Journal of Electroanalytical Chemistry 660 (2011) 230–233
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Estimation of the rate constant for electron transfer in room temperature ionic liquids W. Ronald Fawcett ⇑, Attila Gaál, Daniel Misicak Department of Chemistry, University of California, Davis, CA 95616, USA
a r t i c l e
i n f o
Article history: Received 9 July 2010 Received in revised form 23 March 2011 Accepted 23 March 2011 Available online 30 March 2011 Dedicated to the memory of Alexander (Sasha) Kuznetsov with recognition of his signiﬁcant contributions to the theory of electron transfer in condensed media
a b s t r a c t The kinetic parameters for simple electron transfer are estimated in seven room temperature ionic liquids containing a 1-alkyl-3-methyl-imidazolium cation and an inorganic anion, namely, tetraﬂuoroborate or hexaﬂuorophosphate. The electron transfer system considered is the oxidation of ferrocene. The calculations make use of dielectric relaxation data obtained recently in these ionic liquids. The parameters estimated include the outer sphere Gibbs activation energy for electron transfer, the longitudinal relaxation time, and the standard rate constant. It is shown that the outer sphere Gibbs activation energy is virtually independent of the length of the alkyl chain in the cation. On the other hand, the longitudinal relaxation time increases with increase in the length of the alkyl chain. As a result the standard rate constant for oxidation of ferrocene decreases as the cation in the ionic liquid is made longer. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Ionic liquids Electron transfer kinetics Kinetic parameters
1. Introduction Room temperature ionic liquids (RTILs) have attracted considerable attention as media for electrochemical experiments [1,2]. Many studies have been made in which the thermodynamic and transport properties of simple redox couples have been determined in a variety of RTILs. A popular RTIL involves an imidazolium cation and an inorganic anion such as tetraﬂuoroborate (BF 4 ) or hexaﬂuorophosphate (PF 6 Þ. The physical properties of the RTIL may be varied by varying the length of the alkyl chain in the cation 1-alkyl-3-methyl-imidazolium (RMeIm+). As the length of the alkyl chain increases, the viscosity of the solvent increases and the diffusion of reactants slows down. In more traditional systems in which electron transfer is studied, one has a molecular solvent and a separate electrolyte. In an RTIL, the solvent and the electrolyte are the same. In the traditional system, the solvent’s dielectric properties are important in estimating the Gibbs energy of activation for the electron transfer process. On the other hand, the nature and concentration of the electrolyte can determine the charge on the reactant and thus the double layer effect for the electron transfer process. Obviously, in a RTIL both of the Gibbs energy of activation and the double layer effect are controlled by the properties of the RTIL.
⇑ Corresponding author. E-mail address: [email protected]
(W.R. Fawcett). 1572-6657/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2011.03.019
There are several dielectric properties of the RTIL that are needed to estimate the standard rate constant for electron transfer. These include the static relative permittivity of the RTIL es, the relative permittivity at optical frequencies, eop, and the longitudinal relaxation time of the RTIL, sL. In this regard, an extensive study of the dielectric properties of RTILs containing imidazolium cations in the frequency range from 1 MHz to 20 GHz has been carried out recently by Nakamura and Shikata . Other important data needed to estimate the rate constant for electron transfer in these systems were also summarized by these authors. In the present paper, the rate constant for the oxidation of ferrocene (Fc) to the ferrocinium cation (Fc+) is estimated in several RTILs containing imidazolium cations. The estimates then are compared with kinetic data reported some years ago for this redox couple in non-aqueous solvents  and with the data reported recently in a RTIL by Compton et al. .
2. Dielectric relaxation data Nakamura and Shikata  reported dielectric relaxation data for 11 imidazolium salts at 25 °C. Here attention is focused on seven of these, namely, RMeImBF4 and RMeImPF6, where R is ethyl, butyl, hexyl or octyl. They also gave the molar concentration of the RTIL, its density, molar volume, viscosity, and speciﬁc conductance at 25 °C. These data are summarized in Table 1.
W.R. Fawcett et al. / Journal of Electroanalytical Chemistry 660 (2011) 230–233 Table 1 Fundamental properties of some imidazolium salts. Concentration, M
Density, g cm3
Molar volume, cm3 mol1
Viscosity, Pa s
Conductance, X1 m1
Tetraﬂuoroborates EtMeIm+ 6.46 BuMeIm+ 5.32 HxMeIm+ 4.51 OcMeIm+ 3.91
1.280 1.203 1.146 1.104
154.72 187.98 221.74 255.55
0.0360 0.1000 0.1880 0.2910
1.5165 0.4360 0.1460 0.0752
Hexaﬂuorophosphates BuMeIm+ 4.81 HxMeIm+ 4.14 OcMeIm+ 3.63
1.368 1.294 1.236
207.75 241.42 257.32
0.2480 0.4460 0.6010
0.1650 0.0620 0.0308
After correcting for solution conductivity, the dielectric relaxation results were ﬁtted to a model with two Debye relaxations and one Cole–Cole relaxation. The corresponding equations for the inphase (e0 ) and out-of-phase (e00 ) components of the relative permittivity are
Table 3 Estimated kinetic parameters for the oxidation of ferrocene in some imidazolium salts.
DGos , kJ mol1
ksc, cm s1
Tetraﬂuoroborates EtMeIm+ 15.4 BuMeIm+ 15.2 HxMeIm+ 15.3 OcMeIm+ 15.2
33.7 60.6 112.5 159
4.9 8.7 14.4 21.0
0.48 0.28 0.16 0.12
Hexaﬂuorophosphates BuMeIm+ 15.7 HxMeIm+ 15.7 + OcMeIm 15.5
88.8 154 219
12.5 20.0 28.4
0.165 0.10 0.078
In deriving an effective relaxation time for the solvent all three relaxation modes should be considered. Nakamura and Shikata  have suggested a method of calculating a harmonic average relaxation time sav using an equation of the form.
e1 e2 e3 þ þ þ e1 1 þ x2 s21 1 þ x2 s22 1 þ x2 s23
xs 1 xs2 ðxs3 Þ2b þ þ 2 2 2 2 1 þ x s1 1 þ x s2 1 þ x2 s23
e1 1 e2 1 þ e1 þ e2 þ e3 s1 e1 þ e2 þ e3 s2 e3 1 þ e1 þ e2 þ e3 s3
Here ei is the relative permittivity, and si, the relaxation time in region i. b is the broadening factor in the Cole–Cole region, that is, region 3. The static value of the relative permittivity is obtained in the limit that x goes to zero, so that
es ¼ e1 þ e2 þ e3 þ e1
These data are summarized in Table 2. As can be seen from the data in Table 2, the relaxation time s1 is independent of cation size and anion nature. This is the fastest relaxation process and is attributed to the inter-ion motions of ion pairs. The second relaxation s2 depends more on the nature of the cation and anion. In order to explain this relaxation, Nakamura and Shikata  used a model of an elongated ellipse to represent the RTIL. As the number of carbon atoms in the alkyl group increases, the length of the long axis in the ellipse increases with respect to its width. The value of s2 is attributed to rotation of the ion pair about the longer axis of the ellipse which has a lower moment of inertia . The slowest relaxation is the s3 process. These values clearly depend on the nature of both the cation and the anion, and correlate strongly with the viscosity of the RTIL. This relaxation is attributed to rotation of the cation about the shorter axis of the ellipse corresponding to the higher moment of inertia.
Table 2 Dielectric properties of some imidazolium saltsa.
3.9 2.6 1.6 1.2
21 21 19 20
3.0 2.9 2.3 1.8
160 175 180 200
3.1 7.2 8.2
Hexaﬂuorophosphates BuMeIm+ 14.1 1.9 HxMeIm+ 15.5 1.2 OcMeIm+ 15.2 0.9
21 19 20
2.6 1.8 1.2
210 230 220
4.8 8.2 9.2
Tetraﬂuoroborates EtMeIm+ 13.6 BuMeIm+ 14.1 HxMeIm+ 15.9 OcMeIm+ 15.4
1.9 4.4 5.1
1.00 0.80 0.72
5.45 4.75 4.30
1.991 2.018 2.038 2.051
2.5 5.0 8.0
0.85 0.74 0.67
4.80 4.25 3.90
1.981 2.008 2.024
All of the quantities listed in this table except eop are deﬁned in Eqs. (1)–(3). eop is the square of the refractive index of the RTIL measured at the sodium D line.
Values of sav are considered here to be the relevant quantity for estimation of the longitudinal relaxation time and estimation of the rate constant for electron transfer. They are recorded in Table 3. According to the Stokes–Einstein–Debye (SED) equation [3,6], the effective relaxation time for a solvent is given by
3gV m RT
This suggests that if sav is the effective relaxation time of the RTIL as a solvent, it should be proportional to the product gVm for constant temperature. A plot of sav against gVm for the present data is shown in Fig. 1. An acceptable correlation is found between these quantities but the slope of the best straight line passing through the origin (1.56 103 mol J1) is larger than 3/RT (1.21 103 mol J1). It must be kept in mind that the SED equation does not consider the molecular nature of the environment of the diffusing species so that one cannot expect perfect agreement of the sav estimates with Eq. (5). However, in the following analysis we assume that the average relaxation time is the appropriate quantity for estimating the longitudinal relaxation time, and thus, the rate constant.
τav / ps
Vmη / μJ s mol
Fig. 1. Plot of the average relaxation time sav against the product of the molar volume Vm and the viscosity g for imidazolium salts with the BF 4 anion (d) and the PF 6 anion (N).
W.R. Fawcett et al. / Journal of Electroanalytical Chemistry 660 (2011) 230–233
3. Estimation of the standard rate constant The standard rate constant for adiabatic electron transfer after correction for the double layer effect is given bytpb
exp DGis þ DGos =RT
DGis and DGos are the Gibbs activation energies for the inner sphere and outer sphere contributions, respectively. Kp, the equilibrium constant for precursor complex formation, and sL, the longitudinal relaxation time. ksc depends on the nature of the solvent through the terms DGos and sL, that is, both in the pre-exponential term and in the exponent. First of all we consider the contributions to ksc which do not depend on the nature of the solvent. On the basis of earlier work for the Fc/Fc+ system , the value of Kp is 20 ± 10 pm. The estimate of DGis is 0.5 kJ mol1. Now we consider the estimation of DGos . According to Marcus theory for heterogeneous electron transfer  N L e2 1 1 1 1 ð7Þ DGos ¼ 32pe0 a Re eop es Here NL is the Avogadro constant, e, the electronic charge, e0, the permittivity of free space, a, the radius of the reactant in a spherical representation, and Re, the distance of the reactant from its image in the conducting electrode. The estimate of a for Fc is 370 pm. The distance Re is more difﬁcult to estimate. The lowest value of Re is 740 pm. This would be valid when the reactant is in contact with the electrode. The largest value for this parameter is inﬁnity. This estimate is used when the reaction site is in the diffuse layer well screened from its image in the electrode. Since the ferrocene reaction takes place at positive electrode charge, the reactant is probably separated from the electrode surface by a layer of anions. On the basis of data compiled by Marcus , the effective radii for the BF 4 and PF 6 ions are 230 and 245 pm, respectively. Assuming close packed hard spheres, the distance of closest approach of the center of the Fc molecule to a line through the centers of a close packed monolayer of PF 6 ions is 564 pm. The distance of closest approach of Fc to the electrode surface is therefore 809 pm so that the value of Re in the presence of PF 6 is 1618 pm. In the presence of BF 4 the distance of closest approach of the Fc molecule to the electrode surface is 784 pm, and the value of Re, 1568 pm. These results are used in the following estimates of DGos . Initially, the outer sphere Gibbs activation energy for the Fc/Fc+ system is estimated in RTILs containing the BF 4 anion. Assuming a = 370 pm and Re = 1570 pm, the factor (a1 Re1) is equal to 2.065 109 m1. In the case of EtMeImBF4, the Pekar factor is equal to 0.429. This leads to an estimate of DGos equal to 15.4 kJ mol1. The estimates for the other BF 4 systems which are recorded in Table 3 do not differ signiﬁcantly. In the case of RTILs 1 containing PF Re1) is 2.085 109 m1. 6 , the estimate of (a This leads to slightly higher estimates of DGos in the PF 6 systems. Now the effective longitudinal relaxation time sL and the rate constant ksc may be estimated. The longitudinal relaxation time is given by the equation
eop sav es
In the case of EtMeImBF4, sav is equal to 34 ps, and sL to 5 ps. According to Eq. (6), the value of ksc given that DGos is equal to 15.4 kJ mol1 is 0.5 cm s1. This is a double layer corrected result which is independent of the nature of the metal electrode for reactions like the Fc/Fc+ redox system. The kinetic parameters estimated for the RTILs considered here are summarized in Table 3. It is immediately striking that DGos is essentially independent of the medium. This follows from the fact
that both eop and es vary only by a small amount for the systems considered. Furthermore the estimates of Re for the two sets of electrolytes are quite close to one another. The large variation in the standard rate constant for salts with a ﬁxed anion is due to a corresponding variation in the longitudinal relaxation time sL. Although the magnitude of the values of ksc reported here may certainly be incorrect considering all of the assumptions which are necessary, the trends found in Table 3 should be conﬁrmed by experimental data when they become available. Obviously the easier systems for carrying out kinetic measurements are those with longer chain cations in which the rate constants are smaller. 4. Discussion In the above analysis, we have obtained sav by extending the averaging method of Nakamura and Shikata  to include all three relaxation modes in the RTIL. Other methods of estimating an effective relaxation time are available when all relaxation processes are of the Debye type . In the present case, the second relaxation has a Debye character whereas the third has a Cole–Cole character. However, the values of sav used here seem reasonable. Other studies of dielectric relaxation in RTILs have been carried out [10,11]. The work of Buchner et al.  is especially notable because they were able to study dielectric relaxation up to 3000 GHz. Their study included three of the imidazolium salts considered here. They found four relaxation processes, a Cole–Davidson or Cole–Cole process at low frequencies, then two Debye processes, and ﬁnally a damped harmonic oscillator process at the highest frequencies. Clearly, their ﬁt of the relaxation spectra is very different from that obtained by Nakamura and Shikata . However, since they only obtained data for three of the salts discussed here, their results were not considered further. However, it should be emphasized that data obtained over the wide frequency range used by Buchner et al.  would be welcome. Very few data exist in the literature for electron transfer kinetics. Compton et al.  measured the rate constant for oxidation of ferrocene at a glassy carbon electrode in acetonitrile and BuMeImPF6, obtaining rate constants of 0,03 and 2.3 104 cm s1, respectively. However, at a Pt electrode the rate constant for the same reaction in acetonitrile is 2.6 cm1. Assuming that a very repulsive double layer effect exists at glassy carbon but not at Pt, the estimate of the rate constant at Pt in BuMeImPF6 is 0.02 cm s1. This result is an order of magnitude smaller than our estimate (0.2 cm s1). Another measurement from the same laboratory  involved ferrocene oxidation in EtMeIm+ bis(triﬂuoromethylsulfonyl)imide (TFS) at platinum. Although this RTIL is not considered speciﬁcally here, the rate constant (0.2 cm s1) falls within the range of those estimated for BF 4 and PF 6 salts. As pointed out above, the standard rate constants estimated here correspond to rate constants corrected for double layer effects. Analysis of the double layer effect requires knowledge of the point of zero charge and capacity data as a function of potential. Since the oxidation of ferrocene occurs at positive charge densities, speciﬁc adsorption of the electrolyte is not expected to be important. A reaction such as the oxidation of cobaltacinium cation to cobaltacene takes place at much more negative potentials. Speciﬁc adsorption of the imidazolium cation, especially when the alkyl group is large, is certainly possible. Analysis of the double layer effect in the presence of ionic speciﬁc adsorption is much more complex. Lynden-Bell [13,14] has questioned whether Marcus theory can be applied to electron transfer reactions in RTILs. Simulations to estimate the time-dependent relaxation of the solvent for electron transfer to a monoatomic species with a radius of 180 pm and a charge varying from +3 to 3. The solvents considered were acetonitrile at 298 K and MeMeImPF6 at 450 K. Initially it was
W.R. Fawcett et al. / Journal of Electroanalytical Chemistry 660 (2011) 230–233
concluded that the ionic liquid behaves similarly to the aprotic solvent and that Marcus theory is applicable . This conclusion was later qualiﬁed and situations where deviations from Marcus theory may be found discussed . However, the estimate of DGos given by Eq. (7) does not consider speciﬁc solvation of the reactant by the ions of the RTIL. This would result in an additional contribution to DGos and therefore a smaller rate constant than estimated here. In the above analysis, it was assumed that the electron transfer process is adiabatic. Measurements of the pressure dependence of the standard rate constant provides a means to assess this assumption because the viscosity of the medium varies signiﬁcantly with pressure in organic solvents  and RTILs . Thus, Dolidze et al.  showed that the oxidation of ferrocene is adiabatic in BuMeImTFS by studying the volume of activation for the electrode reaction in the pressure range 1–150 MPa. In conclusion, although the estimated rate constants presented here may be in error by an order of magnitude, the trends predicted by the present calculations should be seen experimentally. Fortunately, new kinetic data are presently being obtained so that the calculations presented here can be compared with experimental data in the near future. Acknowledgements We are indebted to Professor Galina Tsirlina for helpful discussions of an earlier version of this paper. The ﬁnancial
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