (two branches: one each on two carbon atoms, 12.83%), 2,2,3,3-tetramethylpentane (four branches: two each on each carbon atom, 18.62%), 2,4-dimethylheptane (two branches: one each on two carbon atom, 11.684) and 2,2,4-trimethylpentane (three branches: two on one carbon atom and one branch on another carbon atom, 20.283%). The presence of the branches is also evident in other homologous series like 2-methyl propene, 1-1 dimethylcyclopentane, dimethyl cyclohexane, diethyl amine and triethylamine.

T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

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Table 1 Values of r and q with their relative errors for homologous series Molecular ˚ 3) volume (A

˚ 2) Surface area (A

r [44]

q [44]

rpred , Eq. (9)

qpred , Eq. (10)

Er , Eq. (14)

Alkanes Ethane Propane Butane Pentane Hexane Heptane Octane Decane

44.834 62.598 80.368 97.933 115.701 133.287 150.772 186.302

69.871 90.559 110.797 131.458 150.286 172.348 192.693 233.683

1.802 2.477 3.151 3.825 4.500 5.174 5.849 7.197

1.696 2.236 2.776 3.316 3.856 4.396 4.936 6.016

1.780 2.485 3.191 3.888 4.594 5.292 5.986 7.397

1.683 2.182 2.669 3.167 3.621 4.152 4.642 5.630

1.221 −0.323 −1.269 −1.647 −2.089 −2.281 −2.342 −2.779

0.767 2.415 3.854 4.493 6.094 5.551 5.956 6.416

Isoparaffins 2-Methylbutane 2-Methylpropane 2,4-Dimethylhexane 2,2,3,3-Tetramethylpentane 2,4-Dimethylheptane 2,2,4-Trimethylpentane

98.643 81.267 153.097 169.720 170.154 177.069

126.983 108.718 178.292 177.879 200.442 193.531

3.825 3.150 5.847 6.520 6.521 5.946

3.312 2.772 4.928 5.266 5.468 5.863

3.916 3.227 6.078 6.738 6.756 6.324

3.059 2.619 4.295 4.286 4.829 4.663

−2.379 −2.444 −3.951 −3.344 −3.604 −6.889

7.639 5.519 12.845 18.610 11.686 20.283

Alkenes (olefins) Propene Butane 2-Methyl propene Pentene Hexene Heptene

56.391 73.587 74.452 91.243 109.299 126.747

82.702 103.950 102.246 124.398 143.102 165.342

2.247 2.921 2.920 3.595 4.270 4.944

2.024 2.564 2.684 3.104 3.644 4.184

2.239 2.922 2.956 3.623 4.339 5.032

1.992 2.504 2.463 2.997 3.448 3.983

0.356 −0.034 −1.233 −0.779 −1.616 −1.780

1.581 2.340 8.234 3.447 5.379 4.804

Naphthenes Cyclobutane Cyclopentane Cyclohexane 1,1-Dimethylcyclopentane Ethylcyclohexane Cyclooctane

64.244 85.128 101.777 122.274 139.690 136.746

86.284 114.937 131.286 147.022 162.374 159.983

2.698 3.372 4.064 4.947 5.394 5.395

2.160 2.700 3.240 4.084 4.316 4.320

2.551 3.380 4.041 4.855 5.546 5.429

2.079 2.769 3.163 3.542 3.912 3.854

5.448 −0.237 0.566 1.860 −2.818 −0.630

3.750 −2.556 2.377 13.271 9.361 10.787

Aromatics Benzene Aniline Phenol Chlorobenzene Benzonitrile Toluene Benzoic acid Dichlorobenzene Ethylbenzene o-Xylene

82.988 94.863 91.496 97.091 100.239 100.378 110.894 111.016 118.962 118.151

108.356 122.209 118.411 125.760 129.195 128.531 138.648 143.087 148.052 145.170

3.188 3.717 3.552 3.813 3.991 3.923 4.323 4.438 4.597 4.658

2.400 2.816 2.680 2.884 3.008 2.968 3.344 3.288 3.508 3.536

3.295 3.766 3.633 3.855 3.980 3.985 4.403 4.408 4.723 4.691

2.611 2.944 2.853 3.030 3.113 3.097 3.340 3.447 3.567 3.497

−3.356 −1.318 −2.280 −1.101 0.276 −1.580 −1.851 0.676 −2.741 −0.708

−8.792 −4.545 −6.455 −5.062 −3.491 −4.346 0.120 −4.836 −1.682 1.103

Acids Formic acid Acetic acid Propanoic acid Butanoic acid Hexanoic acid Methanol Ethanol 1-Propanol 2-Propanol 2-Methyl-1-propanol 1-Butanol 2-Methyl-2-propanol 1-Hexanol 1-Heptanol 1-Octanol

38.962 55.417 72.092 89.814 126.096 36.654 53.193 71.041 71.899 89.435 88.746 90.153 124.099 168.688 159.174

61.591 83.062 104.283 124.946 163.716 59.362 80.947 101.622 100.038 118.190 120.688 117.249 159.386 191.780 203.846

1.528 2.202 2.877 3.551 4.900 1.431 2.576 3.250 3.249 3.924 3.924 3.923 5.273 5.947 6.622

1.532 2.072 2.612 3.152 4.232 1.432 2.588 3.128 3.124 3.664 3.668 3.744 4.748 5.288 5.828

1.547 2.200 2.862 3.566 5.006 1.455 2.112 2.821 2.855 3.551 3.524 3.579 4.927 6.690 6.320

1.484 2.001 2.512 3.010 3.944 1.430 1.950 2.448 2.410 2.847 2.908 2.825 3.840 4.620 4.911

−1.243 0.091 0.521 −0.422 −2.163 −1.677 18.012 13.200 12.127 9.506 10.194 8.769 6.562 −12.494 4.561

3.133 3.427 3.828 4.505 6.805 0.140 24.652 21.739 22.855 22.298 20.720 24.546 19.124 12.632 15.734

Eq , Eq. (15)

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T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

Table 1 (Continued ) Molecular ˚ 3) volume (A

˚ 2) Surface area (A

r [44]

q [44]

rpred , Eq. (9)

qpred , Eq. (10)

Er , Eq. (14)

Aldehydes Methanal Ethanal Propanal Butanal Hexanal

31.119 47.524 64.646 82.454 117.971

51.090 72.629 93.845 114.401 155.165

1.225 1.899 2.574 3.248 4.597

1.256 1.796 2.336 2.876 3.956

1.236 1.887 2.567 3.274 4.684

1.231 1.750 2.261 2.756 3.738

−0.898 0.632 0.272 −0.800 −1.893

1.990 2.561 3.211 4.172 5.511

Alkynes Ethyne Propyne Butyne Hexyne Heptyne Octyne

34.622 50.839 67.961 103.672 120.621 137.139

54.365 75.940 98.303 139.682 158.339 177.147

1.518 2.193 2.868 4.216 4.891 5.565

1.396 1.936 2.476 3.556 4.096 4.636

1.375 2.018 2.698 4.116 4.789 5.445

1.310 1.830 2.368 3.365 3.815 4.268

9.420 7.980 5.927 2.372 2.085 2.156

6.160 5.475 4.362 5.371 6.860 7.938

Esters Methyl acetate Ethyl acetate Propyl acetate Butyl acetate Hexyl acetate

72.595 89.197 107.168 124.402 161.119

104.974 126.159 147.119 167.265 205.236

2.804 3.479 4.153 4.827 6.176

2.576 3.116 3.656 4.196 5.276

2.882 3.541 4.255 4.939 6.397

2.529 3.039 3.544 4.030 4.945

−2.782 −1.782 −2.456 −2.320 −3.578

1.825 2.471 3.063 3.956 6.274

Ketones Acetone Propanone 2-Butanone 2-Pentanone 3-Pentanone 2-Hexanone 2-Heptanone

83.163 65.091 82.822 100.164 100.778 117.946 135.477

104.076 93.273 112.794 132.082 135.218 153.632 173.867

2.573 2.574 3.248 3.922 3.922 4.597 5.271

2.336 2.336 2.876 3.416 3.416 3.956 4.496

3.302 2.584 3.288 3.977 4.001 4.683 5.379

2.507 2.247 2.717 3.182 3.258 3.701 4.189

−28.333 −0.389 −1.232 −1.402 −2.014 −1.871 −2.049

−7.320 3.810 5.529 6.850 4.625 6.446 6.828

Amines Methylamine Ethylamine Propylamine Dimethylamine Diethylamine Trimethylamine Triethylamine

39.734 56.928 74.756 57.745 92.035 74.324 129.675

64.003 85.235 104.699 85.295 127.127 104.664 158.463

1.596 2.270 2.945 2.335 3.684 2.989 5.012

1.544 2.084 2.624 2.092 3.278 2.636 4.366

1.578 2.260 2.968 2.293 3.654 2.951 5.148

1.542 2.054 2.522 2.055 3.063 2.522 3.818

1.128 0.441 −0.781 1.799 0.814 1.271 −2.713

0.130 1.440 3.887 1.769 6.559 4.325 12.552

Using the above hypothesis, r and q have also been calculated for 24 solvents and compared with their known values. This comparison is presented in Table 2. For the glycols and some solvents, the ‘r’ and ‘q’ are not known so they have been extracted from ASPEN [45], which usually adds the r and q values of the functional sub groups. Except for solvents like propylene carbonate, hexylene glycol, isophorone, water and pyridine, all other values agree very well with the values reported in literature. The overall relative errors in r and q for these solvents were 5.06% and 5.87%, respectively. Comparative studies have been also made with respect to the work carried out by Brelvi [21], as shown in Table 3. This table indicates that our approach gives slightly less relative error for r, and slightly more error for q with respect to Brelvi’s values. In order to explain the deviant components, we have chosen 17 ternary liquid–liquid equilibira systems and 1 quaternary system. The systems selected contained those components, which have the most deviation in their r and q values. This includes systems based on alcohols, the cyclic

Eq , Eq. (15)

compounds which includes both napthenes and aromatics, and olefins. The liquid–liquid equilibrium compositions have been obtained by using a flash calculation. UNIQUAC model (Eqs. (11) and (12)) has been used for the estimation of binary interaction parameters. The minimizing of the residuals between experimental and estimated mole fractions has been used as the objective function in our case. It is defined as: min F =

m  II  c  k=1 j=I i=1

j

ej

cj 2

Wik (xik − xˆ ik )

(16)

In the objective function, ‘W’, the weight factor, has been taken as unity, ‘c’ represents the number of components, ‘m’ the number of tie lines and xe , xˆ c are the experimental and calculated mole fractions, respectively. The details of this calculation can be found from our earlier work [60]. The root mean square deviation (Eq. (13)) has been calculated using the published values of r and q as well as our computed values of r and q. In Table 4, a comparison has been made

T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

71

Fig. 2. Comparison of computed and reported pure component area parameter.

between the two. We observe an overall improvement of 56% in the rmsd values with respect to all the systems studied. Water, having a significant deviation of 8% and 27% in r and q values has been included in nine systems which we have studied. For example, with alcohols one of the system chosen, i.e. water + ethanol + octanol [49], where octanol and ethanol have the most deviations in the r and q values, the rmsd val-

ues obtained gave a better fit as compared with the reported values. For methyl-tert-butyl ether whose percentage deviation is as high as 5.9% and 14.9% in r and q, we have chosen a quaternary system of water + methanol + MTBE + toluene [50] and compared the rmsd values which is 0.004 in our case. For the case of solvent, the nitrogen containing component, pyridine has nearly 17% deviation with respect to q.

Fig. 3. Comparison of computed and reported pure component volume parameter.

72

T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

Table 2 Literature and predicted values of r and q with their relative errors for solvents Solvent

Volume

Sulfolane Dimethyl sulfoxide Furfural Pyridine N-methyl pyrrolidone Morpholine Carbon tetrachloride Triethylene glycola Propylene carbonate Diethylene glycol N-N-dimethylformamide N-methylformamide Hexylene glycola Ethylene glycola Isophoronea Triethylene glycol dimethyl ethera Diethylene glycol dimethyl ethera Diethylene glycol propyl ethera Dimethyl ether Methylethyl ether Diethyl ether Methyl-tert-butyl ether tert-Amyl ethyl ether Water

102.98 74.84 84.47 78.11 100.58 89.28 85.78 147.91 88.63 104.74 76.42 59.04 131.85 61.64 154.07 184.51 141.21 157.45 54.770 71.433 88.078 108.046 168.444 25.246

Area

134.23 104.14 113.44 103.34 131.60 117.34 117.14 199.46 118.72 145.72 108.04 88.10 160.70 91.95 177.13 242.17 188.46 209.04 80.692 102.220 123.135 134.531 187.258 42.746

Literature

Predicted

r [44]

q [44]

rpred

qpred

4.036 2.8266 3.168 2.9993 3.981 3.474 3.39 5.5939 2.9789 4.0013 3.0856 2.4028 4.5742 2.4087 5.9314 6.824 5.2314 5.9651 2.046 2.721 3.395 4.562 5.932 0.92

3.206 2.472 2.481 2.113 3.2 2.796 2.91 4.88 2.28 3.568 2.736 2.192 3.896 2.248 4.94 5.896 4.576 5.152 1.936 2.476 3.016 3.807 4.234 1.42

4.089 2.972 3.354 3.101 3.993 3.545 3.406 5.873 3.519 4.160 3.034 2.344 5.235 2.447 6.117 7.326 5.607 6.252 2.175 2.836 3.497 4.290 6.68 1.002

3.234 2.509 2.733 2.490 3.171 2.827 2.822 4.806 2.860 3.511 2.603 2.123 3.872 2.216 4.267 5.835 4.541 5.036 1.944 2.463 2.967 3.241 4.512 1.03

Er∗

Eq∗

1.307 5.126 5.865 3.402 0.312 2.043 0.472 4.984 18.131 3.963 1.663 2.434 14.451 1.610 3.136 7.354 7.172 4.803 6.276 4.249 3.006 5.968 4.805 8.91

0.877 1.500 10.160 17.838 0.918 1.115 3.018 1.525 25.450 1.604 4.860 3.162 0.620 1.428 13.614 1.041 0.774 2.245 0.416 0.537 1.638 14.863 3.166 27.46

Er∗ , Eq∗ : absolute deviation in r and q. a r and q values from Aspen [45].

Table 3 Comparison with Brelvi’s [21] work Relative error (%)

Hydrocarbons Oxygen containing Nitrogen containing Aromatics

Er (Brelvi)

Er (this work)

Eq (Brelvi)

Eq (this work)

2.49 2.79 7.54 2.66

2.05 3.04 1.69 0.98

1.76 2.74 7.03 3.46

3.14 3.72 5.20 2.24

Table 4 UNIQUAC interaction parameters for deviant ternary systems S. no.

Name of the ternary system, components 1, 2, 3

UNIQUAC interaction parameters A12 A21

A13 A31

A23 A32

rmsda

rmsd (lit)

Imp (%)

Reference

1

Water–methanol–octanol

−153.2 98.6

166.6 686.9

262.1 −176.9

0.0063

0.0060

5.0

[49]

2

Octanol–pyridine–water

115.2 251.8

600.8 247.3

178.0 447.4

0.0040

0.0150

73.3

[50]

3

3-Pentanone–acetic acid–water

−562.6 854.4

638.4 283.7

−77.2 −160.5

0.0018

0.0070

74.3

[51]

4

Water–acetic acid–1-heptanol

539.3 216.7

238.5 538.2

18.1 392.5

0.0050

0.0070

28.6

[52]

5

Water–acetic acid–2-hexanone

224.7 421.1

303.51 1197.9

−94.6 545.4

0.0040

0.0380

89.5

[53]

6

Water–cyclohexane–ethyl acetate

204.4 999.2

993.2 995.6

69.4 −21.1

0.0045

0.0051

11.0

[54]

T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

73

Table 4 (Continued ) S. no.

Name of the ternary system, components 1, 2, 3

UNIQUAC interaction parameters A12 A21

A13 A31

A23 A32

rmsda

rmsd (lit)

Imp (%)

Reference

7

Water–2-propanol–2,2,4-trimethyl pentane

−289.6 995.4

999.4 945.2

−94.6 370.3

0.0067

0.0090

25.6

[55]

8

tert-Amyl ethyl ether–methanol–water

740.9 −129.2

1097.5 571.1

234.1 147.8

0.0030

0.0040

25.0

[56]

9

Acetone–methanol–hexane

−148.8 −512.7

−41.4 −429.3

7.8 540.0

0.0007

0.0050

86.0

[57]

10

Acetone–methanol–heptane

777.4 204.2

−148.9 824.9

19.7 586.7

0.0030

0.0090

66.7

[57]

11

Acetone–methanol–octane

−90.8 154.3

353.1 5.8

18.4 665.7

0.0030

0.0080

62.5

[57]

12

Furfural–benzene–hexane

23.1 9.7

82.7 535.3

−274.1 633.1

0.0040

0.0080

50.0

[58]

13

Furfural–toluene–hexane

650.7 −136.0

100.0 361.5

−157.3 755.7

0.0060

0.0080

25.0

[58]

14

Furfural–xylene–hexane

−103.4 151.5

50.2 333.7

651.6 333.7

0.0020

0.0070

71.4

[58]

15

Furfural–benzene–hexane

−204.6 576.5

41.0 861.2

−349.2 944.3

0.0030

0.0200

85.0

[59]

16

Furfural–toluene–hexane

−43.7 40.0

5.8 776.0

440.5 −372.3

0.0040

0.0200

80.0

[59]

17

Furfural–xylene–hexane

197.1 35.9

3.7 698.5

183.4 −33.5

0.0030

0.0200

85.0

[59]

18

Water + methanol + toluene + methyl-tert-butyl ether

NA

NA

NA

0.0040

0.0100

60.0

[51]

(lit) − rmsda )/rmsd

(lit) × 100.

a

r and q obtained from PCM calculation; imp (%) = (rmsd

The water + pyridine + acetic acid [50] has been used in this case which also showed a similar improvement. This leads us to conclude that r and q values have significant effect on the interaction parameters; additionally, PCM based r and q calculations give consistently better fit with equilibrium data for all the systems studied. 6.2. Prediction of UNIQUAC interaction parameters for imidazolium based ternary systems After validating our method for homologous series and solvents we have applied the same concept for ionic liq-

uids. In order to check for the effect of cation and anion, we have chosen three anions, namely chloride (Cl− ), tetrafluoroborate (BF4 − ), hexafluorophosphate (PF6 − ) and trifluoromethanesulphonate (CF3 SO3 − ) or (TfO). The systems studied are listed in Table 5. The areas and volumes have been computed for varying the alkyl groups within the cation. With the same hypothesis as of Eqs. (9) and (10), we have computed the r and q values for the ionic liquids (Table 6). The r and q values obtained then have been tested on seven ternary systems using UNIQUAC model. The binary interaction parameters have been estimated in Table 7 and the root mean square deviation obtained gives us an

Table 5 List of imidazolium based ionic liquid ternary systems System no.

Name, components 1, 2, 3

Reference

1 2 3 4 5 6 7

[C4 mim]Tfo (1) + ethanol (2) + tert-amyl ethyl ether (3) [C6 mim]PF6 (1) + benzene (2) + heptane (3) [C6 mim]BF4 (1) + benzene (2) + heptane (3) [C6 mim]PF6 (1) + ethanol (2) + heptene (3) [C8 mim]Cl (1) + methanol (2) + hexadecane (3) [C8 mim]Cl (1) + ethanol (2) + hexadecane (3) [C8 mim]Cl (1) + ethanol (2) + tert-amyl ethyl ether (3)

[11] [48] [48] [12] [15] [15] [47]

[Cn mim]X: n, no. of carbons in the alkyl group; mim, methylimidazolium; X, anion.

74

T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

Table 6 r and q values of ionic liquids derived from relations (9) and (10) Volume

Area

rk

qk

c1mimcl c2mimcl c3mimcl c4mimcl c5mimcl c6mimcl c7mimcl c8mimcl

159.60 181.08 201.43 220.11 241.78 262.57 282.08 302.07

177.28 202.38 224.27 239.05 266.42 285.17 307.44 327.33

6.337 7.190 7.997 8.739 9.600 10.425 11.200 11.993

4.271 4.876 5.403 5.759 6.419 6.871 7.407 7.886

c1mimbf4 c2mimbf4 c3mimbf4 c4mimbf4 c5mimbf4 c6mimbf4 c7mimbf4 c8mimbf4

188.10 211.13 232.90 253.26 273.10 293.61 317.80 332.14

201.43 221.26 247.12 264.33 292.23 306.66 336.23 346.85

7.468 8.382 9.247 10.057 10.843 11.658 12.618 13.187

4.853 5.331 5.954 6.368 7.035 7.388 8.101 8.357

c1mimpf6 c2mimpf6 c3mimpf6 c4mimpf6 c5mimpf6 c6mimpf6 c7mimpf6 C8mimpf6

216.40 237.44 256.60 277.91 299.50 324.12 350.16 358.40

230.36 247.35 271.12 288.81 315.03 338.94 365.00 370.85

8.592 9.427 10.188 11.034 11.891 12.869 13.903 14.230

5.541 5.959 6.532 6.958 7.589 8.166 8.794 8.935

c4mimTfO

300.79

313.74

12.460

7.518

Table 7 Interaction parameters of ionic liquid ternary systems using UNIQUAC model System no. of Table 3 1 2 3 4 5 6 7

UNIQUAC interaction parameters A12

A13

A21

A23

A31

A32

−17.751 120.04 133.35 436.38 −960.44 −575.24 516.74

−73.54 153.07 138.39 206.36 −50.468 76.077 600.43

−183.59 50.36 49.96 −192.50 999.98 995 −224.81

226.33 161.47 153.34 18.25 193.83 −68.586 −429.9

824.08 65.11 60.81 6.00 289.47 170.41 −85.229

−205.56 −24.20 −15.16 190.56 948.52 583.41 709.63

overall improvement of 40% to that obtained through NRTL model.

NRTL rmsd (lit) reference of Table 4

UNIQUAC rmsd (pred)

0.0031 0.0150 0.0180 0.0370 0.0001 0.0040 0.0067

0.0040 0.0119 0.0128 0.0116 0.0004 0.0007 0.0058

most important to the ionic liquids. The r and q values, once derived, allow the application of UNIQUAC and UNIFAC models. Our work extends to glycols, glycolic ethers and sulfur containing compounds.

7. Conclusions A novel method for computing pure component volume and surface parameters using PCM calculations has been developed for imidazolium based ionic liquids. It correctly accounts for the intramolecular interactions as well as the atomic shapes by incorporating the hybridization states of the atoms. The method does not require any experimental data like molar volume and is entirely predictive. It does not require the knowledge of critical volume or radius of gyration in contrast to Brelvi’s approach. Its applicability has been demonstrated by computing r and q for several solvents and

List of symbols A surface area of the molecule E relative error G free energy IL ionic liquid l bulk factor n number of slices q surface parameter r volume parameter R radius S surface area

T. Banerjee et al. / Fluid Phase Equilibria 234 (2005) 64–76

V W X z

volume of the molecule weight factor length coordination number

Greek symbol α scaling factor Subscripts c number of components cx convex area cv concave area i number of components, i = 1, 2, 3, . . ., c j number of phases, j = I, II K number of tie lines, k = 1, 2, 3, . . ., m m number of tie lines q surface area S saddle area vdW van der Waals wi given molecule ws standard segment

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