Thermodynamic Modeling of Aqueous Ionic Liquid Solutions Using PC

Article pubs.acs.org/IECR

Thermodynamic Modeling of Aqueous Ionic Liquid Solutions Using PC-SAFT Equation of State Reza Shahriari, Mohammad Reza Dehghani,* and Bahman Behzadi Thermodynamics Research Laboratory, School of Chemical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran ABSTRACT: In this work, an equation of state has been utilized for thermodynamic modeling of aqueous ionic liquid (IL) solutions. The proposed equation of state is a combination of perturbed chain statistical associating fluid theory (PC-SAFT) equation of state and mean spherical approximation (MSA) term. In this model, the ion-based approach has been used to adjust the model parameters. The ion parameters have been estimated through simultaneous fitting to experimental mean ionic activity coefficient and liquid density data of strong electrolytes. Using adjusted ion parameters, osmotic coefficients, mean ionic activity coefficients, liquid densities, apparent molar volume, and water activity of several ILs, assumed as chainlike electrolytes, have been calculated. Results show that PC-SAFT, in combination with the MSA term has acceptable accuracy for prediction of density, apparent molar volume, and activity coefficient of ILs. The average deviations of predicted mean ionic activity coefficients, liquid densities, water activity, and apparent molar volume are 6.43, 0.86, 0.033, and 8.20%, respectively.

1. INTRODUCTION The first ionic liquid (IL), ethylammonium nitrate, was reported as early as 1914.1 Subsequently, thousands of ILs have been synthesized for specific applications in various fields. In the past decade, a considerable amount of research has been focused on the synthesis of ILs for specific applications.2−8 Being nonvolatile with negligible vapor pressure, nonflammable, and thermally stable are the common characteristics of ILs. Similar to salt molecules, ILs are comprised of a cationic part (such as imidazolium, pyridinium, or ammonium) and an anionic part (such as chloride, hexafluorophosphate, or tetrafluoroborate). However, unlike usual salts, their melting point is usually lower than 100 °C.9 The length of the alkyl chain and the size of the anion have a remarkable influence on melting and decomposition temperatures. This property has turned them into attractive solvents for application in chemical processes. Most of the ILs are relatively new compounds, and as a result there are not enough experimental data on their thermodynamics properties, such as density, activity coefficients, and osmotic coefficients in mixtures. Considering the above fact, prediction or even correlation of their thermophysical properties is essential and useful for engineering applications.10 Aqueous solutions of ILs can be categorized as electrolyte solutions in which the ILs dissociate into ionic particles. In such mixtures, the nature of electrostatic ion−ion and ion−solvent interactions is different from that of dispersive and repulsive interactions between neutral particles. Therefore, the electrostatic interaction plays an important role for nonideality of such systems, which must be correctly accounted for in relevant thermodynamic modeling. During recent years, many models have been developed to obtain an accurate description of thermodynamic properties of aqueous electrolyte solutions. There are two kinds of approaches to describe the nonideality of electrolyte solutions; one is the application of activity coefficient models, and the other consists of equation of state (EoS) models. Despite the © 2012 American Chemical Society

simplicity and vast application of activity coefficient models, there are some disadvantages as well; one disadvantage is that the density of the solution cannot be derived from the model itself, and they are pressure-independent. On the other hand, this approach depends on the reference state, the correct choice of which is sometimes difficult. The EoS do not have these disadvantages, but to extend the EoSs to electrolyte solutions, an additional Helmholtz free energy term is needed to consider the electrostatic interaction between charge particles. For this purpose, long-range electrostatic terms such as the Debye− Huckel term11 or the MSA term12,13 can be used. The first attempt to develop an EoS for electrolyte solutions was proposed by Planche and Renon.14 Afterward, the models of Fürst and Renon,15 Wu and Prausnitz,16 Myers et al.17and the electrolyte SAFT (statistical associating fluid theory) EoS18−21 are the most notable in this field. Cameretti et al.18 used the perturbed chain SAFT (PC-SAFT) model22 with a Debye−Hückel term, and Galindo et al.19 extended the SAFTVR model for aqueous electrolyte solution in which a simplified MSA term was used for long-range interactions. The electrolyte equation of state (eEoS) of Tan et al.21 includes the SAFT EoS plus a restricted primitive model of MSA for ionic interactions. All aforementioned models were utilized for modeling typical single strong electrolyte solutions. Recently, Li et al.23 applied an equation of state based on statistical mechanics to describe the thermodynamic properties of aqueous solutions of ILs. In their model, long-range interactions and ion pairing were considered using an MSA term and the shield-sticky approach,24,25 respectively. Using the proposed model, densities and osmotic coefficients of nine aqueous IL solutions were correlated with acceptable accuracy. Received: Revised: Accepted: Published: 10274

May 17, 2012 July 3, 2012 July 6, 2012 July 6, 2012 dx.doi.org/10.1021/ie3012984 | Ind. Eng. Chem. Res. 2012, 51, 10274−10282

Industrial & Engineering Chemistry Research

Article

In another recent work, Belvèze et al.26 applied the electrolyte NTRL27 to model activity coefficients of quaternary ammonium salts in water. Two binary interaction parameters were adjusted for each salt over wide ranges of salt concentration, assuming the ILs were strong electrolytes with complete dissociation, whereas this assumption is not in agreement to pervious investigation on this kind of ILs.28 We have recently modeled gas solubilities in pure ILs using EoS approaches;29 the electrostatic interactions between counterions of pure ILs were ignored, and the ILs were modeled as neutral molecular liquids. On the other hand, the polar and association interactions between IL molecules were taken into account. The results indicated that by considering IL molecules as neutral species, reasonable results could be obtained.29 In this work, the modified PC-SAFT equation of state has been utilized to correlate the thermodynamic properties of several binary aqueous IL solutions. The model is described first, and then the procedure of parameter estimation is presented. Finally, calculation of mean ionic activity coefficients (MIACs), osmotic coefficients, water activities, apparent molar volume, and the densities of ionic liquid solutions are given.

specified, the compressibility factor and chemical potential can be derived by applying standard thermodynamic relations.30 In this work, the contribution due to the electrostatic interactions between ions (Aelec) is calculated using a mean spherical approximation (MSA) term, given as follows:12,13 ⎡ ⎤ Γzi 2 λ ⎢ π Γ3 Aelec 2⎥ =− + Ω + P ∑ n ⎥⎦ NkBT 4πρ ⎢⎣ k =+, − 1 + Γσi 2Δ 3πρ (5)

The MSA parameters are obtained using the following equations: 4Γ 2 = λ

Ω=1+

Pn =

2. ELECTROLYTE PC-SAFT EQUATION OF STATE The PC-SAFT EoS was based on a perturbation theory, where a hard-chain system is used as reference.22 In the modified PCSAFT model for electrolyte solutions, the residual Helmholtz free energy is comprised of four contributions:18 res

hc

disp

elec

λ=

⎤ ⎡ ⎡ Mi ⎥ X Ai ⎤ Ai ⎢ x ln X − + ⎥ ∑ i ⎢∑ ⎢ 2 ⎦ 2 ⎥⎦ ⎣ Ai ⎣ i

1 1 + ρ ∑i xi ∑B X BiΔA iBj j

(1)

k =+, −

1 + Γσk

(7)

ρk σkzk

∑ k =+, −

π 6

ρk σk 3

1 + Γσk

∑

(8)

ρk σk 3

(9)

k =+, −

e2 ϵw ϵ0kBT

(10)

ϵw = 281.67 − 1.0912T + 1.6644 × 10−3T 2 − 9.7592 × 10−7T 3

(11)

3. THERMODYNAMIC PROPERTIES The activity coefficient of electrolyte is one of the important properties for phase equilibrium calculations. When applying an EoS approach, the activity coefficient should be derived from fugacity coefficients. For electrolyte solutions, asymmetrical activity coefficients are derived as follows:

(2)

γi* =

φi(T , P , x) φi(T , P , xe→0)

(12)

The fugacity coefficient of component i in a mixture is derived from the residual Helmholtz free energy, given as

(3)

where Δ is the association strength between two association sites A and B belonging to two different molecules i and j, which is given as AiBj

⎡ ⎛ ε A iBj ⎞ ⎤ ΔA iBj = gijhs(dij+)⎢exp⎜ ⎟ − 1⎥σij 3κ A iBj ⎢⎣ ⎝ kBT ⎠ ⎥⎦

∑

where e is the electronic unit charge (1.60219 × 10 C) and ϵ0 is the vacuum permittivity. The relative permittivity of pure water can be determined from the follow equation:21

where xi is the mole fraction of component i in the mixture, M is the number of association sites per molecule, and XAi is the fraction of molecules not bonded at specific interaction site A. The summation is over all association sites on molecule i. The nonbonded fraction XAi in the mixture is expressed as X Ai =

π 2Δ

(6)

−19

where the superscripts res, hc, disp, elec, and assoc refer to residual, hard chain, dispersion, electrostatic, and association contributions, respectively. Details of the hard-chain term (Ahc) and the dispersion term (Adisp) are given in the original work of Gross and Sadowski.22 Chapman et al.30 used Wertheim’s theory to obtain the contribution due to association, given as follows: Aassoc = NRT

1 Ω

Δ=1−

assoc

A A A A A = + + + NkBT NkBT NkBT NkBT NkBT

⎡ z − (π /2Δ)σ 2P ⎤2 k n ⎥ ∑ ρk ⎢ k 1 + Γ σ ⎦ ⎣ k k =+, −

ln φi = μi res − ln Z

(13)

⎛ ∂(Ares /NkBT ) ⎞ Z = 1 + ρ⎜ ⎟ ∂ρ ⎝ ⎠T , n

(14)

μi is the chemical potential, which is derived from the Helmholtz free energy: ⎡ ∂(na res) ⎤ μi res = ⎢ ⎥ ⎣ ∂ni ⎦T , V , n

(4)

εAiBj and κAiBj are the association energy and volume of interaction between site A of the molecule i and site B of the molecule j, respectively. Once the Helmholtz free energy is

j≠i

⎡ ∂(ρa res) ⎤ ⎥ =⎢ ⎢⎣ ∂ρi ⎥⎦ T ,ρ

μi res = μi hc + μidisp + μielec + μiassoc 10275

j≠i

(15) (16)

dx.doi.org/10.1021/ie3012984 | Ind. Eng. Chem. Res. 2012, 51, 10274−10282


Article

ares is the residual molar Helmholtz free energy. The MIAC (based on mole fraction) is defined as γ± x = (γ+ν+γ

ν

)1/(ν++ ν

)

273.15 and 298.15 K, Held et al. use a temperature-dependent equation for molecular diameter, given as follows: σw(T ) = σw0 + T1 exp(TT2) + T3 exp(TT4)

(17)

Good agreement with experimental saturated liquid density data was reported using eq 21 (see Table 1). 4.2. Strong Electrolytes. In this work, the ion-specific approach has been utilized to obtain a universal set of ionic parameters. Ji and Adidharma32 have reported that, for thermodynamic modeling of electrolyte solutions, regression of activity coefficients is more effective than the application of osmotic coefficient data for the fitting of model parameters. For this purpose, in the first step, eight conventional salts including NaCl, NaBr, NaI, NaF, KCl, KBr, KF, and KI have been selected. Then, the ion-specific parameters have been adjusted by simultaneous regression of experimental data on MIAC and liquid solution densities. The results are reported in Table 2. For the studied electrolyte solutions, short-range dispersion solvent−solvent as well as ion−solvent interactions have been considered in the model. The ions are assumed as spherical segments. Therefore there are two adjustable parameters for each of the ions (i.e., σ and ε). The following objective function has been used for parameter optimization:

where v+ and v− are stoichiometric numbers of cation and anion, respectively. Consequently the osmotic coefficient in an aqueous electrolyte solution is determined from Φ=−

ln(x wγw ) (ν+ + ν−)mM w

× 1000 (18)

where m and Mw are the molality and molecular weight of water, respectively. The mole fraction based activity coefficient of water is defined as γw =

φw (T , P , x) φw (T , P , x w→1)

(19)

The apparent molar volume (Vϕ) of salts in the studied solvent has been calculated from densities of solutions using the follow equation: Vϕ =

1000(ρ − ρ0 ) M − ρ mρρ0

(21)

2 ⎡⎛ m ,exptl ⎛ ρexptl − ρcalcd ⎞2 ⎤ − γ±m, j,calcd ⎞ ⎢⎜ γ±, j j ⎟ ⎜ j ⎟⎥ OF = ∑ ∑ ⎢⎜ m ,exptl exptl ⎟ +⎜ ⎟⎥ γ ρ i=1 j=1 ⎢ ⎠ ⎝ ⎠ ⎥⎦ ±, j j ⎣⎝ i

(20)

Ns

where m and M are the molality and molar mass of salt in binary mixtures and ρ and ρ0 are densities of salts solution and pure solvent, respectively.

Np

(22)

4. PARAMETER ESTIMATION 4.1. Pure Water. In solution thermodynamics, accurate modeling of the solvent is a primary requirement. Here the EoS parameters of water are taken from Held et al.;31 as given in Table 1. For accurate modeling of water density between

where Ns is the number of studied salt solutions, and Np is the number of data points for each salt. Experimental data on electrolyte solution density have been taken from Novotny and Sohnel,33 whereas MIAC is given in Robinson and Stokes.34 The fitted diameters of the cations are generally larger than the corresponding Pauling diameters, while, for the anions, minor differences are observed compared to the Pauling diameters. This could point to the strong hydration capacity of the cations. It must be noted that in order to minimize the aforementioned objective function, 12 parameters have been adjusted simultaneously. Three optimization algorithms were utilized, namely, the Nelder−Mead simplex method,35 a genetic algorithm (GA),36,37 and the shuffled complex evolution (SCE) algorithm.38 In the simplex method, high sensitivity to the initial guesses was observed, leading to multiple local minima. Our comparison of the GA and SCE algorithms for adjusting the parameters has shown that the SCE algorithm gives a

Table 1. Estimated PC-SAFT Parameters for Pure Water

a

parameter

value

parameter

value

m σw0 (A) T1 (A) T2 (1/K) T3 (A) T4 (1/K)

1.204 659 2.792 700 10.1100 −0.017 75 −1.417 00 −0.011 46

ε/kB (K) εAB/kB (K) κAB ARD (%)a psat ARD (%)a ρsat ΔT (K)

353.9449 2425.6714 0.0450 989 0.52 0.06 273.15−373.15

expt sat ARD (%) = (100/N)∑Ni=1|1 − (Ωcalc i /Ωi )|, where Ω is ρ or p .

Table 2. Ion Parameters Obtained for EoS and the Average Relative Deviations (ARDs) in the MIAC, Solution Densities, and Osmotic Coefficient at 298.15 K and 1 bar ARDa (%)

parameters ion Na+ K+ F− Cl− Br− I−

σ(Å)

ε/kB (K)

salt

mmax (mol/kg)

ρ

γ*±

ϕ

2.772 3.358 1.645 2.801 3.222 3.663

412.57 189.85 479.65 69.88 84.62 105.87

NaCl KCl NaBr KBr NaF KF NaI KI

6.0 5.0 4.0 5.5 1.0 6.0 3.5 4.5

0.71 0.64 0.44 0.58 0.34 0.84 0.08 0.12 0.468

1.97 0.98 1.43 0.84 1.07 1.72 1.26 0.77 1.26

1.42 0.78 0.99 0.75 0.84 1.21 0.98 0.57 0.942

av error a

expt ARD (%) = (100/N)∑Ni=1|1 − (Ωcalc *, or ϕ. i /Ωi )|, where Ω is ρ, γ±

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reasonable set of parameters at lower computational time, with the lowest values of objective function and percent absolute relative deviations (ARD %). The results show that the proposed model can correlate the MIAC and densities of alkali halide electrolyte solutions with acceptable accuracy. As shown in Table 2, the average deviations for density, MIAC, and osmotic coefficient are lower than 0.47%, 1.3%, and 0.95%, respectively. Figures 1 and 2 show the calculated results compared to experimental data.

Figure 2. Comparison of the correlated mean ionic activity coefficient with experimental data for aqueous electrolyte solutions at 298.15 K. Solid lines are correlation using proposed model.

Previous studies39,40 have shown that, in IL systems, considerable ion pairing can occur. On the basis of these findings, for all families of ILs studied in this work, the ion pair formation between counterions has been taken into account using an association contribution. An acceptor association site on the charged segment of the cations and a donor association site on the anions have been considered (see Figure 3). In the proposed model, association between charged ions and water molecules has been neglected.

Figure 1. Comparison of the correlated solution liquid densities with experimental data for aqueous electrolyte solutions at 298.15 K. Solid lines are correlation using proposed model.

4.3. Ionic Liquids. Osmotic coefficient experimental data have been utilized for parameters estimation in the case of ILs, as there are not enough experimental data on activity coefficients of the ILs. For this reason, after adjusting the model parameters for the common ions, the modified PCSAFT model has been utilized to correlate the osmotic coefficients of the ILs. The primitive MSA model has been used to apply the effect of electrostatic interactions between counterions. A single charged segment has been assumed for each cation chain, and the electrostatic interactions between charged segments of the cations and those of the anions have been considered.

Figure 3. Schematic representation of the molecular models used in this work. 10277



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simultaneous regression of several salts. The parameter estimation procedure is as follows: in the first step the [C4mim]+ and [C5mim]+ cation parameters as well as the association energy between cation and anion are adjusted by simultaneous fitting for the following ILs: [C4mim][Cl] (1butyl-3-methylimidazolium chloride), [C4mim][Br] (1-butyl-3methylimidazolium bromide), [C5mim][Cl] (1-pentyl-3-methylimidazolium chloride), and [C5mim][Br] (1-pentyl-3-methylimidazolium bromide). Then using [C4mim] cation parameters, the [MSO4] parameters are obtained. For estimation of the [C1mim] (1,3-dimethylimidazolium) and [C3mim] (1propyl-3-methylimidazolium) cation parameters, the adjusted parameters of [MSO4] anion have been used. The same procedure is also used for calculation of [C2mim] (1-ethyl-3methylimidazolium) and [C6mim] (1-hexyl-3-methylimidazolium) cation parameters, and also for the alkylammonium ILs. A wide range of temperature has been considered for fitting of component parameters, resulting in a set of temperatureindependent parameters. Since no experimental data on MIAC and osmotic coefficients of [C7mim][Br] and [C8mim][Br] were available, the cation parameters for these two types of ILs have been obtained by using experimental density data of the relevant aqueous solutions. The following objective function has been utilized for adjusting model parameters for all systems that are considered in this work:

To reduce the number of model parameters, the association volume κAiBj for the cation−anion association is set to 0.03.20 This assumption has been verified by comparing results using different values of this parameter (between 0.001 and 0.5) for several IL solutions, including small and long alkyl chains. Fitting of κAiBj has a minor effect on the modeling results, which could be compensated for by an accurate fitting of the other model parameters .Therefore, by considering ion-pair formation in ILs and by using an association contribution term, the association energy between counterions (εAiBj) should be taken into account as an additional parameter. Meanwhile, to reduce the number of adjustable parameters, the segment number of imidazolium or alkyl−ammonium cations is determined by a simple rule: one segment number is assumed for the imidazolium ring or ammonium group, and the segment number is set equal to 0.3 for each methyl or ethyl group attached to the imidazolium ring or ammonium group. Therefore for each cation, just two adjustable parameters (i.e., σ and ε) have been considered. Using the adjusted parameters of anions given in section 4.2, the parameters of cations have been regressed to experimental osmotic coefficient data, as reported in Table 3. Table 3. Estimated Individual Ion Parameters for Ionic Liquid Solutions ions +

[C1mim] [C2mim]+ [C3mim]+ [C4mim]+ [C5mim]+ [C6mim]+ [C7mim]+ [C8mim]+ [(CH3)4N] + [(C2H5)4N] + [MSO4]− [ESO4]− [BF4]−

m

σ (Å)

ε/kB (K)

1.6 1.9 2.2 2.5 2.8 3.1 3.4 3.7 2.2 3.4 1.0 1.0 1.0

3.990 3.966 4.308 4.732 4.930 4.985 4.887 4.891 1.647 1.631 3.289 3.742 4.864

162.451 125.708 150.021 167.762 160.589 155.104 117.334 119.714 249.886 204.921 68.956 150.785 98.992

⎡⎛ exptl calcd ⎞2 ⎤ ⎢⎜ Ω j − Ω j ⎟ ⎥ OF = ∑ ∑ ⎢⎜ ⎟⎥ Ωexptl ⎠⎦ j i = 1 j = 1 ⎣⎝ Ns

Np

(23)

i

In this equation, the symbol Ω is the osmotic coefficient (or aqueous solution densities in the case of [C7mim][Br]and [C8mim][Br]).

5. RESULTS All regressed parameters have been listed in Tables 2−4. It should be noted that the temperature-independent parameters have been obtained by simultaneously fitting the experimental data for several ILs and that the obtained parameters are saltindependent. In Table 4 the estimated cross-association energy parameters between counterions of the ILs have been presented. The association energies of the ILs show that heavier halides tend to pair more strongly with IL cations. It is worth mentioning there is a consistency between the calculated association constants and corresponding experimental data for some of the ILs such as tetra-n-alkylated ammonium salts.28,41

It should be noted that for some of the ILs that contain methyl sulfate (MSO4) and ethyl sulfate (ESO4) groups, the anion parameters have not been obtained in section 4.2; for these ILs the anion parameters have been fitted after obtaining common cation parameters (see Table 3). It is also worth mentioning that in order to obtain universal parameters, the optimal parameters have been obtained by

Table 4. Estimated Cross-Association Energy Parameter between Counter Ions of Ionic Liquids ions [C1mim]+ [C2mim]+ [C3mim]+ [C4mim]+ [C5mim]+ [C6mim]+ [C7mim]+ [C8mim]+ [(CH3)4N] + [(C2H5)4N] +

Cl−

Br−

[BF4]−

[MSO4]−

[ESO4]−

505.090

1487.670 1709.363

108.239 61.740

1085.596 1843.647 2055.228 1853.103 2142.929 1199.206 1739.924 1955.691 2186.080

1873.014 1510.813

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Table 5. Average Relative Deviation (ARD) Values of Fitted Osmotic Coefficients, Predicted Apparent Molar Volume, MIAC’s, Solution Densities, and Water Activities of Ionic Liquid Solutions ΔT

Npa

a

ARD (%)

component

Φ

ρ or Vϕ

mmax

Φ

ρ or Vϕ

Φ

aw

[C1mim][MSO4] [C2mim][Br] [C2mim][ESO4] [C3mim][Br] [C3mim][MSO4] [C4mim][Cl] [C4mim][Br] [C4mim][BF4] [C4mim][MSO4] [C5mim][Cl] [C5mim][Br] [C6mim][Br] [C7mim][Br] [C8mim][Br] [(CH3)4N][Cl] [(CH3)4N][Br] [(C2H5)4N] ][Cl] [(C2H5)4N] ][Br] Average Deviation

30 12 76 80 39 41 12 68 60 76 80 76 36 22 26 29

72 50

3.29 0.44 2.76 2.70 1.34 4.08 2.02 2.065 2.17 1.9 2.00 2.39 0.24 0.15 19 5.5 9.0 12

313.15−333.15 298.15 298.15−328.15 298.15−328.15 308.15−328.15 298.15−333.15 318.15 298.15−328.15 298.15−328.15 298.15−328.15 298.15−328.15 288.15−328.15

298.15−328.15 293.15−313.15

2.59 0.37 1.85 1.18 3.55 2.78 1.81 2.91 1.98 2.32 2.29 2.38

0.05 0.001

80 56 45

65 60 100

288.15−308.15 298.15−328.15 298.15−318.15

288.15−308.15 288.15−308.15 288.15−308.15

298.15 298.15 298.15 298.15

3.22 3.66 4.90 4.11 2.62

γ±*

ρ

Vϕ

ref

0.61

0.46 1.23

13.8 5.75

45,46 47 48

1.31 2.48

32.3 15.41

1.37

9.33

0.082 0.013 0.007

1.71 0.372 0.730

0.86

8.20

0.1

0.033

5.89 8.87 8.91 7.87 6.43

49,50

46 47,45 51,52 51 45 53 49 49,50 50 50 54 54 54 54

Np number of data points.

The capabilities of the model have also been assessed by predicting water activities, MIAC, and solution densities as well as apparent molar volume (AMV) for several systems. It is worth mentioning that the AMV is very sensitive to the model accuracy and prediction of the mentioned thermodynamic property is proportional to calculated pure solvent density and solution density.42 Results have been presented in Table 5, and it can be seen that the results of the prediction are in good agreement with the experimental data, considering the number of adjusted parameters. In the case of alkyl ammonium salts, our calculations show that the obtained results by PC-SAFT EoS are comparable with the results obtained using the electrolyte NTRL model, which have been reported by Belvèze et al.26 It should be noted that, in the work of Belvèze et al., the parameters were adjusted for each specific IL, whereas, in this work, ion parameters have been obtained through simultaneous fitting of several ILs. In Figures 4−7, calculated osmotic coefficients and solution densities are depicted over a range of concentrations and temperatures; the higher deviations depicted in Figure 5 for the [C4mim][Cl] system at molalities lower than 1 mol/kg might be related to the relevant experimental uncertainties. In the case of the imidazolium based ILs, the maximum error of osmotic coefficient calculations is about 3.5% (ARD). On the other hand, the aforementioned error is about 5% for alkyl ammonium halides. It is noticeable that the correlation of experimental osmotic coefficient data has been applied over a wide range of molalities of up to 19 m (see Figure 6). In Figure 7, calculated densities of [C7mim][Br] and [C8mim][Br] have been compared with experimental data. It can be shown that the presented model gives reasonable results with the ARDs being less than about 0.015%. As an example, a comparison between predicted and experimental liquid density of [C4mim][Br] at different temperature has been depicted in Figure 8. It can be seen that the model can well-predict the experimental data over a wide range of molality and temperature. It must be pointed out, SAFT based EoS exhibit numerical pitfalls that

Figure 4. Comparison of the correlated osmotic coefficient with experimental data for aqueous IL solutions. Experimental data: ◊, [C2mim][Br] at 298.15 K; Δ, [C3mim][Br] at 328.15 K; ○, [C4mim][Br] at 318.15 K; ×, [C6mim][Br] at 298.15 K; □, [C5mim][Br] at 298.15 K. Solid lines are correlation using the proposed model.

lead to unreasonable results of thermodynamic properties.43 In cubic EoS, only three molar−volume roots can be found whereas with the PC-SAFT EoS four or five roots may exist when solving the EoS at a specified temperature and pressure for pure components. Therefore, to get safe results, an appropriate algorithm to solve the mentioned problem must be applied.44 Such pitfalls can appear at ambient temperature for ionic liquids. However, in this work mentioned pitfalls for strong electrolyte as well as aqueous ILs have not been 10279



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Figure 5. Comparison of the correlated osmotic coefficient with experimental data for aqueous IL solutions. Experimental data: ◊, [C4mim][Cl] at 313.15 K; Δ, [C4mim][BF4] at 328.15 K; ○, [C5mim][Cl] at 328.15 K. Solid lines are correlation using the proposed model.

Figure 7. Comparison of the correlated solution densities with experimental data for aqueous IL solutions: (a) [C7mim][Br] and (b) [C8mim][Br]. Solid lines are correlation using the proposed model.

Figure 6. Comparison of the correlated osmotic coefficient with experimental data for aqueous IL solutions. Experimental data: ◊, [(CH3)4N][Cl]; ○, [(CH3)4N][Br] at 298.15 K. Solid lines are correlation using the proposed model.

observed. Therefore, some erroneous results may return to the aforementioned pitfalls.

6. CONCLUSIONS In this study, the electrolyte PC-SAFT has been applied to describe thermodynamic properties of aqueous IL solutions. In the first step, the modified EoS has been applied to strong electrolyte solutions to obtain the general ion-specific parameters. The optimized parameters have been obtained through simultaneous correlation of MIAC and solution densities. The anion parameters that were obtained in the first step were subsequently used for correlation of ILs osmotic coefficients. The predictive capability of the proposed model has been verified as acceptable by calculation of liquid densities, water activities, apparent molar volume, and MIACs of IL aqueous solutions.

Figure 8. Comparison of the predicted solution densities with experimental data for [C4mim][Br] aqueous solutions. Solid lines are prediction using the proposed model.

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REFERENCES

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*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Article

LIST OF SYMBOLS

Roman Symbols

a = Helmholtz free energy per number of molecules (J) a = activity A = Helmholtz free energy (J) D = dielectric constant e = electric charge g = radial distribution function kB = Boltzmann constant, 1.38065 × 10−23 J/K kij = binary interaction parameter m = molality (moles of solute i per kg of solvent, mol/kg) M = number of association sites on the molecule P = pressure T = temperature (K) Vϕ = apparent molar volume (cm3/mol) XA = mole fraction of molecules not bonded at specific interaction site A xi = mole fraction of component i z = charge of ion Z = compressibility factor Abbreviations

ARD = absolute average relative deviation EoS = equation of state MIAC = mean ionic activity coefficient NRTL = nonrandom-two-liquid model OF = objective function PC-SAFT = perturbed chain statistical associating fluid theory SAFT = statistical associating fluid theory SAFT-VR = statistical associating fluid theory variable range Superscripts

assoc = association term calcd = calculated disp = dispersion exptl = experimental hc = hard chain term res = residual Subscripts

i, j, k = component index Greek Symbols

γ = activity coefficient ΔAB = strength of interaction between sites A and B (Å3) ε = dispersion energy parameter (J) εAB = energy parameter of the association between sites A and B (J) η = reduced density κAB = volume of interaction between sites A and B Γ = inverse shielding length μ = chemical potential ξ = packing fraction ρ = number density (number of molecules in unit volume), Å−3 σ = temperature-independent segment diameter (Å) φ = fugacity coefficient Φ = osmotic coefficient 10281



Article

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Thermodynamic Modeling of Aqueous Ionic Liquid Solutions Using PC

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