12968
J. Phys. Chem. B 2008, 112, 12968–12972
Static Dielectric Constant of Room Temperature Ionic Liquids: Internal Pressure and Cohesive Energy Density Approach Tejwant Singh and Arvind Kumar* Central Salt and Marine Chemicals Research Institute, (CSIR), BhaVnagar-364002, India ReceiVed: July 07, 2008; ReVised Manuscript ReceiVed: August 05, 2008
Measurements of the static dielectric constant (ε) of ionic liquids (ILs) are very difficult because of the decay of field by the ionic conductivity of ILs. Herein, we describe an easy method for the prediction of ε of various imidazolium-based ILs [Cnmim] from n, i.e. the ratio of internal pressure (Pi) and cohesive energy density (ced). A calibration curve of n vs ε for conventional organic solvents (mainly the linear alcohols) has been used to estimate the ε of the ILs. Estimated ε values for ILs having the anions [Cl]-, [BF4]-, [PF6]-, [TfO]-, and [Tf2N]- showed a very good comparison with the literature results, whereas ILs having the anions [CnOSO3]- tend to deviate from such correlation. Also, for a series of ILs having a common anion, the ε is shown to follow a very good correlation with the molecular volumes. Predicted values show that both the nature of the anion and alkyl chain length of the cation contribute significantly to the ε of the ILs. The method developed makes use of properties which can be either experimentally determined or estimated with good accuracy and can be extended to the other categories of ILs with ease and reasonable accuracy. 1. Introduction The polarity of ionic liquids (ILs) is a concern in their use in many industrial and other applications. Like any other solvent, the solvation behavior of ILs is governed by their polarity. Therefore, knowledge of polarity is of immense importance for designing task-specific ILs. Polarity of a solvent, in a broader sense, includes the contributions from its electron donor/acceptor capability, hydrogen bonding, and electrostatic interactions. The most common polarity scale relies on the static dielectric constant (ε), which reflects the orientational polarization caused by the molecular dipole moment and the molecular polarizability arising from electrostatic interactions. Despite being an extremely important physical property, the direct determination or estimation of the dielectric constant of ILs has not received much momentum either because of the limitations of the methods or erroneous estimations. As a consequence, the dielectric constant of many of the ILs is not known to date. Conventional methods fail to measure the static dielectric constant of ILs because the samples are short-circuited due to their intrinsic electrical conductance, and the solvatochromic shifts or other chemical probes used in the past to estimate the solvent polarity of ILs reflects the solvent behavior of ILs, which is quite different from the static dielectric constant of ILs.1-6 The empirical scale of polarity has been introduced by using the unusually large negative solvatochromism of a standard betaine dye, referred to as ET(30) scale. The ET(30) value for ILs gave very high dielectric constants when compared to the values measured directly from dielectric spectroscopy. Recently the dielectric constant ε for ILs has been measured by the research groups of Weinga¨rtner,7-9 and Buchner and Hefter10,11 using dielectric spectroscopy with comparatively high accuracy. Although the dielectric spectroscopy give ε values with quite good accuracy, fitting of model equations using certain assumptions and zero-frequency extrapolation of the dispersion curve leads to some experimental errors. Another * Corresponding author. Tel.: +91-278-2567039. Fax: +91-278-256 7562/256 6970. E-mail:
[email protected];
[email protected].
problem encountered in the direct determination of the dielectric constant of ILs is the requirement of sophisticated signal processing techniques to cover a broad range of frequencies.12,13 Looking at the difficulties in direct measurements of the dielectric constant, it is imperative to develop reliable predictive methods. Since a number of ILs with different physical properties can be synthesized by changing the nature of the cation or anion and the number of possible modifications is huge, it is necessary to have some easy predictive models for the estimation of the dielectric constant in order to design new materials without consuming much time in cumbersome measurements. Some attempts have already been made by the research groups of Krossing14 and Ludwig15 for the prediction of ε using a combination of volume-based thermodynamics (VBT) and quantum chemical calculations, and FTIR spectroscopy, respectively. Both methods predict ε values with quite good accuracy but have certain serious limitations. In the former method, knowledge of quantum chemical calculations is a prerequisite and the known dielectric constant of ILs is the input parameter for the calculation of free energies of solvation (∆solvG). The presence of long-lived ion-pairs in ILs also weakens the approximation of ∆solvG required to calculate the dielectric constant, and it is beyond imagination to judge the ion-pairing in many of the ILs because of the strong ion association. The latter method, developed by the research group of Krossing, which is based on the association of water molecules in the ILs and makes use of shifts in the stretching frequencies of the water molecules, is limited to ILs that do not undergo hydrolysis upon contact with water. This method is also limited by the assumption that ILs are homogeneous fluids. However, many ILs are shown to be nanostructured materials and are segregated into polar and nonpolar regions once water is added.16-18 Herein we describe a very simple predictive model for the estimation of ε of ILs, which is based on internal pressure (Pi) and a cohesive energy density (ced) approach. The input parameters required are the accurately determined bulk physical properties such as speed of sound (u), density (F), heat capacity
10.1021/jp8059618 CCC: $40.75 2008 American Chemical Society Published on Web 09/24/2008
Static Dielectric Constant of Ionic Liquids
J. Phys. Chem. B, Vol. 112, No. 41, 2008 12969
(Cp), heats of vaporization (∆vapH), and molar volume (V) of the ILs. The method developed by us overcomes the limitations of the previously existing models and predicts ε values of ILs quickly. Since experimentally determined ε values are available for many imidazolium-based ILs, we have chosen this class of salt for testing our hypothesis. The effect of change of alkyl chain length of the imidazolium cation and different anions on the ε values has been observed and compared. The standard error of the estimate (sest) statistics is 1.5. Estimations of ε have been also made for a few pyridinium- or pyrrolidinium-based ionic liquids, but a comparison with the literature could not be made because of the lack of experimental data. 2. Experimental Section 2.1. Materials. Ionic liquids (ILs): 1-butyl-3-methylimidazolium tetrafluoroborate [C4mim][BF4] (>99 mol%), 1-methyl3-octylimidazolium tetrafluoroborate [C8mim][BF4] (>99 mol%), 1-butyl-3-methylimidazolium hexafluorophosphate [C4mim][PF6] (>99.0 mol%), 1-butyl-3-methylimidazolium chloride [C4mim][Cl] (>98 mol%), 1-methyl-3-octylimidazolium chloride [C8mim][Cl] (>98.0 mol%), 1-butyl-3-methylimidazolium methylsulfate [C4mim][C1OSO3] (98 mol%), and 1-butyl-3-methylimidazolium n-octylsulfate [C4mim][C8OSO3] (>98.0 mol%) were obtained from Solvent Innovation, Germany. All the ILs were purified and dried with the procedures similar to that described in our earlier paper.19 Karl Fisher analysis of the samples indicated the water content to be less than 200 ppm in each solvent. All the solvents were stored in a dry atmosphere. 2.2. Measurements. Density and speed of sound measurements for various ILs at 25 °C were carried out using an Anton Paar (Model DSA 5000) vibrating tube density meter with a resolution of 5 × 10-6 g · cm-3 and 0.01 m · s-1 for the density and speed of sound, respectively. All measurements were performed in at least triplicate in an inert atmosphere. 3. Methodology The concepts of internal pressure (Pi) and cohesive energy density (ced) are used as structural probes.20-22 Pi is a measure of the change in internal energy of 1 mol of solvent as it undergoes a very small isothermal expansion. The most important contributions to the internal pressure come from those interactions which vary most rapidly near the equilibrium molecular separation in the solvent. Dispersion, repulsion, and dipole-dipole interactions all vary rapidly with intermolecular separation, so Pi mainly reflects these interactions. The quantity Pi can be expressed as a function (∂U/∂V)T, where U is the change in internal energy and V is the molar volume of the liquid and can be obtained using the thermodynamic equation of state:
Pi )
∂P ) T( ) - P ( ∂U ∂V ) ∂T T
V
(1)
The thermal pressure coefficient, (∂P/∂T)V can be equated to the ratio of isothermal expansion and the isothermal compressibility (R/κT), which in turn can be calculated from the experimental density (F), speed of sound (u), and heat capacity measurements with very high accuracies. On the other hand, cohesive energy density (ced) is the measure of total molecular cohesion per unit volume contrary to internal pressure. Such a differential change in volume does not necessarily disrupt all the molecular interactions to an equal extent. The interactions most affected include repulsion, dispersion, and weak dipole processes. In other words, a small volume expansion has a
disruptive effect on these interactions, which may be equivalent to that of vaporization.23 Therefore, the ced can be readily obtained from experimentally determined heats of vaporization, ∆vapH using the relationship:
ced ) (∆vapH - RT)/V
(2)
The two quantities, i.e. Pi and ced, can be related using the typical form of van der Waals equation of state in the following manner:
P+
a RT ) V2 V - b
(3)
where a and b are called the attraction and repulsion parameters and other symbols have their usual meanings. The differentiation of eq 3 with respect to T and constant V gives
R ) [ ∂P ∂T ] V-b V
(4)
When we substitute the result of eq 4 into eq 3 and compare with the thermodynamic equation of state (eq 1), we get the following expression:
a ) [ ∂U ∂V ] V
2
T
(5)
Over a small range of volumes, U could be represented in the functional form:
U)-
a Vn
(6)
so that by differentiating eq 6 with respect to volume at constant temperature can be expressed as24
na -nU ) ) [ ∂U ∂V ] V V T
n+1
(7)
Further, Scott states that for most practical purposes, the cohesive energy per mole U may be replaced by (-∆U) so that eq 7 can be written as24
(∂U/ ∂ V)T ) n∆U/V ) n(∆vapH - RT)/V
(8)
The quantity, n, has been correlated to the static dielectric constant ε of the liquids. For the estimation of ε of ILs, we plotted a calibration curve of n vs ε of the linear alcohols. The parameter n for the various ILs was obtained from eq 8. The quantities R, κT required for the determination of Pi were calculated either from our experimentally determined speed of sound and density data or using the literature values. The ced was determined from the heats of vaporization directly measured25 or estimated using the surface tension values reported in the literature from the empirical relation put forward by Zaitsau et al.25
∆vapH(298 K) ) A(γV2⁄3N1⁄3) + B
(9)
where N is Avogadro’s constant, and A and B are empirical parameters, their values being A ) 0.01121 and B ) 2.4 kJ mol-1. Although eq 9 has not been very well tested, it predicts the ∆vapH, which is highly comparable to the experimentally determined values25 and has been successfully used for the estimations of ∆vapH in some recent publications.26,27 4. Results and Discussion Ion identities of the ILs investigated in this study are given in Table 1. The polarity parameter n and ε of the normal alcohols have been used to draw the calibration curve for the estimation
12970 J. Phys. Chem. B, Vol. 112, No. 41, 2008
Singh and Kumar
TABLE 1: Definitions of Acronyms Used in the Present Study for Various ILs Cations [Cnmim] [Cnmpyr] [C4nmpy][BF4]
1-alkyl-3-methylimidazolium, n denotes the alkyl chain length 1-alkyl-1-methylpyrrolidinium, n denotes the alkyl chain length 1-butyl-n-methylpyridinium, n denotes the position of methyl group
Anions BF4 PF6 Tf2N TfO Cl CnOSO3
tetrafluoroborate hexafluorophosphate bis(trifluoromethylsulfonyl)imide trifluoromethysulfonate chloride n-alkylsulfate, n denotes the alkyl chain length
TABLE 2: Cohesive Energy Density (ced), Internal Pressure (Pi), Polarity Parameter (n), and Dielectric Constant (ε) of Alcohols at 25 °C Used in the Calibration Curve solvent
ced,a atm
Pi,a atm
n
ε
1-propanol 1-butanol 2-butanol 1-pentanol 2-pentanol 1-hexanol 1-heptanol 1-octanol 1-nonanol
5915 5370 5045 4950 4555 4617 4476 4215 4195
2879 2961 3065 2972 3072 3153 3233 3226 3322
0.487 0.551 0.608 0.600 0.675 0.683 0.722 0.766 0.792
20.1 17.5 15.8 15.5 13.8 13.3 11.6 10.3 8.7
a
Values of ced and Pi are calculated from ref 28.
Figure 1. n vs ε for the linear alcohols (b) at 25 °C. Predicted ε values of the [Cnmim] ILs: (0) [BF4]-, (O) [PF6]-, (2) [Tf2N]-, (4) [Cl]-, and ()) [TfO]-. (n and ε values for various ILs are given in Table 2).
of ε of the ILs. The ced, Pi, n, and ε for the normal alcohols are reported in Table 2. Figure 1 shows the calibrated plots of n vs ε of the alcohols along with the ILs. A comparison of estimated ε values with the literature results for various ILs is given in Table 3. ILs having the anions [Cl]-, [BF4]-, [PF6]-, [TfO]-, and [Tf2N]- with n values ranging from 0.5 to 0.9 show a good comparison of ε with the available literature data. The standard error of the estimate (sest) statistics, which give an estimate of the average absolute error of the prediction, is 1.5. As the ε data covers a large range of values, the average of the percentage errors for the predictions (
