6563
J. Phys. Chem. 1995, 99, 6563-6569
Solvent Influences on Electron-Transfer Reactions Russell S. Drago* and Donald C. Ferris Department of Chemistry, University of Florida, Gainesville, Florida 3261 I Received: October 24, 1994; In Final Form: February 13, 1995@
The term ( l/n2 - 1ks) for the solvent dependence of electron-transfer rates in transition-state theory is replaced by S', of the united solvation model (USM) to estimate the influence of nonspecific solvation on the rates. The equation A@ = PS' W is derived. Specific interactions with the solvent are accommodated by adding appropriate terms for the donor-acceptor interaction to this equation. In this manner, electron-transfer rates of several metallocene systems are correlated without the need of any solvent friction correction. In addition to fitting the data, the USM analysis results in a detailed model for the influence of solvent on a reaction coordinate with effects from coordination, solvent polarity, and coupled anion migration contributing to the electron-transfer reactions.
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Introduction Solvation Models. Electron-transfer reactions are among the most thoroughly studied and most completely understood reactions. In contrast, the influence of solvent on the rates of electron-transfer reactions is poorly understood. The literature concerning solvent effects on electron-transfer reactions is extensive and the many literature citations will not be repeated here. The interested reader is referred to a recent review by Heitelel for background references. The solvation component of electron transfer has been modeled by transition-state theory, a solvent friction correction to transition-state theory, or the Sumi-Marcus approach. According to transition-state (TS) theory the solvent reorganization energy As is given by2
where Aq is the shift between the free energy minima of the free energy surfaces of the reactants and products. According to the Marcus-Hush theory A, is approximately given2 by eq 2, = e2(1/2r+
+ ~ 2 r -- 1/4(1/n2 - I/CJ
(2)
where r* are the effective radii of the reactants at distance d, n is the refractive index, and E , the static dielectric constant of the solvent. Equation 2 for the reorganization energy, A,, is of the form of the orientational polarization of the solvation energy of an ion pair. The solvent perturbs the product and reactant potential energy surfaces whose intersection is typically used to describe the barrier for electron transfer. Solving for the intersection of the two solvent-perturbed free energy surfaces of products and reactants leads to eq 3 for a solvent-influenced AG*:
A& x (AG
+ A J ~ J ~ -A ,v
(3)
The splitting V, which results from coupling of the product and reactant states, is usually neglected.' In the above discussion, we have assumed that the solvent influence on the reorganization energy giving rise to the electron transfer barrier is constant. In the discussion section, the accommodation of a solventdependent reorganization energy will be discussed.
The rate k for the uniform adiabatic ET process is given' by eq 4.
k =p(kBT/hzo)exp(-AG*/kBr)
M
p(wd2n) exp(-A&/kBr) = pktS(4)
ZJis the partition function for one-dimensional motion in the initial well with a solvent-dependent oscillation frequency corresponding to wo = (~JIV)''~; where M is the effective mass for the motion near the bottom of the well. T is absolute temperature, h is Planck's constant, and k~ is Boltzmann's constant. The term exp(-AG*/kBT) is the probability of the system in the TS region moving in a forward direction (where ke is the Boltzmann constant). jj is the electronic transition probability for a transition from reactant to product FESs whenever the system traverses the TS region. The remaining term is the average inverse time it takes to cross the TS region "in straight flight". In eq 4, k,, is the TS rate. The fluctuating interaction of the solvent medium and the reacting species is proposed' to have a significant influence on the rate under certain conditions, leading to the solvent friction correction to TS theory. Crossing the reaction coordinate q(X) becomes a function of a large number of coordinates X, characterizing the location of interacting solvent molecules. The combination of many solvent molecules into a single reaction coordinate leads to frictional and random forces in the equation of motion for 4.' The effect of friction leads to a correction of k, to yield the ET rate k(5). The ratio k(5)/kl, is defined as a measure of frictional effects. Weak frictional forces result in systems that undergo several oscillations between the reactants and products before locking into one of the wells. The reaction rate is limited only by how fast the required activation energy is obtained and how fast the surplus energy is dissipated to lock the system in either well. In this case k(5)lklsis proportional to 5. As friction becomes dominant, the ratio k(5)/kls approaches the inverse of the ratio of the traversal times of Zt,,diff to tts,unl for diffusive or uniform motion through the TS region, respectively. Using the above considerations, eq 4 is changed to eq 5 for the solvent-controlled adiabatic reaction.' In general, C will
k(rL)= [A,/( 16nkBr)]1/2(l / t J x exp[-(AG -k AJ2/(4A,kBr)] = C/tLeXp(-A&/kBr)
(5)
~~~
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Abstract published in Advance ACS Abstracts, April 1, 1995.
0022-365495J2099-6563$09.00/0
have a contribution from A, and usually is on the order of 0.10 1995 American Chemical Society
Drago and Ferris
6564 J. Phys. Chem., Vol. 99, No. 17, 1995 1. Thus, the rate is inversely proportional to the friction constant
5 or the dielectric relaxation time ZL. Interesting to note, eq 5 sets an upper limit of l / z ~for all reactions. The other major model for electron transfer which offers an extension of dynamic solvent effects is the Sumi-Marcus ( S M) modeL3 In this approach a low-frequency intramolecular vibration mode is assumed to provide a path to cross the activation barrier. Combining this mode and solvent fluctuation, the FES becomes a two-dimensional function of a diffusive solvent (X) and an intramolecular (Y) coordinate. The relaxation along Y is always faster than that of the solvent to maintain thermal equilibrium of X with respect to Y . In contrast to TS theory, the barrier is crossed along the nondiffusive, intramolecular coordinate (Y). The function of solvent coordinate (x) is to change AG*(X) of the reaction. The final conclusions of the Sumi-Marcus and friction corrected TS models are similar. Experimentalists tend to use the solvent friction modified TS theory equations to analyze data because they are easier to manipulate. Exceptions are explained and reactions described with the Sumi-Marcus model. The Unified Solvation Model. A series of articles4 have described an empirical measure of solvent polarity, S‘, which results from fitting a wide variety of spectroscopic shifts, Ax, of probe solute molecules that accompany solvent variation. These spectral shifts (electronic transitions, NMR shifts, EPR coupling constants, etc.) are less likely to be influenced by special effects that can influence rates. Accordingly, these measurements form the basis for 5”. In solvents that do not undergo donor-acceptor interaction with the probe, the data are fit to
Ax = PS’
+W
where Ax is the value of the measured property in the polar solvent, S’ is the measure of solvent polarity, P is the sensitivity of the probe to solvation, and W is the intercept at S’ = 0. Though a variety of spectral transitions for 38 probes of different sizes and shapes could be correlated for 46 different aprotic solvents of varying polarity, no correlation of S’ with the ( l/n2 - lks)solvation term of eq 2 could be found. Furthermore, no measured property of the neat solvent correlates with S’. The nonspecific solvation of a probe molecule is determined by the polarity of the probe and the polarity and dimensions of the solvent cavity formed for the particular probe molecule. Furthermore, electronic polarization of the solvent induced by the probe transition must also be included in S; making it difficult to correlate S’ to other measured solvent properties. When solvation of the solute includes specific donoracceptor interactions, the magnitude of the specific interaction is estimated with the ECW equation:5
(7) The solvation of an acceptor probe in donor solvents is given by4c
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Ax = EA*EB CA*CB PS‘ -I- W The EB and CBand S‘ values are those r e p ~ r t e dfor ~ ~the , ~donor solvent molecules. The values of EA*, CA*. P, and W for the probes are determined from a best-fit, least-squares solution of equations for a series of measurements of Ax in various solvents. The solvation of a donor probe in acceptor solvents is given by4b
Ax = ELEB*
+ CA’CB*+ PS‘ + W
(9)
The reported values of EA‘. CA‘, and S’ for the neat solvent are employed.4b The value of P and W is the same for the probe as in donor solvents, where no specific donor-acceptor interactions occur. A series of measurements of Ax in acceptor and donor solvents (where only the PS W term is used) are solved for EB*, CB*,P, and W. Electron Transfer in Metallocenes. The metallocenes represent one of the most extensively studied solvent-dependent electron-transfer systems.6 The solvent dependence of the electron transfer was analyzed6 in terms of solvent friction. An increase in loglO(k,,) with increasing loglO(zL-I) for the cobalt complexes and less so with the iron complexes was noted. Solvents which showed “scatter” from the trend were D20, acetone, nitromethane, and acetonitrile. This was attributed to the large ZL-~values for these four solvents. It was proposed that these solvents might be approaching a theoretical limit based on solvent friction considerations. Propionitrile, however, which did fit the trend has a similar ZL-’ to acetonitrile. The JC solvents nitrobenzene and benzonitrile yielded faster rates than expected from their ZL-~and this is “justified” by stating6that ZL-~should only provide an approximate description of solvent dynamics. The protic solvents methanol and ethanol, as well as propylene carbonate, were never compared. This was due to the fact that these solvents exhibit “non-Debye” behavior (Le., there is at least one higher-frequency dispersion region in the dielectric loss spectra). Further, NMF was never compared even though it is a “Debye” solvent with a Z L - ~between those of benzonitrile and DMSO. Specific donor-acceptor complexation to the solvent was not considered in this interpretation. The self-exchange reactions of metallocenes6 have been analyzed7by using the Kamlet, Abraham, Abboud, Taft (KAAT) multiparameter a p p r ~ a c h .An ~ earlier metallocene data set was analyzed and few details of the data fit are reported other than correlation coefficients of 0.93 and 0.95 for the cobaltocenes and ferrocene systems. The authors concluded that donor and acceptor properties of the solvent are important and they rejected the dielectric continuum model. No justification for the correlation of electron-transfer data with the KAAT parameters was reported and the probe parameters were not interpreted. In earlier reports,8 the shortcomings of the p-z* models’ have been discussed and these shortcomings will be shown to be evident in their use for the analysis of the electron-transfer data. Charge-transfer interactions are averaged into the KAAT parameters and these effects go undetected when a data set is analyzed with these parameters. The averaging of these specific interactions into the E* parameters also leads to poor parameters for acetonitrile, which are compensated for in the KAAT model by incorrectly attributing acceptor solvent properties, a, to it. A similar problem arises4bwith the acceptor number AN and donor number scales often used in solvent effect interpretations. Both the specific and nonspecific contributions to solvation are averaged into a single parametePb in these scales.
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Results and Discussion The Unified Solvation Model and Electron-Transfer Processes. Our first concern will be with systems in which specific donor-acceptor interactions are absent. Specific interactions introduce added complications that will be treated in a later section. Extensive studies show that physicochemical properties of many systems are better correlated with the S’ parameter of the USM than (lln2 - l/es). Since (lid - 1ks) poorly correlates the many physicochemical properties correlated with USM, ( l h 2 - 1 k s )should also lead to a poor estimate of solvent polarity on ET rates. It is proposed that S’ be used to evaluate nonspecific solvation.
J. Phys. Chem., Vol. 99, No. 17, 1995 6565
Solvent Influences on Electron-Transfer Reactions Equation 3 is the basis for transition-state, solvation contributions to electron-transfer rates. When AG x 0, as in selfexchange reactions, eq 3 becomes similar to eq 10.
A 6 = (1/4)As -V
M+
(10)
Since S’ is a better measure of solvent polarity than (l/n2 l/cs), it replaces this term in eq 2 for As, and neglecting the V term, eq 9 yields eq 11. The slope r is given by e2(1/2r+
+
M
AG* x rs‘
(11)
1/2r- - l/d)B’, where B‘ is the dimensionality constant required by the substitution of S for ( l/n2 - l/cS). The form of eq 1 1 is similar to that of eq 6 where W of eq 6 corresponds to the value of A@ at S’ equals zero. All cases of solvent influence can be accommodated by multiplying the r term in eq 11 by a factor Q, where Q is related to the amount of influence that As, and thus S‘, has on A@. When AG >> As, Q approaches 0, and when AG
