J . Phys. Chem. 1990, 94, 8910-8912
8910
Solvent Effects on Electron-Transfer Kinetics: A Correlation of Rate Constants with Solvent Acidlty, Basicity, and Polarizability Parameters Andrew P. Abbott and James F. Rusling* Department of Chemistry (U-60), University of Connecticut, Storrs, Connecticut 06269-3060 (Received: January 12, 1990; I n Final Form: June 19, 1990)
The solvent-dependent dielectric constant term ( I / c o - l / c ) in the Marcus expression for reorganizational energy can be replaced by a measure of the solventsolventsolute interactions from the Taft theory. The Taft theory describes the acidity, basicity, and polarizability of the solvent. It provides a very good model for the variation of electron-transfer rate constants of outer-sphere reactions in a variety of solvents. The improved correlation between predicted and theoretical rate constants for the Taft model over the dielectric continuum model arises because of the former model's recognition of specific solventsolute interactions in the formation of the transition state. Several examples of electron-transfer reactions in different solvents are analyzed to demonstrate the greatly improved correlation between observed and predicted rate constants using the Taft model.
Introduction The influence of the solvent on the kinetics of electron-transfer processes has received considerable attention recently. Most of the work has centered around a dielectric continuum model (DCM) which was first applied to a quantitative theory of electron transfer by Marcus and Hush.14 When an electron-transfer process occurs in a polar solvent, a reorientation of the solvent dipoles surrounding the reacting species is necessary because of the change in charge distribution that accompanies the process. The DCM theory holds that, for an outer-sphere electron transfer between two hard conducting spheres in a structureless dielectric medium, the reorganizational energy, A,,, is given by equation 1," where e is the electronic charge, Xo = e2(1 / r l
+ 1 / r 2 - 1/ d ) ( 1
/to
- 1 /e)
(1)
r , and r2 are the radii of the two reacting species, d is the distance across which the electron is transferred, to is the optical dielectric constant (equivalent to the square of the refractive index of the solvent), and t is the bulk dielectric constant of the medium. An analogous expression exists for charge transfer between an electrode and a reactant. The value of A,, is generally obtained from the activation free energy AG'
AG' = (AGO
+ x0)~/4&
(2)
where AGO is the free energy change associated with the conversion from precursor into successor complexes. The activation free energy can be obtained from the rate constant for electron transfer
k' k ' = Kz exp(-AG'/RT)
(3)
where K is the transmission coefficient (which in many cases can be assumed to be unity) and z, the collision frequency of the reacting species, is equal to ( k T / 2 ~ m ) ' /where *, k is Boltzmann's constant, Tis the absolute temperature, and m is the reduced mass of the reactants. Equations 1-3 predict that the term In k'should be proportional to the function (1 /to - 1/e), Several investigations of electron-transfer processes in a range of solvents with different dielectric constants have not observed this codependence.5-8 The Marcus, R. A. J . Chem. Phys. 1965, 43, 679. (2) Hush, N. S . T r a m Faraday SOC.1961.57, 557. (3) Dogonadze. R. R. In Reactions of Molecules at Electrodes; Hush, N . S., Ed.; Wiley Interscience: London, 1971; Chapter 3. (4) Marcus, R. A. Annu. Reu. Phys. Chem. 1964, IS, 155. ( 5 ) Yang, E. S.; Chan, M A . ; Wahl, A. C. J . Pkys. Chem. 1980,84, 3094. (6) Kapturkiewicz, A.; Opallo, M. J . Electroanal. Chem. Interfacial Elecfrochem. 1985, 185, IS. (7) Sahami, S.;Weaver, M. J. J . Electroanal. Chem. Interfacial Elecfrochem. 1981, 124, 3 5 . ( 8 ) Sharp, M.; Petersson, M.; Endstrom, K . J . Electroanal. Chem. Inferfacial Elecfrorhem. 1980, 109. 271. (1)
only data sets in which a dependence of In k 'on (1 /e0 - 1/e) was observed were those carried out in a limited range of solvent^,^ i.e. those which are nonpolar and non-hydrogen-bonded, In these cases the solvent approximates a dielectric continuum and the solvent-solute interactions are negligible. It would appear that this discrepancy occurs because of the simplicity of the dielectric continuum model, which assumes a structureless medium between the reacting species. Over the relatively short distances that electron-transfer processes occur in solution, the solvent molecules can be assumed to have a small degree of order, which arises from short-range solventsolvent and solventsolute interactions, such as a solvation sheath surrounding strong electrolytes in aqueous systems. Thus, a bulk solvent property like the dielectric constant will poorly describe the microenvironment around the reacting species, which governs the stability of the intermediate reaction complex and hence the rate of electron transfer. One approach to account for the discrepancies in the DCM is the use of an average dielectric constants in place of t in eq 1. This has had some success, but is totally empirical. Other approaches have been based on nonlocal dielectric theory such as those developed by Ulstrup and Kornyshev,'osll which attempt to account for hydrophobic solvent effects and solvent-induced nonlocal dielectric properties. Attempts have also been made to correlate solvent reorganization energies to empirical parameters such as the acceptor and donor n ~ m b e r s , 'with ~ ~ ' some ~ success but little theoretical justification. The work presented here investigates the solvent-dependentterm of the Marcus theory in terms of the polarizability of the medium and the various solvent-solvent-solute interactions. These are important, as explained above, and any model that attempts to accurately describe an electron-transfer reaction must take these effects into account. The electron-transfer process can be divided into three parts: first, a partial reorientation of the solvent molecules to form the transition state; second, a fast electron transfer step; and finally, another reorientation of the solvent molecules to form the products. Clearly, the first and third steps will depend upon the specific interactions between the solvent and solute species. These are a function of the acidities and basicities of the solvent molecules, reactants, and products. During the electron-transfer step, however, reorientation of the solvent molecules is precluded by the Frank-Condon principle and thus only electronic rearrangements can take place. These will be dependent upon the polarizability of the solvent molecules. (9) Weaver, M. J.; Gennett, T. Chem. Phys. Left. 1985, 113, 213. (IO) Kornyshev, A. A. Elecfrochim. Acfa 1981, 26, I . ( I I ) Kornyshev, A. A.; Ulstrup. J. Chem. Phys. Left. 1986, 126, 74. (12) Fawcett, W. P.; Jaworski, J. S. J . Phys. Chem. 1983, 87, 2972. (13) McManis, G. E.; Gochev, A.; Nielson, R. M.; Weaver, M. J. J. Phys. Chem. 1989, 93, 7733.
0022-3654/90/2094-89 I0$02.50/0 0 1990 American Chemical Society
Solvent Effects on Electron-Transfer Kinetics
The Journal of Physical Chemistry, Vol. 94, No. 26, 1990 8911
Various treatments for these solventsolventsolute interactions based on linear free energy relationships have been developed."I6 The most widely used technique is that of Taft, Kamlet, Abboud, and Abraham,17-19 which measures the absorption spectra of indicator solutes in different solvents. The solutes used have different dipole moments in the ground and excited states. When the indicator solutes become excited, the solvent molecules must undergo electronic rearrangements to stabilize the new charge distribution. The degree to which the solvent can stabilize the excited state is reflected by shifts in the absorption spectrum of the solute. These so-called "solvatochromic" shifts are a measure of the polarizability of the solvent and can be related to various thermodynamic and kinetic properties.18 These solvatochromic shifts have been transformed into a semiempirical equation that produces a value for the local polarizability, T * , between 0 and l :
umax=
00
+
ST*
(4)
where umaxis the value of the absorption maximum in the test solvent, uo is absorption maximum of the standard solvent, and s is a constant obtained from experiment. The values of s and ?r* are documented for a large range of solvents and indicator solutes. This relationship has been found to be valid for a wide range of nonpolar solvents and solvent mixtures.20,2' Deviations from linearity are observed when the solvents are strong hydrogen-bond donors or acceptors. In such cases eq 4 can be modified to account for these effects:
TABLE I: Dielectric Constant Functions and Polarizability, Acidity, and Basicity Values for the Various Solvents Used in the AMIVW no. solvent I / C o - I/€. T*b a p 1 acetonitrile 0.528 0.75 0.19 0.31 2 acetone 0.495 0.71 0.08 0.48 3 methylene chloride 0.382 0.82 0.30 0.05 4 formamide 0.469 0.97 0.71 0.0 5 dimethylformamide 0.463 0.88 0.0 0.69 6 dimethyl sulfoxide 0.438 1.00 0.0 0.76 7 benzonitrile 0.390 0.90 0.0 0.41 8 tetramethylurea 0.433 0.83 0.0 0.80 9 propylene carbonate 0.480 0.83 0.0 0.40 IO methanol 0.538 0.60 0.93 0.62 11 ethanol 0.500 0.54 0.83 0.77 I2 I -propanol 0.473 0.52 0.78 0.78 13 N M P T 0.437 0.87 0.0 1.05 14 water 0.437 1.09 1.17 0.50 15 nitromethane 0.390 0.85 0.22 0.0 16 nitrobenzene 0.385 1.01 0.0 0.39 17 acetic acid 0.371 0.64 1.12 0.0 18 dichloroethane 0.384 0.81 0.0 0.0 19 dimethylacetamide 0.459 0.88 0.0 0.76 "Data from ref 29. *Data from ref 18.
I83l9
u,,,
= uo
+
SA*
+ aa + b@
A
0
DCM
I
0
I
0 0
(5)
where a and p are the acidity and basicity factors of the solvent and a and b are constant^.^^*^^ Values for a, b, and s are obtained by a multiple linear-regression analysis.22 It should be noted, however, that very few solvents have both acidic and basic characteristics, so that eq 5 can often be simplified when one considers only a certain type of solvent.
Tait
-7
1 1
-11' ' -9.00
'
0 "
"
-8.50
"
'
-8.00
.
"
-7.50
"
"
-7.00
Data Analysis
The solvent effect on electron-transfer kinetics was analyzed for several sets of data reported in the literature. The data sets were analyzed in two ways. First, the rate constants were compared to the dielectric constant function shown in eq I . A line of best fit was calculated, by linear regression for the data, of the form In k i x p 3 = j c
+ m ( l / c o j- I / c j )
(6)
From the values of m and c and the dielectric constant data, values for In k:alc,j were obtained. The experimental rate constant was also correlated to the Taft model by forming a set of simultaneous equations of the form
In k i , p , j =
ST*^ + aaj + bPj
(7)
The coefficients a, b, and s were obtained by solving the simultaneous equations by a Gauss-Seidel method24provided by com(14)Allerhand, A.; Schleyer, P. v. R. J. Am. Chem. SOC.1963,85, 374. (IS)Knauer, B. C.; Napier, J. J. J . Am. Chem. SOC.1976, 98. 4395. ( I 6) Brownstein. S. Can. J . Chem. 1960, 38, 1590. (17)Taft, R. W.; Abboud, J.-L. M.; Kamlet, M. J.; Abraham, M. H. J . Solution Chem. 1985, 14, 153. (18)Kamlet, M. J.; Abboud, J. L. M.; Abraham, M. H.; Taft, R. W. J . Org. Chem. 1983, 48, 2877. (19)Kamlet, M. J.; Abboud, J. L. M.; Taft, R. W. J . Am. Chem. SOC. 1977, 99, 6027. (20)Abboud, J.-L. M.; Kamlet, M. J.; Taft, R. W. J . Am. Chem. SOC. 1977, 99, 8325. (21)Abbott, A. P.; Schiffrin, D. J. J . Chem. SOC.,Faraday Trans. 1990, 86, 1449. (22)Kamlet, M.J.; Jones, M.E.;Taft, R. W.; Abboud, J.-L.M. J . Chem. SOC.,Perkin Trans. 2 1979, 342. (23)Kamlet, M. J.; Taft, R. W. J. Chem. Soc.,Perkin Trans. 2 1979,349. (24)Norris, A. C. Computational Chemistry; Wiley: New York, 1981; Chapter 5.
In(k(expN Figure 1. Correlation between the experimental rate constant In kcxpand the calculated rate constant In k,,, for the D C M and the Taft model for data reported in ref 5. The line corresponds to the theoretically perfect correlation.
mercially available software.2s Again, values of In k L l c , jwere obtained by putting the data and coefficients back into eq 7. The solvent parameters used in the calculations are given in Table I. The data were compared statistically in two ways: first, by the familiar coefficient of correlation, r, and second by using the x2 test,26 which is a more accurate measure of the correlation. The value obtained for x 2 can be compared to values listed in standard tables2' at various confidence levels. Sets of data with x2 values less than those listed are said to be correlated with the degree of confidence listed. It appears initially that the correlations of eqs 6 and 7 differ because the former has two solvent parameters whereas the latter has three. For most solvents considered, however, the value of either a or p is zero or close to zero so eq 7 often has only two parameters per solvent (Table I). It is felt, therefore, that a direct statistical comparison of the two fits is justified, especially in view of the large differences that were found in these correlations. It should be noted that the analyses were carried out on the complete data sets reported in the literature, except where a,j3, and T * values were not available for a specific solvent. (25)PC MATLAB; Mathworks, Inc.: Sherborn, MA, 1987. (26)Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, 2nd ed.; Ellis Horwood Ltd.: Chichester, U.K., 1988. (27)Beyer, W. H. CRCStandard Mathematical Tables, 26th ed.; CRC Press Inc.: Boca Raton, FL, 1981.
Abbott and Rusling
8912 The Journal of Physical Chemistry, Vol. 94, No. 26, 1990
TABLE 11: Statistical Correlations of Rate Constant Data with the Dielectric Continuum Model (DCM) and the Taft Model no. of solvents
ref 28" 5b 28' 30d 6' 3 If
IO 9 6 6 6 5
rTafl
PDCM
X'Tafl
X2WM
X29%
0.93 0.95 0.90 0.91 0.8 1 0.89
0.23 0.16 0.73 0.76 0.09 0.34
2.29 0.1 1 1.34 2.15 1.18 0.23
163.85 8.10 5.78 7.83 206.08 3.63
3.33 2.73 1.64 1.64 I .64 1.15
solvents see Figure 2 see Figure 1 I , 5, 6, 8, 9, 1 1 1 , 2, 5, 6, 13, 19 1, 5, 6, 9, 13, 19 1 , 5, 6, 1 I , 14
*
'Cp,Co+ reduction at a mercury electrode. Ferrocene/ferrocenium electron self-exchange. (COT)Fe(CO), reduction at a mercury electrode. "Co(sa1en) oxidation at a Pt electrode. e Nitromesitylene reduction at a mercury electrode. /Ferrocene oxidation at a Pt electrode. A
raft
0
DCM
I
3.60 1 0
-3.60
'
-1.20
0
,
0.00
m
"
1.20
In(k(exp)) Figure 2. As in Figure 1 . Data taken from ref 28
Results and Discussion As described previously, the dielectric continuum model is only valid when the specific solventsolute interactions are negligible. However, when the range of solvents is increased to include those which are polar and strongly hydrogen-bonded (Le. those most commonly used) the DCM fails to predict the observed electron-transfer kinetics. FIgures 1 and 2 show the correlation between the observed rate constants and those predicted by the Taft model and the DCM for data given in the literature.5+28It can clearly be seen that in both examples the values calculated from the Taft model are very close to those observed, whereas those from the DCM are almost randomly distributed. This (28) McManis, G.E.; Golovin, M. N.; Weaver, M. J. J . Phys. Chem. 1986, 90, 6563. (29) Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic soluenfsPhysical properties and methods of purification, 4th ed.; Wiley: New York, 1986.
(30) Kapturkiewicz, A.; Behr, B. J. Electroanal. Chem. Interfacial Electrochem. 1984. 179. 187. (31) Diggle, J . W.; Parker, A. J. Electrochim. Acta 1973, 18, 915.
improvement clearly comes from the Taft model's recognition of the important solvent-solute interactions in the formation of the transition state. Improved correlations with solvent properties using the Taft model are observed for several other electron-transfer reactions. The statistical analyses of these systems are shown in Table 11. It is evident from the x2 tests of the two models that a very strong correlation exists for the Taft model at the >95% confidence level. All of the correlations predicted by the DCM, however, must be rejected at this level and some must be rejected at all reasonable levels of confidence. It is evident from Figure 2 that ignoring several of the data points predicted by the DCM (generally those with a higher dielectric constant) produces a better correlation. This probably arises because solvents with high dielectric constants are usually those which are highly polar and strongly hydrogen-bonded and hence deviate most strongly from the DCM. It was found that with these limited data sets equally good correlations can be achieved with just the A* parameter from the Taft theory. This is logical, as the low dielectric constant solvents are generally less polar and so the acidic and basic terms of eq 7 can be largely ignored. For most solvents commonly used in kinetics studies, a more sophisticated model than the DCM is required to account for the important solvent-solvent-solute interactions. I t can be seen from the results in Table I1 that the model proposed here is valid both for electron transfer at a solid electrode and for electron-self-exchange reactions in solution. It is difficult to predict how universally the Taft theory applies, because of the lack of reliable data in the literature for electron-transfer processes in a wide range of solvents. It must be concluded however that, for the limited range of solvents studied in this work, the Taft model for the solvent-dependent term in the Marcus theory provides a more practical and experimentally more useful approach to predicting electron-transfer kinetics than the dielectric cont i n u u m theory. Acknowledgment. This work was supported by US.PHS Grant ES03 154 awarded by the National Institute of Environmental Health Sciences.
