J . Phys. Chem. 1986, 90, 6563-6570
6563
Role of Solvent Reorganization Dynamics in Electron-Transfer Processes. Anomalous Kinetic Behavior in Alcohol Solvents George E. McManis, M. Neal Golovin, and Michael J. Weaver* Department of Chemistry, Purdue University, West Lafayette, Indiana 47907 (Received: May 8, 1986)
Electrochemical rate constants are reported for electron exchange of Cp2Co+/o(Cp = cyclopentadienyl) and (COT)Fe(CO)?l(COT = q4-cyclooctatetraene)in 13 organic solvents at mercury electrodes, with the objective of probing further the influence of solvent dynamical properties on the electron-transfer kinetics. Comparisons are made between the solvent-dependent rate constants and the theoretical predictions based on a dielectric continuum treatment of overdamped solvent relaxation. While most solvents, including amides and nitriles, yielded relative rate constants that are approximately as predicted by the solvent dynamical model, the observed rate constants for both reactions in methanol, ethanol, and 1-propanol are at least ca. SO-fold larger than anticipated on this basis. Likely reasons for this anomalous behavior in the alcohol solvents are discussed, including the influence of high-frequency dielectric relaxation associated with rotation of solvent monomers and with translational motion of solvent dipoles.
There is currently rapid development occurring in theoretical treatments of solvent effects upon the barrier crossing frequency for simple electron transfer as well as for other charge-transfer processes.’ This theoretical activity has given rise to a number of experimental studies aimed a t testing the underlying physical model^.^,^ The experimental tests so far are chiefly of two broad types: measurements of time-dependent fluorescence for, presumably, excited-state intramolecular electron (or charge) transfer2 and solvent-dependent rate parameters for outer-sphere electron-exchange reactions in homogeneous solution and at metalsolution interface^.^ While the former can in principle provide a more direct real-time probe of solvent dynamics, the latter processes are of broad fundamental interest as well as being free from driving-force effects. In connection with our ongoing interest in unraveling structural and environmental factors in electron-transfer reactions, particularly for electrochemical systems, we have recently been undertaking measurements of their solvent-dependent kinetic^.^"*^,^ We have selected as reactants simple metallocene redox couples, in particular of the type Cp2M+/0 (where Cp = cyclopentadienyl and M = Co, Fe, or Mn). Their virtues include the presence of only small or negligible inner-shell (Le., bond distortional) barr i e r ~ the , ~ ~presence of surprisingly small double-layer (Le., work-term) effects upon the electrochemical rate constants$b and electrochemical behavior that is uncomplicated by coupled chemical steps in a wide range of protic as well as aprotic media. Measurements of rate constants for electrochemical exchange, k,, (i.e., “standard” rate constants) in polar nonaqueous solvents have been made by means of phase-selective ac polarography, primarily at the dropping mercury electrode (dme), in order to ~
(1) (a) Calef, D. F.; Wolynes, P. G.J . Phys. Chem. 1983,87, 3387. (b) Calef, D. F.; Wolynes, P. G.J . Chem. Phys. 1983, 78, 470. (c) Alexandrov, I. V. Chem. Phys. 1980,51, 499. (d) Zusman, L. D. Chem. Phys. 1980,49, 295. ( e ) Friedman, H. L.; Newton, M. D. Faraday Discuss. Chem. Soc. 1982, 74, 73. ( f ) van der Zwan, G.; Hynes, J. T. J . Chem. Phys. 1982, 76, 2993. (9) van der Zwan, G.; Hynes, J. T. Chem. Phys. Lett. 1983, 101, 367. (h) van der Zwan, G.; Hynes, J. T. J. Phys. Chem. 1985,89,4181. (i) Sumi, H.; Marcus, R. A. J . Chem. Phys. 1986,84,4272, 4894. (2) For example: (a) Kosower, E. M.; Huppert, D. Chem. Phys. Lett. 1983, 96, 433. (b) Kosower, E. M.; Huppert, D. Annu. Rev. Phys. Chem., in press. (3) (a) Weaver, M. J.; Gennett, T. Chem. Phys. Lett. 1985.113, 213. (b) Gennett, T.; Milner, D. F.; Weaver, M. J. J. Phys. Chem. 1985,89, 2787. (c) Harrer, W.; Grampp, G.; Jaenicke, W. Chem. Phys. Lett. 1984, 112, 263. (d) Kapturkiewicz, A,; Opallo, M. J . Electroanal. Chem. 1985, 185, 15. (e) Grzeszczuk, M.; Smith, D. E. J . Electroanal. Chem. 1986, 198, 245. ( f ) Opallo, M.; Kapturkicewicz,A. Electrochim. Acta 1985,30, 1301. (9) Opallo, M. J . Chem. Soc., Faraday Trans. 1 1986, 82, 339. (4) (a) Farmer, J. K.; Gennett, T.; Weaver, M. J. J . Electroanal. Chem. 1985, 191, 357. (b) Gennett, T.; Weaver, M. J. J . Electroanal. Chem. 1985, 186, 179. (c) Sahami, S.; Weaver, M. J. J. Electroanal. Chem. 1981, 124, 35.
0022-3654/86/2090-6563$01.50/0
achieve reproducible and well-defined interfacial condition^.^^,^ Prominent in our original studies is the finding that reconciliation of the experimental rate parameters with the theoretical predictions for outer-sphere processes requires the inclusion of a solvent-dependent preexponential factor. This term was f o ~ n d ~ ~ , ~ to be roughly consistent with the outer-shell (Le., solvent) nuclear frequency factor, vas, as determined from solvent relaxation time data using a dielectric continuum model in the so-called “overdamped” limit. l a The present report contains an extension of these studies for C P ~ C O +to/ ~several other solvents, including methanol, ethanol, and 1-propanol, and to the redox couple tricarbonyl(q4-cyclooctatetraene)iron(O/ 1-) [(COT)Fe(CO),O/-]. The alcohols are of particular interest in comparison with amide, nitrile, and other aprotic solvents also examined here in view of the presence of multiple dielectric relaxation behavior for the first type;5 a recent theoretical study suggests that the faster relaxation modes may provide disproportionately large contributions to v , . ~ While the (COT)Fe(C0)30/- couple is not as structurally simple as the metallocenes, the kobsd values for the former are significantly smaller. This eases considerably the task of determining accurate electrochemical rate data using phase-selective ac polarography, enabling reliable kokd values to be determined for a wider range of solvents. The C P ~ C O +couple /~ was selected for further study in view of its simple structure, the ready solubility of the cobalticinium cation in a range of solvents, and its suitability for examination a t the dme. The solvent-dependent rate data presented here indicate that the alcohols exhibit surprisingly rapid solvent dynamics for both C ~ , C O +and / ~ (COT)Fe(CO),O/-.
Experimental Section Dimethyl sulfoxide, dimethylformamide, dimethylacetamide, propylene carbonate (Burdick & Jackson), N-methylformamide, formamide, and tetramethylurea solvents (Fluka) were used as received. Acetonitrile, methylene chloride (Burdick & Jackson), 1-propanol (Baker), and benzonitrile (Fluka) were distilled over calcium hydride, the last in vacuo. Methanol (Burdick & Jackson) and ethanol (US)were distilled over activated magnesium turnings. These solvent purification procedures generally followed Perrin et aL8 Cyclooctatetraeneiron tricarbonyl was synthesized as described in ref 9a. Cobalticinium hexafluorophosphate was obtained from (5) (a) Garg, S . K.; Smyth, C. P. J . Phys. Chem. 1965, 69, 1294. (b) Saxton, J. A.; Bond, R. A,; Coats, G . T.; Dickinson, R. M. J . Chem. Phys. 1962, 37, 2132. (6) Hynes, J. T. J . Phys. Chem. 1986; 90, 3701. (7) Tulyathan, B.; Geiger, W. E., Jr. J . Electroanal. Chem. 1980, 109, 325. (8) Perrin, D. D.; Armarego, W. L. F.; Perrin, D. R. Purification of Laboratory Chemicals, 2nd ed.; Pergamon: New York, 1980.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, NO. 24, 1986
6564
Strem Chemicals. The fluoroborate salt was employed in some solvents, including the alcohols, in view of its greater solubility. This was prepared by oxidizing an ethereal solution of cobaltocene (Strem) with tetrafluoroboric acid (Alfa) a t room temperature and precipitating the product a t 0 O C over a I-h period. Tetraethylammonium perchlorate (TEAP, G. F. Smith) was recrystallized twice from water. n-Tetrabutylammonium fluoroborate (TBAFB, Eastman Kodak) was recrystallized twice from acetone-water. n-Tetrabutylammonium hexafluorophosphate (TBAH) was prepared from tetrabutylammonium iodide (Eastman Kodak) and ammonium hexafluorophosphate (Ozark-Mahoning) in acetone. The observed “standard” rate constants, koM, for both CP,CO+/~ and (COT)Fe(CO),O!- in each solvent were determined by using phase-selective ac polarography largely as outlined in ref 3b and 10. This utilized a PAR 173/179 or PAR 273 potentiostat, a PAR 175 potential programmer, a PAR 5204 lock-in amplifier with an IEC F52 a c function generator, and a Fluke 1900A frequency counter. In-phase and quadrature currents were displayed as a function of potential for ac frequencies, between 100 and 1000 Hz (peak-to-peak amplitude 8 mV) using a H P 7045B X-Y recorder. Sufficiently small lock-in amplifier time constants (0.03 and 0.1 s) were employed so to minimize damping effects on the measured currents a t the end of drop life. The dme had a flow rate of 0.5 mg s-’ with mechanically controlled drop times between 2 and 4s. The ratio of the in-phase to quadrature currents. corrected for the residual currents on either side of the a c polarographic peak. yielded the required frequency-dependent cot values. The slope of the resulting cot 4 - ul/, plots combined with the diffusion coefficients obtained from the dc polarograms (see Results) and transfer coefficients, CY,provide a simple and direct route to the ,~~ IR required k,, values.Iob As p r e v i ~ u s l y positive-feedback compensation was employed in order to eliminate as completely as possible the influence of solution resistance, R,. However, as outlined elsewhere,10a,” the effect of residual uncompensated resistance, R,,, can be significant for systems displaying rapid kinetics, especially in aprotic media. and can provide an upper limit to the reliable determination of k,, using a given experimental arrangement. Since this condition commonly applies for Cp2Co+/’ (and other m e t a l l o ~ e n e s )it, ~is~ necessary to establish procedures by which the influence of R,, upon the kinetic measurements can be both diagnosed and incorporated into the kinetic data analysis. The primary procedure employed here is outlined in ref 10a. This entails applying the usual analysis based on the cot 4 - u l / * slopelobto find an “apparent” value of the observed rate constant, kObsd(app).For rate constants that approach the measurement limit. k,b,d(app) will be systematically smaller than the actual (“true”) rate constant, kobsd, by a factor that depends upon R,,, the reactant diffusion coefficient, D.and to a lesser extent on the double-layer capacitance, Cdl.loa The appropriate correction was determined by digitally simulating polarograms using trial values of kobsd along with known values of C,,, D,and estimates of R,, so to find a kobSd(app)value that matches that obtained from the cot 4 - u i j 2slope over the given frequency range (100-IO00 Hz). Approximate estimates of R,, for this purpose were obtained in two ways. The first involved employing various RC “dummy cells” in place of the electrochemical cell. with R, and Cd]values that mimicked the measured values; R,, was found from the in-phase and quadratic currents, I, and Io. obtained for the optimal IR compensation setting by using R,, = EZ,/(I, + Io),. where E is the root-mean-square amplitude of the ac potential. (This procedure could not readily be employed with the electrochemical cell since the I , values a t the optimal compensation settings were sufficiently small so to be obscured by the presence of Faradaic currents associated with trace impurities.) The resulting R,, estimates varied somewhat with R, but lie in the range 5-1 5 Q ~~
~~~
R.A,; Stone, F. G . A. J . Am. Chem. SOC.1960, 82, 366. (b) Dickens, B.; Lipscomb, W. N. J . Chem. Phys. 1962, 37, 2084. (IO) (a) Milner, D. F.; Weaver, M . J. J . Elecrroanul. Chem. 1985, 191, Weaver, M. J. Anal. Chem. 1984, 56, 1444. 411. (b) Gennett. T.; ( I 1 Cfilner, D. F.; Weaver, M . J., submitted to J . Electroanai. Chem. (9). (a) Manuel.
McManis et al.
-
for R, 500-2000 Q. The second method utilized the polarographic response for C ~ , C O +in/ ~methylene chloride containing 0.1 M Bu4NBF4. This system almost certainly yields k,, values that lie beyond the upper measurement limit of the present techniqut. (k,,,, > 5 cm s-I 3b). given the high solvent resistance (typically R, 2500 2). This enables R, estimates to be obtained directly from the cot I#J - u I / , slopes. These two methods yielded concordant results. The protocol employed to achieve optimal IR compensation is clearly critical; this involved gradually increasing the resistance setting for an electrode potential 200-250 mV either side of the polarographic peak until the in-phase current at the end of drop life is as small as possible, just below the point where potentiostat oscillation occurred. Some measurements were made with a PAR 273 potentiostat in place of the PAR 173/179. Although significantly smaller R,, levels could be achieved for the former when R, < 1 kQ, stable IR compensation could not be achieved for higher R, values. The PAR 173/179 was therefore preferred for most measurements. The cell configuration employed included a platinum wire counter electrode and an aqueous saturated calomel electrode (SCE), the latter being placed in a separate compartment separated by a glass frit. Although this arrangement necessarily entailed substantial reference electrode resistances, essentially the same results were obtained with a silver wire “quasi-reference’’ contained in the working compartment instead of the SCE. All measurements were made a t 23 & 1 “C and all potentials were measured and are reported vs. the aqueous SCE.
-
Results Table I summarizes electrochemical rate and other pertinent data for the CP,CO+’~and (COT)Fe(CO),O/- couples in 13 solvents, comprising the alcohols methanol (MeOH), ethanol (EtOH), and 1-propanol (1-PrOH); the amides formamide, Nmethylformamide (NMF), N,N-dimethylformamide (DMF), and tetramethylurea (TMU); the nitriles acetonitrile and benzonitrile, dimethyl sulfoxide (Me,SO); propylene carbonate (PC); and methylene chloride (CH,CI,). The dielectric and related properties of these solvents relevant to the present purpose are summarized in Table 11. (Attempts to measure kobsd for C ~ , C O +in/ ~aqueous media were thwarted by erratic current-time curves apparently due to precipitation of electrogenerated Cp,Co on the mercury surface.) For Cp,Co+’O, rate data are reported for only a single electrolyte since, as noted previously, the parameters are almost independent of the electrolyte composition.Jb However, (COT)Fe(C0)30’- exhibits the usual sensitivity of kobsd to the supporting electrolyte cation as well as to the ionic strength seen in nonaqueous media for anionic couples.4b Values of kobsd are therefore given for this couple in both 0.1 M tetraethylammonium and tetrabutylammonium electrolytes. The kobsdvalues in Table I were obtained from the kobSd(app)values listed along side by correcting for the residual uncompensated resistance as described above. These data for C ~ , C O +differ / ~ slightly from those reported earlier.jb Transfer coefficients, cy, required in order to extract k,,,,(app) values from the cot 4 - u’‘’ slopes were obtained from the dc polarographic half-wave potentials, E,,, (Table I), in comparison with the potentials a t which cot 4 reaches a maximum;Iob generally a = 0.5 f 0.05. The diffusion coefficients, D, also required for the kinetic analysis were obtained from the cathodic dc polarograms. I n aprotic media, the (COT)Fe(C0)3 system displays two cathodic dc polarographic waves, separated by 200-250 mV, corresponding to the formation of the monoanion and dianion, re~pectively.~ The a c amplitude of the latter, however, is markedly smaller than the first in most solvents, indicating the occurrence of an irreversible following chemical step; this was supported by the virtual absence of an anodic return peak corresponding to the second cathodic peak on the cyclic voltammograms. In protic media (formamide, N M F , and the alcohols), the second step became entirely irreversible, being absent on the ac polarograms, and merged into the first dc polarographic wave so that the height of the latter corresponded to a net two-electron process. Thus the diffusion coefficients calculated from the dc and ac polarographic currents
The Journal of Physical Chemistry, Vol. 90, No. 24, 1986 6565
Solvent Reorganization Dynamics TABLE I: Summary of Electrochemical Kinetic and Related Data solvent' electrolvteb E l ,: mV vs. SCE
acetonitrile CH2C12 formamide NMF DMF DMA TMU Me2S0 benzonitrile PC MeOH EtOH 1-PrOH
TEAP TEAP TEAP TEAP TEAP TEAP TEAP TEAP TEAP TEAP TBAFB TBAFB TBAFB
D,d cm s-l cp2co+/o
-900 -900 -1050 -920 -890 -810 -730 -900 -840 -970 -950 -900 -875
knhd(auu),' cm s-'
3.3 1.35 0.22 0.5 0.8 0.7 0.35 0.22 0.32 0.3 1.3 0.35 0.25
knkA/cm s-I
-5
kzF.g cm s-I
35 >5 0.6 0.45 2.0 2.0 0.35 0.75 0.6 1.o -6 34
1.5 0.35 0.35 1.3 1.25 0.25 0.5 0.4 0.55 2.6 0.6 -0.2
>I
(COT)Fe(C0)30/acetonitrile
TEAP -1320 1.45 1.6 2 17 TBAH -1340 1.5 0.95 1.1 9.5 NMF TEAP -1 300 0.4* 0.7 1.1 5.5 TBAH -1300 0.4* 0.18 0.19 1.o DMF TEAP -1 160 1.05 0.58 0.7 6 TBAH -1 180 1.15 0.15 0.16 1.5 -1 170 0.5 0.33 0.5 (4) TMU TEAP TBAH -1 160 1.o 0.07 0.07 (0.6) 0.35 0.34 0.45 4 Me2S0 TEAP -1210 TBAH -1 190 0.15 0.06 0.07 0.6 PC TEAP -1330 0.25 0.24 0.35 2.5 TBAH -1290 0.2 0.09 0.10 0.7 EtOH TBAFB -1350 0.88* 3.1 >3 >30 1-PrOH TBAFB -1410 0.35* 0.35 >2 (>20) NMF = N-methylformamide, DMF = N,N-dimethylformamide, DMA = N,N-dimethylacetamide, TMU = tetramethylurea, Me2S0 = dimethyl sulfoxide, PC = propylene carbonate, MeOH = methanol, EtOH = ethanol, and I-PrOH = I-propanol. bElectrolytes were 0.1 M in each case. TEAP = tetraethylammonium perchlorate, TBAH = tetrabutylammonium hexafluorophosphate, and TBAFB = tetrabutylammonium fluoroborate. 'Dc polarographic half-wave potential for redox couple vs. SCE in indicated solvent and supporting electrolyte. For some solvents values reproducible only to ca. 20-40 mV due to variable liquid-junction potentials. dDiffusion coefficient in medium indicated for oxidized forms of redox couple, obtained from dc polarographic limiting current. For systems marked with asterisk, values obtained by assuming that polarographic wave involved a net two-electron process (see text). e"Apparent" observed electrochemical rate constant for electron exchange at mercury-solvent interface, obtained from frequency-dependent phase-selective ac polarography with positive-feedback IR compensation but without correction for residual uncompensated solution resistance (see text). 'Observed electrochemical rate constant for electron exchange, obtained from kOhd(app)by correction for residual uncompensated solution resistance (see Experimental Section for details). 9 Double-layer corrected electrochemical rate constant for electron exchange, obtained from eq 1 using Z,, = 0 and a,,, = 0.5, and 4:' values extracted with the Gouy-Chapman model and electrode charge-potential (4" - E ) data extracted from potential of zero charge (pzc) and capacitance-potentia1 values for 0.1 M electrolytes given in the following references: acetonitrile, ref 4c, 15; NMF DMF, ref 4c, 16; DMA, ref 16; Me2S0, PC, ref 4c, 17; EtOH, ref 18 (pzc determined to be ca. -250 mV vs. SCE in present work). For TMU and 1-PrOH for which double-layer data are unavailable, 4"' - E data assumed to be similar to those for DMF and EtOH, respectively; hence k,, values for these solvents given within parenthesis. [Although published double-layer data mostly refer to electrolytes other than those employed here, this leads to negligible errors in the double-layer corrections since the 4'" - E curves are insufficiently sensitive to the supporting electrolyte cation over the range of electrode charges (-4" 0-12 pC of interest here.19]
-
TABLE 11: Solvent Dielectric and Related Properties at 25 'C
solvent acetonitrile CH2C12 formamide NMF DMF DMA TMU Me,SO benzonitrile PC MeOH EtOH 1-PrOH
1.80 2.03 2.09 2.04 2.04 2.06 2.10 2.18 2.33 2.029 1.76 1.85 1.92
c?
emc
103 q,d P
37.5 9.0 110 182 36.7 37.8 23.1 46.7 25.6 6 58 32.7 24.5 20.4
2h.i -2.5' 7.0' 5.4 4.Y -4.5' -4.5' 5.7) 3.9k 4.19 5.6' 4.2' 2.2m
3.5 4.1 33.9 16.5 8.0 9.2" 14 22.4 12.2 25.3" 5.2 10.8 20
10-12
TD,e
3.3h 1.5" 37j 129 11.0' 12.V 314 19.9 38k 4 38 48' 130' 390"'
s
10-12
T , / S
-0.2 -0.4 2.35 3.7 1.3 1.5 -6 2.4 5.8 2.7 8.2 22 42
'Optical dielectric constant, from refractive index data given in ref 20 unless otherwise noted. bStatic dielectric constant, from ref 20 and 21 unless otherwise noted. C"Infinite frequency" dielectric constant, from sources noted. dViscosity, from ref 20 or 21 except where noted. CDebye relaxation time, from references cited. /Longitudinal solvent relaxation time, determined from corresponding values of q,.es, and c, by using eq 4a. #Reference 22. *Reference 23. 'Estimated value. 'Reference 24. 'Reference 25. 'Reference 5b and 26. "Reference 5a. "Reference 27. Reference 28. P Reference 29. 4 Reference 30.
for (COT)Fe(C0)3 reduction in protic media are in agreement if they are assumed to refer to net two- and one-electron processes, respectively. The validity of this procedure was confirmed by the observation of a n approximate correlation between D and v-l, where 7 is the solvent viscosity (Table 11), as predicted by the
Stokes-Einstein equation. A n exception is methanol, for which the ac polarographic currents are markedly (ca. fivefold) smaller than expected on this basis, indicating that the first cathodic wave also experiences a n irreversible coupled chemical step in this solvent. Kinetic data for (COT)Fe(C0)30/- in methanol are
6566
The Journal of Physical Chemistry, Vol. 90, No. 24, 1986
therefore omitted from Table I. These coupled chemical steps apparently involve reaction with solvent protons.I2 It is clearly desirable to correct the observed rate constants for the effects of the electrostatic double layer. This is conventionally accomplished by means of the coupled Gouy-Chapman-Frumkin (GCF) treatment, whereby a “double-layer corrected” rate constant, k,,, can be obtained from /cow at a given electrode potential byI3
where Z,,is the charge number of the oxidized form of the redox couple, acoris the double-layer corrected cathodic transfer coefficient, and dCcis the potential across the diffuse layer as estimated from equilibrium electrode charge-potential (qm- E ) data using the Gouy-Chapman model. One difficulty with this approach is that it fails to account for the sensitivity of kohsd for anionic redox couples in nonaqeuous media to the nature of the supporting electrolyte cation, especially at negative potential^.^^.^^ This is reflected in the k,, values for (COT)Fe(C0)30/- listed in Table I obtained from the corresponding koM values measured in 0.1 M tetraethylammonium and 0.1 M tetrabutylammonium electrolytes. (The sources of double-layer data required for these corrections are listed in the footnotes to Table I.) Thus the markedly (ca. 4- to IO-fold) larger koM values seen in the former electrolyte are also reflected in the corresponding k,,, values since the extent of the double-layer corrections according to eq 1 is almost independent of the supporting electrolyte cation.4h The magnitude of these corrections is also comparable in all the solvents in Table I, since the electrode charges, qm,corresponding to the E , j 2values all lie in the region -10 to -13 pC cm-2, and dGCis insensitive to qm and the solvent dielectric constant under these conditions. A recent analysis based on a comparison of cationic double-layer effects for C ~ , C O +and / ~ Cp2Coo/-in aprotic media suggests that the “true” double-layer corrections for the latter anionic redox couple are intermediate between the k,,, values extracted from rate constants in tetraethylammonium and tetrabutylammonium electrolyte^.^^ In other words, the magnitude of the G C F correction appears to be too large for the former and too small in the latter electrolyte. However, given that the magnitude of the correction is unlikely to vary markedly with the solvent, at least the relative values of k,,, for (COT)Fe(CO)30’- for a given electrolyte probably reflect the corresponding solvent dependencies of the true double-layer corrected values. In contrast, CP,CO+/~and other Cp2M+/0couples display little or no sensitivity to the interfacial composition, indicating that the double-layer corrections upon kohsdare much smaller than given (12) Murr. N. El.; Riveccie. M.; Laviron. E. Tetrahedron Letr. 1976. 37. 3339. (13) For example: Weaver, M. J . J . Electroanal. Chem. 1978, 93, 231. (14) (a) Baranski, A,; Fawcett, W. R. J . Electroanal. Chem. 1977, 100, 185. (b) Corrigan, D. A,; Evans, D. H. J . Elecrroanal. Chem. 1980, 106, 287. ( 1 5 ) Fawcett, W. R.: Ikeda, B. M.; Sellan, J. B. Can. J . Chem. 1979. 56. 2268. (16) Payne, R. J . Phys. Chem. 1969, 7 3 , 3598. (17) Payne, R. J . A m . Chem. SOC.1967, 89, 489. (18) Minc, S.; Jastrzebska, J. J . Electrochem. Soc. 1960, 107, 135. (19) Peover, M. E. In Reactions of Molecules at Electrodes: Hush, N. S., Ed.; Wiley: New York, 1971; p 259. (20) Riddick, J . A.; Bunger, W. B. Organic Soluenrs; Wiley-Interscience: New York, 1970. (21) Janz, G. J.; Tomkins, R. P. T. Nonaqueous Electrolytes Handbook; Academic: New York, 1972; Vol. 1. (22) Cavell, E. A. S. J . Chem. Soc., Faraday Trans. 2 1974, 70, 78. (23) Eloranta, J. K.; Kadaba, P. K. Trans. Faraday Soc. 1970, 66, 817. (24) Behret, H.; Schmithals, F.; Barthel, J. Z . Phys. Chem. 1975, 96, 73. (25) Poky, J. P. Appl. Sci. Res., Sect. E . 1955, 4, 337. (26) Lane, J . A.; Saxton, J . A. Proc. R . Sac. London, A 1952, A213,400. (27) Covington, A. K.; Dickinson, T. In Physical Chemistry of Organic Solvent Sysrems; Covington, A. K., Dickinson, T., Eds.; Plenum: New York, 1973; Chapter 1. (28) Evans, M. W.; Ferrario, M. Ada. Mol. Relax. Processes 1982, 23, 113. (29) Brownsell, V. L.; Price, A. H . J . Phys. Chem. 1970, 7 4 , 4004 (30) Hoigne, J.: Gaumann, T . Hela. Chim. Acra 1958, 41, 1933.
McManis et al. I
Ma OH
/ OEN
DMSO
0
NMF
0
TMU
-0 5 -
I
where K p is an equilibrium constant for forming the precursor state prior to electron transfer, ti,, is the electronic transmission coefficient, Y , is the nuclear frequency factor, and AGO,*and SI,* are the outer-shell (solvent) and inner-shell (reactant bond distortion) barriers associated with the electron-transfer step. The focal point of attention here is in identifying and understanding the manner and extent to which solvent dynamics influence Y , by analyzing the solvent dependence of k,,,. For this purpose it is clearly necessary to estimate. at least approximately, the solvent dependence of the outer-shell barrier. This is most reliably obtained for outer-sphere electrochemical exchange reactions from the well-known relation3*
c( 1a - I Re ) ( ;
AGO,* = 8
-
t)
(3)
where topand t, are the optical and static solvent dielectric constants, a is the reactant radius, and Re is the reactant-electrode image distance. In the event that the dynamics of solvent reorganization do not influence the barrier-crossing frequency, the combined term tielun (31) Hupp, J. T.; Weaver, M. J. J . Electroanal. Chem. 1983, 152, 1. (32) Marcus, R. A. J . Chem. Phys. 1965. 43, 679.
Solvent Reorganization Dynamics
The Journal of Physical Chemistry, Vol. 90, No. 24, 1986 6567
TABLE III: Calculated Rate Parameters for Cp2Co+loin Various Solvents k&,d
AG,*,b solvent
10I2 v _ , ~s-’
kcal mol-
cm s-’
acetonitrile CH2C12 formamide
5
5.75 4.2 5.1 5.3 5.05 5.0 4.75 4.8 4.2 5.25 5.85 5.45 5.15
0.45 6.5 1.45 1.o 1.55 1.70 2.6 2.45 6.55 1.1 0.40 0.80 1.3
NMF DMF DMA TMU Me2S0 benzonitrile
PC MeOH EtOH 1-PrOH
-2 0.35 0.23 0.6 0.55 0.15 0.34 0.13 0.31 0.11 0.04 0.02
cm
s-l
1.O 6.5 0.26 0.12 0.45 0.45 0.20 0.4 1 0.42 0.16 0.022 0.016 0.013
Outer-shell solvent nuclear frequency factor according to overdamped continuum model, obtained from eq 4 using T~ values listed in Table I1 and AG,* values given in adjacent right-hand column. Intrinsic outer-shell free-energy barrier, obtained from e 3 by using the cop and cQ values listed in Table I1 and taking a = 3.8 and Re m (see text). cCalculated “fixed frequency” rate constants for electrochemical exchange of Cp2Co*/oin solvent indicated, obtained from eq cm, and 2 by assuming that AGi,* = 0.25 kcal mol-’, K p ~ c=l 6 X Y, = 2 X 10l2 s-I in each solvent (see text) and from the AGO,*values given in the left-hand adjacent column. dCalculated rate constants for electrochemical exchange, obtained as indicated for kgidin footnote c but with V , set equal to the outer-shell frequency factor v, listed in far left-hand column.
1
-
in eq 2 will likely be independent of the solvent, so that according to eq 2 and 3 the solvent dependence of k,, should be dominated by the variations in top (and to a much lesser extent in ts since Figure 1 contains a comparison between log usually copi>> €[I). k,, in each solvent for C ~ , C O +(assumed /~ to equal kObdin Table I, vide supra) and the corresponding values calculated from eq 2 and 3, log kcal. Following the procedure in ref 3b, the latter were obtained as follows. The a and Re values required along with the cop and tSvalues (Table 11) in order to estimate AGO,* from eq 2 are assumed to equal 3.8 A and m, respectively. (The former is appropriate for C P ~ C O + 3b / ~the ; latter is tantamount to neglecting the influence of reactant-electrode imaging on the transition-state stability, vide infra.3b) The “effective electrontunneling” distance, Kflel,is assumed to equal 0.6 A;33,34 v, is taken to be 2 X 10I2s-l in each solvent.35 Most importantly, then, the preexponential factor is taken to be independent of the solvent. The resulting k,, values, labeled k:,, (for “fixed frequency factor”), are listed for each solvent along with the AGO,*values in Table 111. Inspection of the log kow - log kfialcdplot for C P ~ C O +(Figure /~ 1) shows that the solvent dependence of kow is utterly inconsistent with the theoretical predictions based on eq 2 and 3 since the rough correlation provided by the experimental points has a slope of opposite sign to the straight line corresponding to the theoretical anticipated behavior. A similar finding for a smaller range of solvents is noted in ref 3b. This suggests that either the solvent dependence of AG,* is quantitatively different than that predicted by eq 3 or that the net preexponential factor K p ~ , p ,depends markedly upon the solvent. The former possibility is rendered unlikely on the basis of the following arguments. First, while the solvent continuum model on which eq 3 is based ignores shortrange solute-solvent interactions and other specific molecular effects, we have recently d e m ~ n s t r a t e dthat ~ ~ their presence is liable to yield only small errors in the estimation of AG,* using eq 3.37 (33) Hupp, J. T.; Weaver, M. J. J . Phys. Chem. 1984, 88, 1463. (34) See footnote 42 of ref 3b. (35) This estimate of Y,, although somewhat arbitrary, was chosen since similar values are obtained by inserting typical estimates of Y, Corresponding to solvent rotation times (“solvent inertial” rela~ation)~) along with AGis*for Cp2Co*’0, 0.25 kcal mol-’,3balong with the inner-shell vibrational frequency, vis = 9 X s - ’ , ~into ~ the usual transition-state formula (eq 7 of ref 3b). (36) Hupp, J. T.; Weaver, M. J. J . Phys. Chem. 1985,89, 1601.
Second, the solvent-dependent absorption energies for optical electron transfer within a variety of symmetrical binuclear complexes in homogeneous solution,38including b i f e r r ~ c e n e s cor,~~ relate well with the dielectric continuum predictions. Similar results have been obtained in a variety of solvents, including the amides, nitriles, and alcohols of interest here.38.39These data are of particular importance since they provide direct evidence that the functional form of the reorganization energy as given by the continuum model is at least approximately correct in such solvents. Admittedly, there is a substantial uncertainty in the effective “scaling factor” (a-l - R;’) in eq 3, arising from imperfections of the continuum modelIb as well as from incomplete knowledge of Re. However, such factors will only act to alter somewhat the slope of the line shown in Figure 1; the qualitatiue nature of the discrepancy between the solvent dependence of koM and kgalcdwill remain irrespective of the scaling factor chosen. Consequently it is likely that the solvent dependence of /cobs,, is in part due to variations in the preexponential factor K,K,~Y,. For reactions having small or negligible inner-shell barriers, v, equals the “outer-shell” frequency factor, vas, that describes the overall dynamics of solvent motion. Several authors have outlined in detail quantitative relations that relate solvent dynamics to u,.la4 One simple relation for v, is obtained for electron-exchange processes from these treatments,Ia4 which can be expressed (4) where kB is the Boltzmann constant and T L is the longitudinal (or “constant charge”) solvent relaxation time. The latter quantity can be extracted from experimental values of the Debye relaxation time, T D , by ~ s i n g ~ ~ , ~ ’ TL = ( L / c s ) T D (4a) where em is the high (‘‘infinite”) frequency dielectric constant. Experimental values of E,, ts, 70, and hence rLare listed for each solvent in Table 11. The theoretical treatment leading to eq 4 treats the solvent polarization dynamics leading to successful barrier crossing as occurring by so-called “overdamped” solvent motion whereby the friction exerted by surrounding molecules causes the rotational motion of individual solvent dipoles to be severely impeded.Ia This situation corresponds to deviations from the usual transition-state theory (TST) case, since the system is obliged to cross and recross the barrier top rather than undergoing a smooth passage from reactants to product^.^.^^ With the possible exception of acetonitrile and methylene chloride, each of the solvents in Table I is anticipated to refer here to the “overdamped” 1 i m i t . l ” ~(For ~ ~ these two solvents, the friction exerted by surrounding molecules is sufficiently small that v, may be determined by the rotation time of individual solvent dipoles, corresponding to the “solvent inertial” v, should be controlled limit where TST a p p l i e ~ . ’ ~ Although .~~) by vis rather than v, when ACi,* is large, in the overdamped limit typically it is predicted that v, ‘v v, for ACis*5 1 kcal mol-’.3b Another assumption made in deriving eq 4 is that the barrier top forms a cusp, specifically that the energy region corresponding to curvature of the barrier top, Ifl2,is much less than kBT(“case B” in ref l a and lb). This therefore limits the strict applicability of eq 4 to nonadiabatic (or at least weakly nonadiabatic) processes, for which K,’ < 1, since H I 2equals the electronic matrix coupling element, and K , ~< 1 when Hi,
