4664
J . Phys. Chem. 1986, 90, 4664-4665
Solvent Effects on Back-Electron Transfer by the trans-Stllbene-Fumaronitrile Ion Pair Orland W. Kolling Chemistry Department, Southwestern College, Winfield, Kansas 67156 (Received: March 28, 1986)
An analysis of standardized residuals demonstrates that correlations of In k,, with the Dimroth ET value for the solvent are inadequate as quantitative descriptions of the role of the solvent in the electron-transfer process. A more appropriate model for the solvent influence on the back-electron transfer by the contact ion pair is constructed by resolving the contributions of the solvent into separate dipolarity-polarizability and hydrogen bond donor terms. The resulting linear free energy relationship can be extended to the prediction of shifts in the free energy gap between the ground state and the ion pair which occurs with changing solvents.
Introduction The very rapid reaction kinetics for single electron transfer within large ion pairs provide important and potentially usable systems for the quantitative testing of major theoretical and semiempirical representations of solvent effects on redox processes. Recent investigations of Peters et al.1-5using picosecond absorption spectroscopic techniques have extended the data base for the examination of the dynamics of the electron-transfer process by charge-transfer complexes in nonaqueous media. The one case that these investigators have most fully studied is the trans-stilbene-fumaronitrile (TS/FN) ion pair,s and its partial scheme involving the essential electron-transfer step is given in eq 1. Here, s (kp)
TS FN
(4s)
k*
TS+.FNTS+(S)FN(CIP) * (-b) ' (SSIP)
kd
(1) Simon, J.; Peters, K. Acc. Chem. Res. 1984, 17, 277. Simon, J.; Peters, K. J . Am. Chem. SOC.1981, 103, 6403. Simon, J.; Peters, K. J . Am. Chem. SOC.1983, 105, 4875. Goodman, J.; Peters, K. J . Am. Chem. SOC.1985, 107, 1441. Goodman, J.; Peters, K. J. Am. Chem. SOC.1985, 107, 6459. Kleinbaum, D.; Kupper, L. Applied Regression Analysis and Other Multivariable Methods; Duxbury: North Scituate, MA, 1978; pp 236-241. (7) Kamlet, M.; Abboud, J.; Abraham, M.; Taft, R. J . Org. Chem. 1983, 48, 2877.
solvent polarization and reoganization energy as expressed in eq 3.
dissoc ions (1)
either the contact ion pair (CIP) may interact with the solvent (S) to form a solvent-separated ion pair (SSIP) that further dissociates as solvated free ions or the contact ion pair will back-electron transfer to the ground state (GS).4 It is the latter process which exhibits the most systematic and dynamic shift with changing solvent and is the center of interest of this communication. Goodman and Peters4 have observed that the condition in nonpolar solvents in k,, > kips and that the magnitude of k,, approaches kipsas the polarity of the solvent increases. In support of that conclusion a seemingly linear correlation between In k,, vs. ET (Dimroth polarity parameter) was noted.5 However, their reported correlation coefficient of 0.971 ( N = 9) appears to be fortuitous since a reexamination of calculated standardized residuals ( e i / s )as well as the experimental uncertainties in In k,, lead to other conclusions. As shown in Figure 1, a linear function for In k,, vs. E T is an inappropriate model even though none of the standardized residuals exceeds the limiting interval of f 1.96 associated with a normal distributiom6 This quantitative failure of the ET parameter most likely arises from the Dimroth scale being a nondifferentiating scale formed from the variable mixing of at least two solvent characteristics, i.e., dipolarity and hydrogen bond donor acidity.' Since the solvents used as S in eq l'have included the full range from the nonpolar (benzene) to strongly polar aprotics (Me2SO) and from the weak hydrogen bonding (acetonitrile) to the strong hydrogen bond donating (HBD) (ethanol), the correlation of ET values with experimental solvent influences on the TS/FN reaction scheme can be only qualitative at best.
(2) (3) (4) (5) (6)
Results and Discussion In seeking other more satisfactory theoretical models for the solvent effects on electron-transfer reactions like that for TS/FN, one usually returns to the predictions from the Marcus relationship as given in eq 2. Here vet is the frequency factor and A, is the
A G O is the Gibbs free energy change for the electron transfer within the ion pair along the line of centers (DA), and like A, it exhibits solvent influences. The solvent dependence for X, is related to the hard-sphere radii ( r ) of the electron donor and acceptor along with the donor-acceptor distance IDA, the static dielectric constant (e,), and the index of refraction (n,) of the s o l ~ e n t . ~ + ~ However, since the A, function has the same algebraic form as the Born solvation energy, it follows that eq 3 will carry the same limitations as similar f(es,ns2)relationships, i.e., best for fitting solvent dipolarity-polarizability factors to r, AGO, and k,, in polar and nonpolar aprotic media but unreliable for highly structured and HBD solvents.lO~" Trial correlations between In k,, and variousf(c,) factors including the Born function in (e, - 1)/(2e, 1) are poor, with correlation coefficients below 0.78. To deal with such specific solvation effects not included in the Marcus theory, LayI2 has proposed replacing the single reorganizational energy term X, by an additive series and for the electron transfer by the C I P for TS/FN; this would take the form of eq 4. The final term (&)i is used here as a composite one including A', = + AH X p (A,)i (4) solvent-dependent inner-sphere reorganizational effects as well as those that are intrinsic (in the absence of solvent).12 Then, X, retains the dielectric continuum model assumed in eq 3 and represents only one of the contributions to the outer-sphere reorganizational energy along with those assigned to polarity (A,) and changes in hydrogen bonding (A,) during the electron-transfer process. Although eq 4 formalizes an improved physical model for the solvent influences on the net reorganizational energy, only approximate and indirect estimates of the relative magnitudes of the nonclassical terms in eq 4 can yet be made.I2 However, it should be noted that A, should be greater than XH in electron
+
+ +
~~
(8) Miller, J.; Beitz, J.; Huddleston, R. J . Am. Chem. SOC.1984, 106, 5057.
(9) Brunschwig, B.; Ehrenson, S.;Sutin, N. J . Am. Chem. SOC.1984, 106, 6858.
(10) Brady, J.; Carr, P. J. Phys. Chem. 1984, 88, 5796. (1 I ) Chastrette, M.; Rajzmann, M.; Chanon, M.; Purcell, K. J . Am. Chem. SOC.1985, 107, 1 . (12) Lay, P. A. J . Phys. Chem. 1986, 90, 878.
0022-3654/86/2090-4664$01 SO10 0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 19, 1986 4665
Solvent Effects on Back-Electron Transfer
Figure 2. Scatter diagram comparing the experimental In k,, (I) with the calculated In k,, (11) obtained from eq 6 .
Figure 1. Trend in the standardized residuals (e/s) for In kd as a function of the ET value of the solvent. The upper portion is the initial scatter diagram showing uncertainties in the In k, data based on T , .
transfer within the C I P according to the Lay model.12 As another alternative, one may isolate the contribution from hydrogen bonding by the solvent as an additive perturbation term in a free energy statement, and following the FiguerasMcRae formalism,13 the total solvent effect on the free energy change for the electron transfer may be represented by the sum in eq 5. Since this in its net effect is a linear free energy function
having AG,,O as the reference state intercept, appropriate semiempirical solvent parameters representing the dipolarity-polarizability component and the hydrogen bond donor function can be substituted into eq 5 . Instead of using the Marcusf(A,) for f(c,,n,2), the Kamlet-Taft function appears to be a better choice because of the large data base used to derive reliable values for a range of solvent types7 and the fundamental support provided for its connection with f(e,,n:) by the various modifications of reaction field theory developed by Brady and C a d o as well as by Abboud and Taft.14 Similarly, the a-scale of Kamlet and Taft can be substituted forf(AH). When the LFE function is written as a “linear solvation energy relationship”,’ the specific free energy function for the electron-transfer step for T S / F N is as follows: In k,, = 4 . 5 9 ( ~ *- 0.386)
+ 3.87a + 17.72
(6)
The normal computational sequence was used to establish the regression, Le., initial correlation with T* and then a for the HBD solvents and finally including the minor d6 term as the only adjustable and optimized parameter.’ Based on the original data of Goodman and Peters4q5and the Kamlet-Taft parameters,’ eq 6 conforms to the usual statistical tests for linearity and has a correlation coefficient of 0.986 ( N = 9) for the uniform scatter diagram shown in Figure 2. (The Polarizability correction in eq 6 assumes a 6 value of unity for all aromatic solvents.) The uncertainty in In k,,(calcd) is i0.22(SD) and falls well within the interval of the stated experimental uncertainty in kct.5 If one assumes a value of 1.31 kcal/mol for the GS-CIP energy separation corresponding to the emission A,,,,,(TS/FN) in benzene and chloroben~ene,~ then the net decrease d(AG,) in the free energy gap can be recalculated for select pairs of solvents with the conditions stipulated by eq 6. For the case of benzene and (13) Figueras, J. J . Am. Chem. SOC.1971, 93, 3255. (14) Abboud, J.; Taft, R. J. Phys. Chem. 1979, 83, 412.
ethanol involving both d6 and a contributions, the newly computed decrease of 8.59 can be compared to the 7.5 kcal/mol reported by Goodman and Peterss In addition it is noteworthy that the reduction in the GS-CIP free energy gap by 3.52 kcal/mol is now predicted for the benzene-dimethoxyethane pair, and since substantial salt effects were found in dimethoxyethane, the solvent influence here becomes simpler when AH = 0 as well as much smaller in magnitude than in ethanol. For the LFE function in eq 6, the intercept (AG,,’) refers to cyclohexane as the standard medium. From the ratio of 1.2/1.0 for the coefficients of the polarity and hydrogen bond donor acidity terms in eq 6, it is clear that dipolarity of the solvent species is still the dominant variable decreasing the free energy separation between the contact ion pair and the ground state for TS/FN in all nonaqueous media, and this parallels the observation that A, > AH. It is also now clear that HBD acidity significantly accelerates the back-electrontransfer step as k,, N k,, for the polar alcohols and nitriles, an effect not accurately anticipated by the qualitative trends in ET and &.. Previously linear solvation energy relationships have been useful in the interpretation of solvent influences on values for the less complex and slower reductions of organometallic species at the DME in polar s01vents.I~ It is probable that a similar resolution of solvent influences into dipolarity and hydrogen bonding components can be made for the observed changes in the electron-transfer rates for porphyrin-quinone coupled molecules in nonaqueous solvents.i6 For the latter, the roughly estimated ratio for the coefficients of r * and a is larger (-2.1/ 1 .O) and lends support to the argument above the hydrogen bonding by the solvent makes the smaller but yet significant contribution to solvent reorganizational energetics determining the reaction rate for the back-electron-transfer process. One final point can be made concerning the relationship of k, to the Hildebrand solubility parameter .,6 Kamlet et aL7 have concluded that h6, corresponds to a differential cavity term in free energy correlations with kinetic rate constants, and sitace the volume change for the ion pair during the back-electron-transfer process would be expected to be minor, the In k,, should exhibit little dependence on h6H for “select solvents”. Although only limited test data are available for the T S / F N system in polar aprotic solvent^,^ the cases of acetone, dimethoxyethane, and dimethyl sulfoxide (with T* values ranging from 0.53 to 1.00) confirm that such a cavity term in bH is indeed a minor variable ( h = 0.3) when compared to the other factors identified by eq 6. Registry No. TS,103-30-0; FN, 764-42-1. (15) Kolling, 0. Anal. Chem. 1982, 54, 260. (16) Leland, A,; Joran, A. D.; Felker, P. M.; Hopfield, J. J.; Zewail, A. H.; Dervan, P. B. J . Phys. Chem. 1985,89, 5571.
