J. Phys. Chem. B 2007, 111, 13047-13051
13047
Solubility of Small Molecule in Ionic Liquids: A Model Study on the Ionic Size Effect Ping Lou,†,‡ Sunwoo Kang, Kyung Cheol Ko, and Jin Yong Lee*,† Department of Chemistry, Sungkyunkwan UniVersity, Suwon, 440-746, Korea, and Department of Physics, Anhui UniVersity, Hefei 230039, Anhui, People’s Republic of China ReceiVed: July 21, 2007; In Final Form: September 8, 2007
Recently, the solvent power of ionic liquid (IL) has been described based on Flory-Huggins (FH) theory assuming that the volumes of the components are the same (J. Phys. Chem. B, 2006, 110, 16205). Here, we extended the FH theory to derive the solvent power in the case of different sizes (molar volumes) of the IL’s components based on “polymer-like” model. Applying this model, the effect of ionic size on the solvent power of ionic liquids has been investigated. It was found that the effect of size can be characterized by introducing the effective volume (V h + and V h -) of each site of the ion, and for the equivalent ionic liquid, the larger effective volume of the ionic liquid has the larger solvent power. Our results are in excellent agreement with the experimental solubility data in various ionic liquids.
I. Introduction The term “ionic liquid” (IL) is used for a novel class of green benign solvents, which promise to have widespread application in industry, possibly replacing currently used organic solvents, due to unique properties such as negligible vapor pressures, broad liquid temperature ranges, and high specific solvent abilities. In particular, after the IL EMIBF4 (1-ethyl-3-methylimidazolium cation + BF4 anion) was synthesized in 1992,1 many new air- and water-stable ionic liquids (ILs) were reported and have recently received increased attention from both the industrial and academic communities.2-6 ILs are generally nonvolatile at room temperature, and some of them can be evaporated at very rigorous conditions (high temperature and low pressure).7 Despite such properties, ILs have been served as good solvents for a wide range of organic and inorganic compounds. A number of cation and anion pairs can form ILs; thus one can choose a proper cation and anion pair to find an IL that has desirable properties. However, accurate experimental values for many of the fundamental physical-chemical properties of this class of ILs are either scarce or even absent.8-9 Instead, a number of molecular dynamics simulation studies have been applied to predict the structural and dynamical features.10-16 Flory theory of mixtures has been useful in interpreting the chemical properties of binary systems composed of molecular species with different size and shape. Although hydrogen bonds and strong electrostatic interactions are excluded in the original Flory theory and the ILs may have strong ionic characters, the Flory theory has proven to be successful in predicting the properties and fluid-phase behavior of IL mixtures.17-18 Very recently, a new theoretical model for ILs9 has been put forward in which they used a three-component model consisting of anions, cations, and neutral solute molecules and successfully explained why ILs possess extra solvent power. They used the Flory-Huggins (FH) theory to express the free energy of the system, and they modeled the system by three components. * Corresponding author. Ph: +82-31-299-4560. Fax: +82-31-290-7075. E-mail:
[email protected]. † Sungkyunkwan University. ‡ Anhui University.
Figure 1. Polymer-like lattice model of ionic liquid.
However, for the sake of simplicity they assumed that the particles of all of the species have the same size (excluding volume). Therefore, it is our motivation to investigate the size effect of the constituents on the solvent power of ionic liquids. In this paper, we extend the FH theory based on the polymerlike lattice model to describe the solvent power of ILs. II. Theory and Equation We follow the “polymer-like lattice” approach as treated in previous papers.17-19 As seen in Figure 1, we assumed that the solution of ILs is composed of solute molecules, polymer-like cations, and polymer-like anions. Each solute molecule occupies one lattice site, while each polymer cation and anion occupy r+ and r- lattice sites, respectively. It was also assumed that
10.1021/jp0757311 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/20/2007
13048 J. Phys. Chem. B, Vol. 111, No. 45, 2007
Lou et al.
the volumes of each of the lattice sites are equal to υ for solute, cation, and anion. In other words, each lattice site is considered to be occupied by one molecule of solute or by one monomer of the polymer cation or anion where the volumes of solute molecule, monomer cation, and monomer anion are equal to υ. If the solution contains Ns solute molecules, N+ solvent polymer cations, and N- solvent polymer anions, the total number of lattice sites is given by N ) Ns+r+N++ r-N-. Then, the entropy before mixing is the sum of Ss ) kBNs ln(Nsυ), S+ ) kBN+ ln(r+N+υ) and S- ) k-N- ln(r-N-υ). Defining V h s, V h +, V h - as molecular volumes of solute, polymer cation, and polymer anion, respectively, we have the following relations: V h s ) υ, r+ ) V h +/υ, and r- ) V h -/υ. In this model, the size effect of cations and anions can be described by r+ and r-. The entropy after mixing is Smix ) kBNmix ln(Nsυ + r+N+υ + r-N-υ), and here Nmix ) N+ + N- + Ns. Thus, the mixing entropy can be written as
φs ) Vs/Vtot, where V+, V-, Vs, and Vtot are volumes of cation, anion, solute, and total, respectively. Then, the volume fraction of the solute can be written as φs ) 1 - φ+ - φ- (the mixture is assumed to be incompressible). They have the following relationships:
∆Smix/NmixkB ) (Smix - Ss - S+ - S-)/NmixkB
1 ∆G h 1 ) φ ln φ+ + φ- ln φ- + (1 - φ+ - φ-)ln(1 kBT r+ + r1 1 1 φ+ - φ-) + χ+,- φ+φ- + χ-,+ φ-φ+ + χ+ φ+(1 2r+ 2rr+ 1 φ+ - φ-) + χ- φ-(1 - φ+ - φ-) (7) r-
( ( ) ( ))
(1)
( )
NN+ r+N+ r-Nln + ln + Nmix N Nmix N Ns Ns ln Nmix N
)-
) -(n+ ln φ+ + n- ln φ- + ns ln φs) On the other hand, for the enthalpy of mixing,9,17-19 we assumed that cations and anions are not coupled into pairs and move independently as treated in a previous paper.9 Then, there are three different interactions in the system, cation-anion, cationsolute, and anion-solute interactions, as shown in Figure 1. The enthalpy of mixing using these interactions can be expressed as eq 2
1 1 ∆Hmix/NmixkB ) T χ+ -n+φ- + χ+ -n-φ+ + 2 2
(
)
1 ∆G ) n+ ln φ+ + n- ln φ- + ns ln φs + χ+,-n+φ- + kBT 2 1 χ n φ + χ+n+φs + χ-n-φs (3) 2 -,+ - + The subscripts “+”, “-”, and “s” refer to cation, anion, and solute, respectively. n+/n-/ns represent the number of moles of cation/anion/solute, and φ+/φ-/φs represent the volume fractions of cation/anion/solute, that is, φ+ ) V+/Vtot, φ- ) V-/Vtot, and
r+n+ r+n+ + r-n- + ns
(4)
φ- )
r-nr+n+ + r-n- + ns
(5)
φs )
ns r+n+ + r-n- + ns
(6)
By using eqs 4, 5, and 6, eq 3 can be expressed as
where ∆G h ) ∆G/(r+n+ + r-n- + ns). It is noted that eq 7 is equal to eq 1 of ref 9 when V h+ ) V h- ) V h s, that is, r+ ) r- ) 1, which is the case of Aerov’s assumption. This is the result of nature. For the polymer-cation and polymer-anion that occupy only one lattice site, that is, r+ ) r- ) 1, volume fraction (V h +/V h -) is equal to mole (or number) fractions (N+/N-), and the above equation recovers to the usual equation obtained from the ideal mixing theory. A condition of macroscopic charge neutrality imposes restriction on the values of the number of moles of the cation and anion, that is, n+ ) n- ) m/2. Here m is the number of moles of the ionized IL. Then, the φ+ and φare expressed as follows:
χ+n-φs + χ-n-φs (2) Here, we used the mean-field treatment for the system. In eq 2, the first two terms of eq 2 are originated from the cation-anion interactions. The cation-anion or anion-cation interaction Flory-Huggins parameter is χ+ - and the probabilities of the cation-anion and anion-cation interaction are φ- and φ+, respectively. Then, the total number of such interactions is 1/2(N+φ- + N-φ+) (here the fraction 1/2 is used to avoid the double counting for the cation-anion and anion-cation interactions). Therefore, the enthalpy of mixing produced by the cation-anion, cation-solute, and anion-solute interactions is TkB(1/2 χ+ -N+φ- + 1/2 χ+ -N-φ+) ) TNmixkB(1/2 χ+ -n+φ+ 1/2 χ+ -n-φ+). The last two terms of eq 2 result from cationsolute and anion-solute interactions. Thus, by using the thermodynamic equation Nmix∆G ) ∆Hmix - T∆Smix, the free energy change of the mixture of the system can be written as
φ+ )
m r+m r+ 2 ) ) φ φ+ ) m 2(rm + ns) 2r (r+ + r-) + ns 2 r+
φ- )
rφ, 2r
φ+ + φ- ) φ
where
r ≡ (r+ + r-)/2 and φ ≡
mr mr + ns
The physical meaning of the parameter φ is the volume fraction of the ionic species (sum of cation and anion). Then, the free energy of the mixture takes the following form:
( ( ) ( ))
r+ r1 ∆G h ) φ ln φ + ln φ kBT 2r 2r 2r
+ (1 - φ)ln(1 - φ) + φ(1 - φ) φ2 χ+,- + χ (8) 4r r
where χ ≡ (χ+ + χ-)/2, which is the Flory-Huggins parameter that describes interactions of the ionic species with the neutral solute. r ) (r+ + r-)/2 (r ) (V h+ + V h -)/2V h s) is the parameter
Ionic Size Effect on Solubility of Molecules in Ionic Liquids
J. Phys. Chem. B, Vol. 111, No. 45, 2007 13049 each curve, below the curve a homogeneous mixture of the ionic and nonionic liquid is stable (solubility region). Above the curve, the macroscopic phase separation of the ionic and nonionic liquids occurs. Therefore, it is very interesting to find that the χ value corresponding to the phase separation increases as the r value increases at the given φ. This implies that for the ionic liquids with equivalent cation or anion, the solvent power of h -) of the ionic liquids with the larger effective volume (V h + or V the counterions is larger. This can be seen by the following simple calculation:
∆χ ) χ(r,φ,χ+ -)|r)r - χ(r,φ,χ+ -)|r)1 ) Figure 2. Interaction model of an ionic liquid.
Figure 3. Spinodal of the macroscopic phase separation of a mixture of the ionic and nonionic liquids.
that describes the effective size including the structural information of ionic species. The interaction model of an IL is shown in Figure 2. It may be argued that eq 8 seems to violate the virial expansion as φ+ f 0 or φ- f 0 due to the Coulomb interaction, χ+ -φ2, between the cations and anions. However, it should be noted that the system studied here is the three component mixture, which is different from the general three component mixture. In the mixture of small solute molecule and ionic liquid, the finite dilute solution means φs f 0 and φ+ + φ- f 1, and the reversed cases (φs f 1 and φ+ = φ- f 0) are not our concern because the ionic liquid is working as solvent and the cations and anions are abundant in the solution. Therefore, the inconsistency of eq 8 with the virial expansion when φ+ = φf 0, may not be a critical problem. As a matter of fact, this equation has been used to study the IL systems similar to our systems.9,20,21 III. Results and Discussion From eq 8, the spinodal toward macrophase separation can be found by a standard procedure, ∂2∆G h /∂φ2 ) 0, which leads to
1 + (r - 1)φ χ+,χ(r,φ,χ+ -) ) + 4 2φ(1 - φ)
(9)
It is noted that when r ) 1, that is, r+ ) r- ) 1, eq 9 is just equal to eq 5 of ref 9. The dependence of χ on φ is represented in Figure 3 for the value of the parameter χ+,- equal to 12 as taken in ref 9 and different r values, r ) 10.0, 5.0, and 1.0. For
(10)
(r - 1) 2(1 - φ)
This equation implies that the solvent power increases as r increases. In the present model, r > 1 always because both r+ and r- are greater than 1. It is unable to anticipate the size effect distinctly because there are many sets of r+ and r- to give the same r value. For example, both r+ ) r- ) 10.0 and r+ ) 5.0 and r- )15.0 sets give the same r value of 10.0. To testify the reliability of our theoretical result on the solvent power of ILs with different size, we collected previously reported experimental solubility data for various ILs with different cations and anions. Especially for ILs studied in the previous experiments, the cationic species are limited, while the anionic species are abundant. The solubilities of difluoromethane in various ionic liquids were recently measured by Shiflett et al.22 At room temperature and atmospheric pressure (298.15 K and ∼0.1 MPa), the solubility (100x) was measured to be 4.1, 6.8, 7.2, 9.2, 9.6, and 10.2 for ILs composed of 1-butyl-3-methylimidazolium (bmim) and SCN, MeSO4, TFES, FS, TTES, and TPES, respectively. Here, the following abbreviations have been used: thiocyanate (SCN), methyl sulfate (MeSO4), 1,1,2,2-tetrafluoroethanesulfonate (TFES), 2-(1,2,2,2tetrafluoro-ethoxy)1,1,2,2,-tetrafluoroethanesulfonate (FS), 1,1,2trifluoro-2-(trifluoro-methoxy)ethanesulfonate (TTES), and 1,1,2trifluoro-2-(perfluoroethoxy)ethanesulfonate(TPES).Unfortunately, their ionic sizes were not known in the previous studies. Thus, we have carried out the ab initio calculations to obtain the volumes of the anions at the level of B3LYP/6-31G* theory using the Gaussian 03 programs.23 The calculated volumes of SCN, MeSO4, TFES, FS, TTES, and TPES are 448.4, 880.4, 854.5, 1148.0, 1185.3, and 1442.7 bohr3, respectively. From the straight line in Figure 4, it is obvious that the experimental solubility has a good linear relationship with the anion’s volume. This is an excellent agreement with our theoretical results. To give more clearly an example that can be directly applied to the experimental study, we finished the following calculation based on the model of gas solubility in liquids. In general, the equilibrium condition for gas (solute) and ionic liquid can be expressed as24
ysPΦs ) nsγsPss
(11)
where ys is the mole fraction of solute in vapor phase, ns is the mole fraction of solute in liquid phase, P is the pressure, Pss is the saturated vapor pressure of solute, γs is activity coefficient of solute in the IL, and Φs is a correction factor for solute. It is reasonable to assume the ionic liquid is nonvolatile and the ideal vapor phase Φs ) 1, ys ) 1 (yIL ) 0, and PsIL ) 0) for
13050 J. Phys. Chem. B, Vol. 111, No. 45, 2007
Lou et al. solvent power of IL. Let R ) 4χ - 4 - χ+ -, then as long as |R/4r| < 1, eq 16 can be changed into
1 r R R ) 1+ +O ∞ 2.71828 4r 4r γs
[
( )]
(17)
Substituting eq 17 into eq 15 and omitting O(R/4r), we have
ns = )
Figure 4. Solubility of difluoromethane in ionic liquids composed of 1-butyl-3-methylimidazolium (bmim) and various anions (SCN-, CH3SO4-, TFES-, FS-, TTES-, and TPES-) as a function of anion’s volume. See text for the abbreviations.
sufficiently low pressures, and at temperature of our interest then we have
P ) nsγsPss
(12)
On the other hand, the Henry’s law constant is defined as
H ) lim
P
nsf0ns
(13)
It is noted that for a fixed pressure P that the larger H is, the smaller ns is. Thus, the Henry’s law constant can be used to characterize the solvent power of IL. In general, the Henry’s law constant depends on the temperature but is relatively insensitive to the pressure, especially over the pressure ranges examined in our interest. Given the assumptions used for eq 12, we also have
H ) lim γsPss nsf0
(14)
) γ∞s Pss where γ∞s is the activity coefficient of solute in the infinite dilute solutions. It is noted that for the fixed Pss, the larger that H is, the larger γ∞s is, which implies that γ∞s also can be used to characterize the solvent power of IL. Comparing eq 13 and 14, we have
ns )
P/Pss γ∞s
(15)
It is shown that for the fixed P/Pss, the larger γ∞s is, the smaller the solubility of solute ns is. Using the definitions of activity as ) ∂∆G/∂ns and activity coefficient of solute γs ) as/ns, the following equation can be easily derived from eq 8
P P R r+ s s 4 2.71828Ps 2.71828Ps
(18)
R 1 P P h -) + (V h +V s 4 s 2V + 2.71828Ps 2.71828Ps
It is clearly shown that ns ∝ V h+ + V h - for the fixed P/Pss, which is consistent with Figure. 4. It should be remarked that the theoretical model can describe the experimental results quite well because it captures key factors of experimental interests. However, the quantitative prediction of any experimental observables compared with experimental measurement should be very interesting and will be studied in the near future. It is well known that the chemical behavior and stability of the ionic liquid have been drastically affected by changing the anions. The change of cation has also a profound effect on the physical properties of ILs such as melting point, viscosity, and density. For example, for the hexafluorophosphate salts,25 their melting point showed a pronounced dependence on the chain length in the cation. A recent molecular dynamics simulation26 has shown the Henry’s constants for five gases in roomtemperature ionic liquids, which are in agreement with our results. IV. Conclusion On the basis of the extended Flory-Huggins theory, we have investigated the effect of size on the solvent power of ionic liquids. It is found that for the ionic liquids composed of various cations and anions, the ionic liquid having the larger effective volume (V h + and V h -) of the ionic components has the larger solvent power. To test the reliability of our theory, we have carried out ab initio calculations to obtain the volume for various anions that were used in the previous experimental solubility study for various ionic liquids. The experimental solubility was fitted in a good straight line as a function of the calculated volume of the anions. Our result is in excellent agreement with the previously reported experimental solubility data for various ionic liquids composed of various anions and a fixed cation component. Further study on thermodynamic properties of ionic liquids is currently being investigated in our group based on our theory and molecular dynamics simulation. Acknowledgment. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-311-C00082) and BK21. This work was also supported by the Korea Research Foundation and the Korean Federation of Science and Technology Societies Grant funded by Korean Government (MOEHRD, Basic Research Promotion Fund). References and Notes
χ+ - χ 1 + ln γ∞s ) 1 - - ln r r 4r r
(16)
It is noted that the eq 16 provides important information that χ+ - enhances the solvent power of IL, while χ reduces the
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