Density and isothermal compressibility of ionic liquids from perturbed hard-sphere chain equation of state

Journal of Molecular Liquids 174 (2012) 52–57

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Density and isothermal compressibility of ionic liquids from perturbed hard-sphere chain equation of state S.M. Hosseini a,⁎, M.M. Papari b, J. Moghadasi a a b

Department of Chemistry, Shiraz University, Shiraz 71454, Iran Department of Chemistry, Shiraz University of Technology, Shiraz 71555-313, Iran

a r t i c l e

i n f o

Article history: Received 24 April 2012 Received in revised form 17 June 2012 Accepted 5 July 2012 Available online 17 July 2012 Keywords: Yukawa hard-sphere chain Ionic liquid Density and isothermal compressibility

a b s t r a c t This paper addresses the modeling of the density and isothermal compressibility of some ionic liquids (ILs) using Yukawa hard-sphere chain (YHSC) equation of sate (EOS) plus first-order perturbation theory. Three pure-component parameters that appeared in the EOS have been optimized for the representation of volumetric data. These parameters reflect the segment number, non-bonded segment–segment interaction energy, and the segment size. This work considered the chains that interact through a range-parameter of Yukawa potential with 1.8. The reliability of the proposed model has been assessed by comparing the results with 3153 experimental data points over a broad range of pressures and temperatures for which, their measured values were available in the literature. Our calculations on the density of studied ILs reproduced very accurately the experimental PVT data. The overall average absolute deviation (AAD) of the calculated densities from literature data was found to be 0.54%. We have also assessed the proposed perturbed hard-sphere chain equation of state (PHSC EOS) in the prediction of isothermal compressibility (κT) by taking 1043 literature data points; AAD was found to be 1.63%. Furthermore, the linear dependency of the liquid bulk modules (BT) on pressure has been successfully investigated via the proposed model. © 2012 Elsevier B.V. All rights reserved.

1. Introduction This work provides impressive progress in the theoretical study of perturbed hard-sphere chain equation of sate (PHSC EOS) and its application to ionic liquids (ILs). The prediction of density and isothermal compressibility of ILs is an important issue with relevancy in many applications where ILs are considered as a fluid design. Generally, they came into focus recently as new materials offering several highly promising applications. The unique properties of these liquids such as, non flammability, electric conductivity, thermal stability, low vapor pressure and high heat capacity lead to the fact that they are more considered than conventional organic solvents. Their potential applications can be briefly cited: 1) they are favorite electrolytes in lithium rechargeable batteries [1] and super-capacitors [2], 2) these materials can also be used as thermal fluids for heat storage by combining their high heat capacity, thermal stability and negligible vapor pressure [3] and 3) their interesting solvent properties make them well-known as green solvents for the future [4]. Therefore, accurate knowledge of their thermophysical properties is valuable as it is required to decide whether the use of ILs could be extended from the laboratory level to large-scale industrial applications. Under this circumstance, the development of PVT equations of state (EOSs) and PVT correlation methods

⁎ Corresponding author. Fax: +98 711 726 1288. E-mail address: [email protected] (S.M. Hosseini). 0167-7322/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2012.07.007

for modeling their thermophysical properties such as density and isothermal compressibility can be considerably useful. Physically based equations of state, derived by applying principles of statistical mechanics, have continuously been developed and improved upon over the past 4 decades. Modern equations of state aim at highly non-ideal systems, such as ILs or associating compounds [5,6]. Yukawa hard-sphere chain (YHSC) model has been widely used to theoretically model a broad range of fluids in liquid state physics [7]. It has found success in modeling the thermodynamics of simple liquids, colloidal suspensions, electrolytes, and molten salts. This model involves hard-sphere repulsion as well as a long-range attraction and that it describes many physical phenomena involving screened interactions, for example, strong electrolytes and polymer solutions. On the other hand, by varying a single parameter λ, the range of attractive forces can be easily adjusted with different interactions encountered in many physical systems. It should be noted that the local structure of the Yukawa hard-sphere chains is strongly affected by the range parameter of the potential [8]. With 1.8, the model has been well-found to approximate the tail of the Lennard-Jones potential. In the perturbative scheme studied in this work, the segments are already connected. The hard-sphere chains are chosen as the reference system and the reference system will be perturbed by a potential. In this work, three pure-component parameters that appeared in the PHSC EOS must be determined by regressing vapor pressures or liquid densities using well-established approach for compounds of low molecular weight. However, the first approach cannot be held

S.M. Hosseini et al. / Journal of Molecular Liquids 174 (2012) 52–57

for ILs: neither the accurate vapor–pressure data nor heats of vaporization are accessible for most ILs in the lithe literature. In this regard, we aim to utilize the liquid density data for the determination of three pure-component parameters of the proposed model. The knowledge of only three adjustable pure-component constants is sufficient to present an accurate PHSC EOS for ILs. We aim to employ the proposed PHSC EOS for predicting the mass density and some thermodynamic coefficients of ILs containing various cations and anions. The predicted results for the density of ILs will be compared with those estimated by the work of Gardas and Coutinho [9]. They have extended the Ye and Shreeve [10] group contribution model to estimate the density as a function of the temperature and pressure of a variety of ionic liquids. Furthermore, some regularities (linearities) that appeared in the PVT behavior of ILs will also be investigated via the proposed model. 2. Theory In the framework of the first order perturbation theory, the compression factor of perturbed Yukawa hard-sphere chain (PYHSC) may be written as [8,11]: Z PYHSC ¼

P HSC Pert: ¼Z þZ ρkT

HSC Z SAFT

Pert:



 1 þ η þ η2 −η3 1 þ η−η2 =2     −ðm−1Þ ¼m ð1−ηÞ 1−η2 =2 1−η3

¼−

12mη Iðη; m; λÞ Tr

ð2Þ

ð3Þ

where, m is the segment number and η is the segment packing fraction of Yukawa hard-spheres, defined as: η ¼ πρσ

.

3

6

:

a0i

a1i

a2i

0 1 2

0.864000 0.697700 −0.239900

−0.444375 0.866242 −0.638078

−0.170239 0.713402 −0.845203

So far, different methods have been developed in the literature to estimate thermodynamic properties of fluid systems. One possible method for deriving thermodynamic properties from speed of sound data is that which allows, by integration, the calculation of (P, ρ, T) and (P, CP, T) surfaces from merely one isobar of both density and isobaric heat capacity [17–20]. In the absence of the speed of sound data and the heat capacity the above-mentioned methods are not straightforward, but it is easy to use fitting of isobaric density data using a Tait equation [21]. Recently, the Tait equation has been widely used in the literature for the description of density behavior with temperature at constant pressure. Generally, the isothermal compressibility was calculated using the isothermal pressure derivative of density according to the following equation:   1 ∂ρ : ρ ∂P T

ð6Þ

The liquid bulk modulus, BT, a dimensionless property, is defined as [22]: " BT ¼

 # ∂P RT 1 : ¼ ρ κ T RT ∂ρ

ð7Þ

T

In this work, we have utilized the improved PHS EOS to estimate the above-mentioned thermodynamic properties for some ILs, using the following equation:

 η þ η2 =2−η3 =12 1 þ η þ η2 =6−η3 =36 2     þ 3m RTη κ T ¼ ð1=ρÞ m ρRT 1−η3 1−η4

 2 3 2 1 þ η þ η =6−η =36 1 þ η−η =12 2 þ mðm−1ÞηRT þm RT ð1−ηÞ3 ð1−ηÞ2 ð1−η=2Þ

 1 þ η−η2 =12 η−η2 =6 þmðm−1ÞRT ð 1 þ η=2 Þ þ m ð m−1 ÞρRT ð1−ηÞð1−η=2Þ ð1−ηÞð1−η=2Þ2 (

2

)

 −1  −1 2 3 3 : þT r m ρRTη 12 M1 mπσ =3 þ ηM2 mπσ −24ðM0 þ 2ηM1 þ ηM2 =2Þ

ð8Þ

ð4Þ

Here, ρ is the number density of Yukawa segments with diameter σ. I(η,m;λ) is related to the average inter-chain function for hard sphere chains in the context of the Percus–Yevick (PY) integral equation theory by Chiew [13,14]. Using the PY numerical solution, I(η,m;λ) has been previously evaluated in the literature [15,16] for a value of λ = 1.8 as a power series in segment packing fraction, η: Iðη; m; λ ¼ 1:8Þ ¼

i

κT ¼

and Z

Table 1 Value of the coefficients in Eq. (5) for the case of λ= 1.8.

ð1Þ

where P is the pressure, ρ is the number density of segments, and kT is the thermal energy per segment. The first and second terms in the right hand of Eq. (1) represent contributions to the compressibility factor from the reference hard-sphere chain potential (Z HSC) and the attractive perturbation potential (Z Pert.), respectively. Regarding the ILs, we aim to combine the YHSC model as the reference system by the contribution from the attractive Yukawa tails as the perturbation part. The HSC compression factor for the reference model is that of Statistical Associating Fluid Theory (SAFT) equation previously developed by Chapman et al. [12], which is given below:

53

2 X

i

ai ði þ 1Þη :

ð5Þ

i¼0

The values of coefficients ai have been provided in Table 1. After that, the experimental values of liquid density data have been well-represented by the proposed PHS EOS; we evaluate the proposed model in the estimation of some thermodynamic coefficients. In this study, isothermal compressibility, κT and liquid bulk modulus, BT have been chosen as target thermodynamic coefficients for ILs.

3. Results and discussion To utilize the proposed EOS three pure-component parameters of ILs must be characterized. In this regard, Table 2 has been presented to provide the required pure-component parameters (m, σ, ε/k) for the studied systems as well as their molecular weight. The pure-component parameters have been adjusted over the appropriate temperature and pressure range covered by the available PVT data. To show the distinctive feature of the present model, a typical plot for the calculated values of P/ρRT using Eq. (1) versus temperature for [C4mim][NTf2] has been presented in the supporting information (see Fig. 1 in supporting information file). As it is impressive from Fig. 1, the data at constant T will be nearly proportional to P. Since the variable on the vertical axis is P/ρRT, this proportionality would imply ρ to be independent of P, which means incompressibility of ILs. So, the PHSC EOS could successfully describe a liquid state

54

S.M. Hosseini et al. / Journal of Molecular Liquids 174 (2012) 52–57

Table 2 Pure-component parameters of the PYHSC EOS for the entire ILs studied in this work. Ionic liquid

Mw (g/mol)

ε/kB (K)

σ (nm)

m

[C2mim][OcSO4] [C2mim][BF4] [C2mim][NTf2] [C4mim][NTf2] [C2mim][EtSO4] [C4mim][CF3SO3] [(C6H13)3P(C14H29)][Cl] [(C6H13)3P(C14H29)][NTf2] [(C6H13)3P(C14H29)][Ac] [N1114][NTf2] [C3mpip][NTf2] [C4mpy][BF4] [C4mpyr][NTf2] [C3mpy][NTf2] [C6mim][NTf2] [C4mim][PF6] [C6mim][PF6] [C4mim][BF4] [C4mim][SCN] [C2mim][SCN]

320.45 198.0 391.3 419.2 236.3 288.2 519.3 374.3 542.9 396.4 422.41 237.11 422.4 416 447.3 284.2 312.2 226.2 197.3 169.3

874.40 781.07 749.77 509.99 709.40 750.93 681.60 742.80 690.10 639.77 679.17 539.50 669.077 638.77 603.00 600.90 687.49 728.07 604.50 625.50

0.59020 0.50291 0.57060 0.52975 0.46400 0.53480 0.67060 0.72600 0.67700 0.56349 0.57604 0.49991 0.57564 0.56430 0.59760 0.54140 0.60099 0.51391 0.51099 0.49964

2.807 2.293 2.639 3.406 3.550 2.752 3.601 3.563 3.673 2.894 2.907 2.802 2.919 2.920 2.760 2.366 4.083 2.602 2.483 2.201

Table 3 Average absolute deviation (AAD in %) of the predicted densities of ILs studied in this work using the proposed PHSC EOS, compared with the measurements. Ionic liquid

ΔP (MPa)

ΔT (K)

NPa

AADb

MDc (%)

Ref.

[C2mim][NTf2]

0.1–30 0.1 0.1–60 1–40 0.1–35 0.1 0.1–35 0.1–40 0.2–65 0.2–65 0.1–35 0.2–65 0.1–65 0.1–60 0.1–10 1–200 0.1–60 0.1–20 1–200 0.1–30 0.1–35 0.1 0.1 0.1 0.1–35 0.1–35 0.1–35 1–200 0.1–100 0.7–40 0.1–202 1–200 0.1–10 0.1 0.1–10 0.1–10

293–393 293–391 298–328 322–415 293–393 293–392 283–333 293–415 298–333 298–333 273–318 298–333 293–338 293–333 293–393 313–452 298–332 293–353 313–472 293–393 298–353 293–391 273–363 293–363 293–393 293–393 293–393 313–472 298–323 298–373 298–343 313–472 293–393 283–393 298–338 298–338

096 009 168 030 080 009 063 041 144 134 072 126 155 163 077 168 068 020 180 096 036 009 018 071 091 091 091 178 144 045 014 180 077 096 065 065 3153

0.15 0.19 0.56 0.82 0.45 0.64 0.90 1.29 0.24 0.22 0.15 0.23 0.32 0.28 0.27 1.05 0.42 0.37 0.71 0.36 0.51 0.48 0.18 0.21 0.27 0.25 0.22 1.25 0.27 0.89 0.88 0.86 1.80 0.99 0.43 0.44 0.54

0.71 0.45 0.98 1.53 1.21 1.04 1.47 2.63 0.58 0.56 0.43 0.51 0.72 0.76 0.69 2.31 1.19 0.66 1.68 0.89 1.17 1.06 0.33 0.41 0.77 0.73 0.68 2.49 0.75 2.16 1.89 1.86 2.78 2.66 0.71 0.84

[23] [24] [25] [26] [27] [24] [28] [26] [29] [29] [30] [29] [23] [27] [31] [32] [33] [34] [35] [23] [36] [24] [37] [38] [39] [39] [39] [32] [33] [40] [41] [35] [31] [42] [43] [43]

[C4mim][NTf2] [C2mim][EtSO4]

[(C6H13)3P(C14H29)][Ac] [(C6H13)3P(C14H29)][Cl] [(C6H13)3P(C14H29)][NTf2] [C6mim][NTf2] [C4mim][CF3SO3] [C4mim][BF4]

[C2mim][BF4] [N1114][NTf2]

which apparently is not far from incompressibility. The only way we can see is that, according to the fitted parameters, the segment packing fraction, η comes out to be slightly close to unity. Some assessments were made on the performance of the proposed PHSC EOS for ILs. At first, we examine PYHSC EOS in the prediction of the mass density of studied ILs. For this purpose, 20 systems including imidazolium-, phosphonium-, ammonium-, pyridinium-, pipyridinium-, and pyrrolidinium-based ILs have been chosen, for which their measured values were available in the literature. Our calculation results were summarized as average absolute deviation percent (AAD in %) from literature data [23–43] in Table 3. As Table 3 shows from 3153 data points examined for studied ILs over a broad pressure ranging from the ambient-pressure up to 200 MPa and the temperature ranging from 273 to 472 K, AAD was found to be 0.54%. It should be mentioned that, the accuracy of our calculations was of the order of ±2.78%. In order to assess further the reliability of the PHSC EOS, we have compared the present model with the group contribution method of Gardas and Coutinho (G & C) [9] as well as our previous model [44]. The outcomes of the calculations are given in Table 4. From 960 data points taken from the literature [23,25,27–29,31,34,36], the AAD of the predicted densities using the present model and those obtained from the work of G & C was found to be 0.46% and 0.53%, respectively. As it is clear from Table 4, the predicted densities

1600

ρ calc. / kg. m-3

1400

1200

1000

800 800

1000

1200

ρ exp. / kg. m-3

1400

1600

Fig. 1. Deviation in predicted densities (3153 data points) vs experimental densities [23–43] of all ILs studied in this work.

[C2mim][OcSO4] [C3mpy][NTf2] [C4mpyr][NTf2] [C3mpip][NTf2] [C4mim][PF6]

[C6mim][PF6] [C4mpy][BF4] [C4mim][SCN] [C2mim][SCN] Overall a b

NP represents the number of data points examined. NP P ρ i;Calc: −ρi;Exp: =ρi;Exp: .

AAD ¼ 100=NP

i¼1 c MD represents an absolute value for the maximum deviation percent of the calculated density.

obtained from the proposed PHSC EOS have almost the same accuracy as those obtained from Ref. [9]. The interesting point of the present study is that the average AAD of the present model is 0.46% which is somewhat lower than those obtained from our previous model (i.e. 0.67%) which used the surface tension properties. This is slightly due to the fact that despite the microscopic scaling parameters (i.e., m, σ, ε/k), the determination of surface tension is strongly influenced by the uncertainty of the experimental method and so it is not seemed to be a more appropriate scaling parameter than microscopic ones. In the next session of the present study, we have utilized the proposed model for the estimation of some thermodynamic coefficients such as the isothermal compressibility (κT) and liquid bulk modulus (BT), both of which are closely related to the derivative of the pressure with respect to density at constant temperature. We have summarized our calculation results as AAD (in %) of the calculated thermodynamic coefficients using PHSC EOS from those obtained by the use of Tait equation [28,31,34] in Table 5. We have assessed the proposed PHSC EOS by taking 1043 literature data points [27,29,33,43] over the temperature and pressure range within 293–338 K and 0.1–150 MPa, respectively. The overall AAD for the tested data points calculated by PHSC EOS was found to be 1.63%. It should be mentioned that the uncertainty of the calculated thermodynamic coefficients (κT and BT) by the present work was of the order of ±5% which is lower than those obtained by the work of

S.M. Hosseini et al. / Journal of Molecular Liquids 174 (2012) 52–57 Table 4 Comparison of AAD calculated high-pressure densities of some studied ILs in this work, the preceding work of Hosseini et al. [44] and those estimated by the method of G & C [9], compared with the experiment.

55

0.8 0.7

Ionic liquid

ΔT (K)

NPb

Hosseini et al. [44]

This work

G&C [9]

Ref.

[C2mim][NTf2] [C4mim][NTf2] [C6mim][PF6] [C2mim][EtSO4] [C4mim][CF3SO3] [(C6H13)3P(C14H29)][Cl] [(C6H13)3P(C14H29)][NTf2] [C6mim][NTf2] [C4mim][BF4] [N1114][NTf2] Overall

293–393 298–333 293–393 293–393 293–393 298–333 298–334 293–338 293–353 293–415

096 168 077 063 077 134 126 163 020 036 960

0.51 0.96 0.89 0.34 0.27 0.71 0.75 0.71 0.27 0.50 0.67

0.15 0.56 1.80 0.90 0.27 0.22 0.23 0.28 0.37 0.51 0.46

0.09 0.36 0.04 0.46 0.03 2.43 0.24 0.26 0.24 0.23 0.53

[23] [25] [31] [28] [31] [29] [29] [27] [34] [36]

a

AAD ¼ 100=NP

NP P

0.6 0.5 0.4 0.3 0.2 0.2

0.3

0.4

0.5

0.6

0.7

0.8

κT exp. / GPa-1 Fig. 2. Deviation in estimated isothermal compressibilities, κT (1043 data points) vs experimental ones [27,29,33,43] for 8 ILs studied in this work.

jρi;Calc: −ρi;Exp: j=ρi;Exp: .

i¼1 b

κT calc./ GPa-1

AAD (%)a

NP represents the number of data points examined.

Goharshadi and Moosavi [45] (± 8%) (to see how the proposed PHSC EOS passes through the literature data, Fig. 2 has been provided in the supporting information file). The precision of the proposed EOS can be discussed in detail; some systematic errors have been observed in the current prediction. To show graphically the performance of the model, some comparisons of the calculated densities and isothermal compressibilities with their experimental values have been made in Figs. 1 and 2. They illustrate the good harmony between the experimental data [23–43] and the estimated densities (for 3153 data points) and the isothermal compressibility coefficients (for 1043 data points) using the proposed PHSC EOS. Another feather of this paper is to show some regularities (linearity) that appeared in the PVT behavior of the studied systems; the liquid bulk modulus (reciprocal compressibility) of ILs was well-found to be linear in the pressure [46] and the second is the linear relation between temperature and density at unit compression factor (Z = P/ ρRT = 1). It has been found that along the contour defined by Z = 1, where the compression factor is the same as for an ideal gas, the density of studied liquids is nearly linear function of temperature [47]. Along the entire Zeno contour, the attractive and repulsive contributions to

the intermolecular potential are in dynamic balance. This regularity can be as a means for testing equations of state. Because the Zeno behavior appears to be a generic property of pure fluids, it is advantageous to incorporate it in the EOS models. In the present study, we have examined the PHSC EOS whether Zeno condition can be satisfied (to illustrate the Zeno contour for several ionic liquids Fig. 3 has been presented in the supporting information file). The present work demonstrated that the above-mentioned linearity can be properly predicted by PHSC EOS for some studied ILs. Besides, the linear dependency of the reduced bulk modulus, BT of IL on pressure at several isotherms has also been successfully investigated (see Fig. 4 in the supporting information file). This is the advantageous feature of Eq. (1) because most of the equations of state cannot describe the compressed liquid region over a wide range of pressure. Finally, we have extended perturbed Yukawa hard-sphere chain EOS for modeling the density and derived thermodynamic properties of ILs. Knowing only three pure-component parameters, ε/k, σ, and m is sufficient to present an accurate PHSC EOS for ILs. The equation of state works over a wide range of temperatures from roomtemperatures up to 472 K and pressures from ambient pressure up to 200 MPa with very good accuracy. 4. Conclusion

Table 5 AAD (in %) of the estimated isothermal compressibility (κT) and the liquid bulk modulus (BT) using the proposed PHSC EOS for some ILs studied in this work, compared with the literature data. Ionic liquid

ΔT (K)

ΔP (MPa)

AADb

MDc (%)

AADd

Ref.

[(C6H13)3P(C14H29)] [Cl] [(C6H13)3P(C14H29)] [NTf2] [(C6H13)3P(C14H29)] [Ac] [C6mim][NTf2] [C4mim][PF6] [C4mim][NTf2] [C4mim][SCN] [C2mim][SCN] Overall

298–323

0.2–65

134

2.32

5.00

2.50

[29]

298–334

0.2–65

126

1.54

3.64

1.47

[29]

298–334

0.2–65

143

2.25

4.24

2.49

[29]

293–338 298–233 293–393 298–338 298–338

0.1–60 0.1–100 0.1–150 0.1–10 0.1–10

163 144 204 065 065 1043

1.30 1.38 1.95 0.98 1.28 1.68

3.35 3.18 3.96 1.88 2.96

1.21 1.20 – 0.54 0.88

[27] [33] [27] [43] [43]

a b

NPa

NP represents the number of literature data points examined. NP P AAD ¼ 100=NP κ Ti;Calc: −κ Ti;Lit: =κ Ti;Lit: . i¼1

c d

MD represents an absolute value for the maximum deviation percent of the κT. NP P AAD ¼ 100=NP BTi;Calc: −BTi;Lit: =BTi;Lit: . i¼1

In conclusion, the work presented here showed the entire success in describing PVT properties and some regularities of ILs using a simplified PHSC EOS combined from the Yukawa hard-sphere chain plus the first order perturbation theory with sufficiently few parameters. The calculated results revealed that the predictive PHSC EOS can well represent the thermodynamic coefficients as well as densities of the studied systems. The predicted results were compared with those obtained from other approaches developed in the literature; e.g., the method of G & C [9] based on the use of group contributions and PHS EOS previously developed by Hosseini et al. [44]. It should be mentioned that the extension of the proposed EOS for modeling the PVTx properties of mixtures involving ILs remains for future work. Abbreviations [C2mim][OcSO4] 1-ethyl-3-methylimidazolium octylsulfate [C2mim][BF4] 1-ethyl-3-methylimidazolium tetrafluoroborate [C2mim][NTf2] 1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide

56

S.M. Hosseini et al. / Journal of Molecular Liquids 174 (2012) 52–57

[C4mim][NTf2] 1-butyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide [C2mim][EtSO4] 1-ethyl-3-methylimidazolium ethylsulfate [C4mim][CF3SO3] 1-butyl-3-methylimidazolium trifluoromethanesulfonate [(C6H13)3P(C14H29)][Cl] 519 trihexyltetradecylphosphonium chloride [(C6H13)3P(C14H29)][NTf2] trihexyltetradecylphosphonium bis[(trifluoromethyl)sulfonyl]imide [(C6H13)3P(C14H29)][Ac] tetradecyl(trihexyl)phosphonium acetate [N1114][NTf2] butyltrimethylammoniumbis(trifluoromethylsulfonyl)imide [C3mpip][NTf2] 3-methyl-1-methyl pyrrolidiniumbis[(trifluoromethyl)sulfonyl]imide [C4mpy][BF4] 1-butyl-1-methylpyridinium tetrafluoroborate [C4mpyr][NTf2] 1-butyl-1-methylpyrrolidinium bis[(trifluoromethyl)sulfonyl]imide [C3mpy][NTf2] 3-methyl-1-propylpyridinium bis[(trifluoromethyl)sulfonyl]imide [C6mim][NTf2] 1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide [C4mim][PF6] 1-butyl-3-methylimidazolium hexafluorophosphate [C6mim][PF6] 1-hexyl-3-methylimidazolium hexafluorophosphate [C4mim][BF4] 1-butyl-3-methylimidazolium tetrafluoroborate [C4mim][SCN] 1-butyl-3-methylimidazolium thiocyanate [C2mim][SCN] 1-ethyl-3-methylimidazolium thiocyanate

Nomenclature and units List of symbols ai values of coefficients in Eq. (5) for the case of λ = 1.8 AAD average absolute deviation (%) kB Boltzmann's constant (J K −1) NP number of data points P pressure (Pa) T absolute temperature (K) MW molecular weight (g mol −1) m segment number R universal gas constant (J mol −1.K −1) BT liquid bulk modulus

Greek letters isothermal compressibility κT η Yukawa hard-sphere chain packing fraction ρ segment density (mol m −3) σ segment diameter (nm) ε non-bonded segment–segment interaction energy (J) λ range parameter of attractive forces

Superscripts HSC hard-sphere chain reference system Pert. perturbed system Lit. literature data Calc. calculated Exp. experimental

Acknowledgment We are grateful to the research committee of Shiraz University and Shiraz University of Technology for supporting this project.

Appendix A. Supplementary data Supplementary data to this article can be found online at http:// dx.doi.org/10.1016/j.molliq.2012.07.007.

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