Rate Dependence of Electron Transfer on Donor-Acceptor Separation

Rate Dependence of Electron Transfer on Donor-Acceptor Separation and on Free. Enthalpy Change. The Ru( bpy),2+/Viologen2f Systemt. H. Rau,* R. Frank,...
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2476

J. Phys. Chem. 1986, 90, 2476-2481

Rate Dependence of Electron Transfer on Donor-Acceptor Separation and on Free Enthalpy Change. The Ru(bpy),2+/Viologen2f Systemt H. Rau,* R. Frank,* and G. Greiner FG Physikalische Chemie, Institut f u r Chemie, Universitat Hohenheirn, 7000 Stuttgart 70, West Germany (Received: September 27, 1985)

By attachment of hydrocarbon chains of different lengths to the bipyridyl ligands in Ru(bpy)32+we have adjusted the donor-acceptor separation in the electron-transfer system Ru[(C,Hz,,)zbpy]~+/methylviologen. Two electron-transfer reactions with different AG are investigated in fluid solution: the quenching of the excited complexes by methylviologen (MV”) which is exergonic with -0.4 eV and the thermal back electron transfer which is exergonic with -1.7 eV. We observe an exponential decrease of the quenching rate on distance. The back electron transfer is independent of donor-acceptor separation; electron transfer is found to take place at distances of 1.5 nm and more. The results are discussed in terms of a hypothesis on the interdependence of transfer distance and free enthalpy change and compared with current theories. In the framework of the simple classical Marcus model, the Marcus equation relating transfer rate and free enthalpy change is transposed into the Rehm-Weller equation by simple mathematical manipulations and the implications of this are discussed.

1. Introduction

The rate of transfer of an electron from a donor molecule to an acceptor molecule is subject to various parameters. One is the donor-acceptor distance. Another is the change of free enthalpy during the transfer reaction. Several geometric configurations can be employed to test the dependence of the rate or efficiency of electron transfer between two partners on their separation. The monolayer and donor molecules may be separated in layered structures, e.g. in monolayer assemblies by fatty acid or in vesicles by the surfactant b i l a ~ e ror , ~ in molecules containing donor and acceptor moieties separated by a stiff or a flexible spacer D-SP-A.~.’ The monolayer assemblies and stiff donor-acceptor molecules eliminate the influence of diffusion and maintain a uniform distance. Another approach makes use of the concentration dependence of emission quenching of codissolved donor and acceptor molecules in rigid solutions.6-10 This configuration has no diffusional component either. The experimental results in monolayer and vesicle arrangements and in rigid solution indicate an exponential decrease of the transfer rate constant with D-A separation. This may be rationalized by an electronic tunnel mechanism in the monolayer assemblies* and by a kind of Perrin model of static quenching” in the frozen solutions. The donor-acceptor do not seem to conform to an exponential decay rate vs. distance relation. Numerous studies on the dependence of the rate of light-induced electron transfer on the free energy change are found in the l i t e r a t ~ r e , ~ ~ Jmany ~ J ~ -of’ ~them aiming at the question whether there should exist a Marcus inverted region15 at highly negative cases RehmAG values (see section 4.4). In most, but not Weller behavior1*is found, i.e. the bimolecular quenching rate constant reaches the diffusion limit at highly negative AG values. In these papers collisional quenching is assumed and consideration of changes in transfer distance is not given explicitly. As a fourth kind of geometric arrangement a fluid solution of acceptor and donor can be used in which the diameter of the donor has been changed by substitution. We have started a systematic investigation according to this principle and have made use of the well-known tris(bipyridyl)ruthenium/viologen system (Figure 1a). By substitution of the bipyridyl ligands with hydrocarbon (HC) chains with increasing numbers of carbon atoms in all six ligand positions para to the N atoms (Figure lb) we have constructed “hedgehog” complexes with isolating spacers of increasing thickness around the donor complex which limits the approach of the acceptor methylviologen to an adjusted minimum distance (Whitten et al. have synthesized similar hydrophobic “spider” ‘Dedicated to Prof. F Lingens on the occasion of his 60th birthday ‘Present address, Battelle Institut, Frankfurt, W Germany

0022-3654/86/2090-2476$01.50/0

complexesI6 because of their solubility properties). Diffusion is retained. The forward electron transfer (light-induced) and the back electron transfer (thermal) can be studied separately. These two electron-transfer processes have quite different free enthalpy (1) Seefeld, K. P.; Mobius, D.; Kuhn, H. Hela Chim. Acta 1977.60, 2608. Penner, T. L.; Mbbius, D. J . Am. Chem. SOC.1982, 104, 7407. (2) Kuhn, H.; Mobius, D.; Biicher, H. In Physical Methods ofchemistry, Weissberger, A., Rossiter, B. W., Eds.; Wiley-Interscience: New York, 1972; Part 111 B, Chapter VII, p 645 ff. (3) Hidaki, S.; Toda, F. Chem. Left. 1983, 1333. Ford, W,,E.; Otvos, J. W.; Calvin, M. Nature (London) 1978, 274. Ford, E. W.; Otvos, J. W.; Calvin, M. Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 3590; Tunulli, S.; Fendler, J. H. J . Am. Chem. SOC.1981, 103, 2507. (4) (a) Miller, J. R.; Calcaterra, L. T.; Closs, G. L. J . Am. Chem. SOC. 1984, 106, 3047. (b) Joran, A. D.; Leland, B. A.; Geller, G. G.;Hopfield, J. J.; Dervan, P. B. J . Am. Chem. SOC.1984, 206, 6090. (c) Bolton, J. R.; Ho, T.-F.; Liauw, S.; Siemiarczuk, A.; Wan, C. S.; Weedon, A. C. Chem. Commun. 1985,559. (d) Wasiliewski, M. R.; Niemczyk, M. P. J. Am. Chem. SOC.1984, 106, 5043. (e) Hush, N. S.; Paddon-Row, M. N.; Cotsaris, E.; Oevering, H.; Verhoeven, Z. W.; Heppener, M. Chem. Phys. Lett. 1985, 117, 8. (5) McIntosh, A. R.; Siemiarczuk, A,; Bolton, J. R.; Stillman, M. J.; Ho, T.-F.; Weedon, A. C. J . Am. Chem. SOC.1983, 105, 7215. Siemiarczuk, A,; McIntosh, A. R.; Ho, T.-F.; Stillman, M. J.; Roach, K. J.; Weedon, A. C.; Bolton, J. R.; Connolly, J. S . J . Am. Chem. SOC.1983, 105, 7224. (6) Strauch, S.; McLendon, G.; McGuire, M.; Guarr, T. J . Phys. Chem. 1983, 87, 3579, and literature cited therein. (7) Butty, E.; Suppan, P.; Haselbach, E. EPA European Postgraduate Svmuosium on Photochemistrv. London. A d . 1984. ‘ (8’) Miller, J. R.; Peeples, J.X.; Schmidt, M. J,; Closs, G.L. J . Am. Chem. s o c . 1982,104.64aa. (9) Miller, J. R.; Hartmann, K. W.; Abrash, S. J . Am. Chem. SOC.1982, 104, 4296. (10) Guarr, T.; McGuire, M.; Strauch, S.; McLendon, G. J . Am. Chem. SOC.1983, 105, 616. ( 1 1) Perrin, J. C. R . Acad. Sci. Paris 1924, 178, 1978. See also Turro, N. Modern Molecular Photochemistry; Benjamin/Cumrnings: Reading, 1978; p 317 ff. (12) Rehm, D.; Weller, A. Ber. Bunsenges. Phys. Chem. 1969, 73, 834. (13) Ballardini, R.; Varani, G.; Indelli, M. T.; Scandola, F.; Balzani, V. J . Am. Chem. SOC.1978, 100, 7219. Whitten, D. G. Acc. Chem. Res. 1980, 13,83. Balzani, V.; Bolletta, F.; Scandola, F.; Ballardini, R. Pure Appl. Chem. 1919.51, 299. Balzani, V.; Scandola, F.; Orlandi, G.;Sabbatini, N.; Indelli, M. T. J . Am. Chem. SOC.1981,103,3370. Bruce, B.; Sutin, N. J. Am. Chem. Soc. 1978, 100, 7568. Balzani, V.; Bolletta, F.; Gandolfi, M. T.; Maestri, M. Top. Curr. Chem. 1978, 75, 1. Nagle, J. K.; Dressieck, W. 2.: Meyer, T. J J . Am. Chem. SOC.1979, 101, 3993. (14) Vogelmann, E.; Schreiner, S.; Rauscher, W.; Kramer, H. E. A. Z . Phys. Chem. (Wiesbaden) 1976, 101, 321. Scheerer, R.; Gratzel, M. J . Am. Chem. SOC.1977,99,865. Tamura, S.; Kikuchi, K.; Kokubun, H . Z . Phys. Chem. (Wiesbaden) 1978, 1 1 1 , 7. Vogelmann, E.; Rauscher, W.; Traber, R.: Kramer, H. E. A. Z . Phys. Chem. (Wiesbaden) 1981, 124, 13. Iwa, P.; Steiner, U. E.; Vogelmann, E.; Kramer, H . E. A. J. Phys. Chem. 1982, 86,

1277. ( 1 5) Marcus, R. A. Discuss.Faraday SOC.1960, 29, 21. Marcus, R. A. J . Chem. Phys. 1965, 43, 679. (16) DeLaive, P. J.; Lee, J. T.; Abruna, H.; Sprintschnik, H. W.; Meyer, T. J.; Whitten, D. G. Adc. Chem. Ser. 1978, No. 168, 28.

0 1986 American Chemical Society

Dependence of Electron Transfer on R and AG

Tht? Journal of Physical Chemistry, Vol. 90, No. 11, 1986 2477 monochromator for variable wavelengths of the spectroflash or with a Spectra Physics He-Ne C W laser. The absorption was monitored with the fast 1P28 photomultiplier and the Tektronix 7834 scope and recorded on Polaroid film. The emission decay curves were analyzed with a program affording one or two exponential terms by a Tektronix 4052 computer and found to be monoexponential. The Polaroid photographs of the traces of the oscilloscope taken in the flash absorption measurements were evaluated according to second-order kinetics.

Figure 1. (a, top) Donor and acceptor molecules. (b, bottom) Kinetic scheme for the light-induced and thermal electron-transfer reactions.

changes of -38.6 and -164 kJ mol-’. Therefore we can study the distance dependence of the transfer rate with AG as a parameter. If the two processes had favorable rates the system we use could be of relevance for studies of light-driven hydrogen production.

2. Experimental Section 2.1. Chemicals. Ru(bpy),Clz and R ~ [ ( C H , ) ~ b p y l , Cwere l~ purchased from Strem Chemicals and used as obtained. The bipyridyl ligands (C,H,,,,),bpy, for n = 5, 7, 11, and 13, were prepared in a two-step synthesis: 4-methylpyridine was reacted with N a N H , and with the respective bromoalkane without a solvent.” The alkylpyridine was converted to the dialkylbipyridyl by boiling with W-7J Raney nickel in xylene.18 The yields were low (ca. 15%); impurities in the pyridines severely reduce the yields of the bipyridyls, especially in the case of short-chain molecules. The ruthenium complexes R U [ ( C , H ~ , + ~ ) ~ were ~ ~ ~prepared ]~*+ from the ligands and RuCI3-H20in boiling EtOH. The ruthenium complex R ~ [ ( C , , H , ~ ) ~ b p( yb l~ y was ) ~ prepared from cis-Ru(bpy)zC1219and (C13H27)2bpy in EtOH under argon. During the course of the experiments it became apparent that the lifetime of the hexamethyl-substituted complex did not fit into the homologous series (cf. Figure 2); therefore the complex was prepared also starting from 4-methylpyridine. All complexes were purified by column chromatography on Sephadex L H 20 with EtOH as an eluent.18 For solubility reasons spectrograde E t O H / H 2 0 (85:15, v:v) and C H 3 C N / H 2 0 (5050,v:v) were used as solvents. The solutions were deaerated by flushing with argon for at least 2 h. Typical concentrations in the quenching experiments were to IOd to M of the ruthenium complexes and 5 X M methylviologen. The concentrations used in the flash-absorption M M Ru complex and 4 X measurements were 8 X methylviologen. 2.2. Apparatus. Absorption and emission spectra were taken with a Zeiss D M R 10 spectrophotometer and a Farrand spectrofluorimeter, respectively. Lifetime measurements were made with an Ortec 6220 single-photon-counting equipment with an Applied Photophysics gated lamp or by a Lambda Physik excimer pulse laser E M G 500 (operated with nitrogen) with a suitably wired 1P28 photomultiplier and a Tektronix 7834 storage oscilloscope as a recording unit. This setup and the fluorimeter were also used for the dynamic and stationary quenching experiments. The disappearance of the methylviologen radical cation as mon+ itored at 610 nm and the repopulation of the R ~ ( b p y ) , ~ground state as monitored at 450 nm were observed in a flash apparatus: The E M G 500 was combined either with a pulsed Garching Instrumente xenon flashlamp with Bausch & Lomb high-intensity (17) Valenty, S.; Behnken, P. E.; Gaines, Jr., G. L. J . Inorg. Chem. 1979,

3. Results The absorption spectra of the HC-substituted R u complexes are very similar in shape to the spectrum of the unsubstituted complex, but they are red shifted by IO nm, the maxima being at 460 nm. The emmission spectra are red shifted, too, by 5 nm with the maxima at 625 nm. The emission decay of all complexes is monoexponential (Figure 2a). The emission lifetime is independent of H C substitution and amounts to 800 ns in E t O H / H 2 0 and 900 ns is C H 3 C N / H 2 0 (Figure 3). The methyl-substituted complex is exceptional in C H 3 C N / H 2 0 . Its lifetime is lower than that of the other complexes. This is not an artifact; complexes from different sources (Strem and our laboratory) give identical results. Figure 3b shows the plot of the quenching constants with methylviologen as a quencher vs. the number of carbon atoms in the HC chain of the complexes in the ethanol/water and the acetonitrile/water solvents. In the quenching experiments all decay curves of the complex emission are purely monoexponential, too (Figure 2a). The experiments are evaluated according to Stern-Volmer;20the k, values (Figure 3b) decrease with increasing chain length. The decrease is exponential up to C7. The point for the C1 complex does not fit very well to the experimental graph for the acetonitrile/water solution. This results from the lower lifetime of this complex: the Stern-Volmer constant fits very well to a corresponding logarithmic plot. The quenching constant of the complex bearing but one bipyridyl ligand with two C13chains and two unsubstituted ligands is not on the EtOH/water curve. Here the Ru(bpy)?+ unit is still partly open to the quencher, and the quenching constant is higher than observed for the complexes with a closed H C sheath. Figure 4 shows the rate constant kb of the bimolecular electron back transfer in acetonitrile/water. The disappearance of the MV+’ absorption is second order and in the microsecond region (Figure 2b). The half-lives t 1 / 2correlate with the l/E(O) values. N o data for EtOH/water solution can be reported as MVZ+still absorbs 337 nm radiation weakly and then oxidizes the solvent EtOH in an irreversible redox reaction. The kb values are higher than the k, values and independent of chain length. Again the C1 complex is exceptional. We have no explanation for this observation. To ascertain that the disappearance of the MV” radical is not due to, e.g., an impurity in the solvent, the rate of repopulation of the complex’s ground state was checked2‘ and found to agree. 4. Discussion The independence of the absorption spectra, the emission spectra, and the emission lifetimes of the length of the H C chain indicates that H C substitution does not influence the electronic states and dynamics of deactivation of the complexes. As the lifetimes of the complexes are long k will be a constant independent of time, On this basis we can compare the quenching behavior of the different ruthenium complexes. 4.1. The Quenching Reaction in C H , / C N . Quenching experiments in fluid solution often show exponential emission decay curves and linear I,JI (and r o / r )dependence on quencher concentration: they show Stern-Volmer behavior. In this model a “collision” follows a diffusional process. If the probability of the event (electron transfer, energy transfer, etc.) following the collision

18. 2160. ~~

1

~~~

(18) Johansen, 0.;Kowala, G.; Mau, A. W. H.; Sasse, W. H. F. Aust. J . Chem. 1979, 32, 1452. (19) Sprintschnik, G.; Sprintschnik, H. W.; Kirsch, P. P.; Whitten, D. G. J . Am. Chem. SOC.1977, 99, 4941.

(20) Stern, 0.;Volmer, M. Phys. Z. 1919, 20, 183. (21) We thank Dr. D. Mobius, Gottingen for pointing out the necessity of this experiment.

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The Journal of Physical Chemistry, Vol, 90, No. 11, 1986

Rau et al.

Holblog P l o t

I

I

-

0.50 1 .OO 1.50 2.00 2.50 E+2 S +4.302E-001 1.327E-802 = 3.08 % A = +4.08QE+001 : 2.008E+000 Mf E= * 2.775E+000 M of R ~ [ ( C ~ H , ~ ) ~ b pand y l ~2.1C Xl ~lo-) M MVCI,. y axis in Figure 2. (a, top) Emission decay (640 nm) of a solution of 1 X (b, bottom) Decay of the MV+' absorption (610 nm) in the system R U [ ( C ~ H , ~ ) ~ ~ ~ ] ~ ~ + / M V ~ + .

(which means a very narrow distribution of transfer distances) is 100% the quenching reaction is diffusion controlled. Smoluchowski has developed a theory for diffusion-controlled reactions22and calculated the bimolecular rate constant

k,,ff = 106-

2RT ( T A +- r d 2 3 0 0 0 ~ rArD

(1)

(SI units). If ions are involved eq 1 needs a modification according to D e b ~ and e ~ Eigen24 ~ for the electrostatic interaction 7 being the bulk viscosity

(22) Smoluchowski, M. Z . Phys. Chem. 1918, 92, 129. (23) Debye, P. Trans. Electrochem. SOC.1942, 82, 265. (24) Eigen, M. Z. Phys. Chem. (Wiesbaden) 1954, I . 176.

arbitrary units.

with b = Z ~ Z $ ~ / ( 4 a t o € k ~ T (rrD~) ) . Usually the radii of donor and acceptor are assumed to be equal m is assumed as a rough value to and r = ( r A r D ) = calculate the rate constant of electron transfer in the diffusion limit with the proper dielectric constant E . In this paper we are primarily interested in the effects of transfer distance brought about by varying only rD. For unmodified Ru(bpy),2+ the experimental and calculated quenching data are compared in Table I. The calculated values of kdiffin water and in acetonitrile/water are close to the experimental k, values suggesting diffusion-controlled quenching. Electron transfer in molecular systems is discussed in terms of separated electronic and nuclear (vibrational) factors as an approximation (as are energy transfer or radiationless transitions) although for large distances the separability of these factors has been called into question.2s For the calculation of the electronic

+

~~

~

(25) Beratan, D. N.; Hopfield, J. J. 1.Chem. Phys. 1984, 81, 5753.

Dependence of Electron Transfer on R and AG

The Journal of Physical Chemistry, Vol. 90. No. 11. 1986 2419

TABLE I: Expriarnt.l Qumebing Cmtanb for the Ru(bpy),'+/MV" System a d Values of k m Calculated according to F . q I8 by Assuming Vanom Donor a d Acceptor Mametema k d d d )

solvent water CH,CN/H,O EtOH/H,O

q.

78.3' 5 9 30'

r, = 0.4 nm.

rA = 0.3 nm.

kq(evtl)

rD = 0.6 nm

r D = 0.5 nm

5 x 108d to 1.03 x 109' 2.3 X IO' 2.7 X IO'

1.1 x 109 7.6 X IO' 1.5 x IO'

7.8 X IO' 3.7 x IO' 2.8 X IOb

N I m"

0.9 X 7.6 X IO4* 1.9 x i(r"

'Rate constanu are in M-'P'.b H o n d b w k of Chemistry ond Physics; Chemical Rubber Co.; Boca Raton, FL. 52nd ed. p E-49. 'Ibid., p F-36. ' B I U ~ ~ CP.I . A.; Critul. M.J. A m . Chem. Sm. 1980. 102. 2461. 'Amouyal. E.: Zidler, B.: Keller, P.;Moradpour. A. Chem. Phys. Lett. 1980. 74. 314. IMorcau, C.; Douhcret. G . J. Chem. Thermodyn. 1916. 8. 403. ~Lnndoll-B8mnsleinin. Zohlenwerte und Funktionen: Springer Verlag: Berlin. 1969; 1I.Band. 5.Teil. p 714. "bid.. p 775. 'Ibid., p 367.

Figure 5. Spacefilling model of the trir(d.d'-dihcpi?1-2.2'-bipyridyl)ruthcnium complex and methylviologen.

\ 10

5

C atoms in HC chain F@rr 3. Lifetimes and quenching constants of Ru(HCybpy)," wmplerrs VP. HC chain length 0. CH,CN/H>O. 0. EtOH/H,O 0. Ru-

l(C,,H,,)ybpy(bPy),I''

0

01,

'

0

5

10

Catoms in HCchain Figm 4. The bimolecular rate mnstants of the thermal back electron transfer vs. chain length. factors there a n two models: the tunneling model' and the Dexter modeLm Both models give an exponential dependence of transfer probability or rate

P = C exp(-or)

Using the tunneling model and the ionization potential of hydrocarbons" one calculates a = 1.38 X 1O'O m+. This is the same as the value of 1.1 X IO'O to 2.6 X IOto m-l used by Dexter,'6 McLendon,l0 and Marcusz8 and those found by Miller and u)workers when electron transfer was initiated by pulse photoly,iS,29.3O Unfortunately, we do not really know the distance increment effected by one CH, group in the HC chains of the ligands although we may infer that for short HC chains the number of C atoms is representative of the minimum distance in an encounter process. From inspection of space-filling CPK models (Figure 5 ) we see that short HC chains up to C7 do not bend back effectively." Longer chains fold back. Therefore the leveling off of the plot of In kq vs. the number of C atoms is easily understood. But the CPK models also show that the distance of 0.1 1 nm per CHz group which has been found to be adequate when HC chains are used in monolayer assemblies',2 cannot hold for our system. The complex core still is partly accessible for the quencher even with solvent molecules intercalated between the chains and even when rotation is rapid. Therefore we cannot plot In k VI r But we can try to find dlculated from the tunnel out whether the use of the value model leads to a reasonable value for the distance increment of a CH2 group. The diffusion-limited rate constant kdimfor acetonitrile/water calculated according to eq l a plotted vs. distance gives a graph slightly bent upward. It can be well approximated by a straight line between 1.0 and 2.0 nm with a slope of 3.7 M-'

(2)

(27) Mann. D.; Kuhn. H. 1.Appl. Phys. 1971. 42. 4398. (28) Mareus. R. A.; Sidcrs. J. P. 1.Phys. Chem. 1982.86.622. (29) Huddleston. R. K.; Miller. J. R. 1. Chrm. Phyr. 1983, 79. 5337. (30) Miller, 1.R.: BeiY 1.V.; Huddlaton, R. K. 1. Am. Chem. Soc. 1984. I("'7 I106.5057. -, (31) This is in agreement with findings of Demas. 1. N.. wwnal vewnal mm_1"_11.

(26) Dexter. D. L. 1.Chem. Phys. 1953.21.836. Inohti. M.: Hirayama. F. 1. Chcm. Phys. 1965.43. 1978.

munication

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The Journal of Physical Chemistry, Vol. 90, No. 11, 1986

nm-I. If we plot In kq = In [kd,fdCakd)P(calcd)] vs. (rA + rD) we still obtain a good straight line with the slope of a = 1.2 X 1Olo m-l. This value of a is calculated for a distance increment of 0.03 nm per C H 2 group. Thus the quenching seems to be controlled by diffusion32and dominated by the electronic factors in the distance dependence of the transfer step. 4.2. Thermal Back Electron Transfer in CH3CN/H20. The thermal back transfer of the electron from MV+' to R ~ ( b p y ) , ~ + in acetonitrile/water is faster by a factor of 10 than the quenching of the R ~ ( b p y ) , ~ +emission. * This corresponds to the findings in water.34 Surprisingly, the rate is independent of the length of the H C chains; this is in accord with Whitten's findings.I6 This phenomenon could be due to a sort of static mechanism if the ions after the first light-induced electron transfer did not really separate, if e.g. the MV" radicals were held back by the H C chains. Such a mechanism can be ruled out, however, as the disappearance of MV+' clearly is second order and on a microsecond time scale (Figure 2b) indicating the encounter of free particles. The electron-transfer reaction is highly exergonic (G = -1.7 eV); according to most experiments and the Rehm-Weller equation,'* we would expect diffusion control. If we use eq l a to calculate the diffusion-controlledrate constant of the 3+ charged acceptor and the 1+ charged donor with the values for 7, t, rA, and rDof Table I, we find that the calculated diffusion-limitedrate constant of the thermal back electron transfer is smaller than that observed in the experiment. In order to obtain the experimental value for kdiff(calcd),we have to assume an electron-transfer probability of unity at a distance of 1.6 to 1.8 nm. Fortunately, the Smoluchowski-Debye formula l a for a charged donor and acceptor gives this information that cannot be obtained from experiments with one or two neutral reaction partners according to formula 1. If we used a lower dielectric constant in the transfer volume according to an increasing H C content or a value of kq smaller than that of kdiffthen the transfer distance according to eq l a would increase further. From simple geometric reasoning one would not expect any influence of the size of the acceptor complex on the transfer rate at transfer distances of this magnitude since all the changes of radii with H C substitution are within the transfer distance. The diffusion limit is the upper limit of the transfer rate. If we accept a high reorganization energy in the highly polar solvent used and a most effective electron transfer process at high exergonicity, still the limit of kb would be the diffusion limit. As the experimental kb value indeed is higher than that of kdlfl calculated for the contact distance the idea of electron transfer at larger distances seems to be a solution to the problem. The large transfer distances are on the order of those observed by Miller et al.839 But why should there be such a difference in the transfer distances of the light-induced complex MV2+and Ru complex transfer reactions? The systems the thermal MV" differ in the driving forces of electron transfer (AG = -0.4 and -1.7 eV, respectively). McLendon et a1.I0 have found a correlation between a "Perrin critical distance" and the A E values (which can be set equal to the AG values) by varying the bidentate pyridine-type ligands of the ruthenium complex. This correlation has been set up for light-induced electron-transfer reactions; the transfer distance of our thermal reaction fits into this pattern. At transfer distances larger than the contact distance of the donor and acceptor, solvent molecules must be participating in the electron-transfer process. Two theoretical approaches have been taken. Closs and co-workers have done calculations comparing D-A pairs at fixed distances in vacuo and D-Sp-A

-

-+

(32) In this calculation we have assumed diffusion-controlled quenching. Evidence from previous asymmetric electron transfer experiments)) in the Ru(bpy)32+/MV2+system made us believe that quenching should not be controlled by diffusion. Unfortunately, although we got additional evidence from an independent experiment, we could not reproduce asymmetry with newly synthesized material. (33) Rau, H.; Ratz, R. Angew. Chem., Int. Ed. Engl. 1983, 22, 550. (34) Brugger, P. A.; Gratzel, M. J. Am. Chem. SOC.1980, 102, 2461. Amouyal. E.; Zidler, B.; Keller, P.; Moradpour, A. Chem. Phys. Left. 1980, 74, 314.

Rau et al. molecules with hydrocarbon spacers. They find highly increased transfer rates in the "spacered" pairs.35 Indeed, Hush et al. have demonstrated extremely fast electron transfers over large distances in spacered Solvent molecules might do the same job as a hydrocarbon spacer in highly exergonic transfer reactions, e.g. via virtual state^.^'^^^ If such a solvent participation is important then the dependence of k, on the nature of the solvent is expected. In a different approach Sutin and c o - ~ o r k e r have s ~ ~ analyzed the distance dependence of electron transfer in bimolecular reactions. They have separated and calculated the electronic coupling and reorganizational energies for the normal (AG = +0.25 eV) and the Marcus inverted region (AG = -2.0 eV). They find a nearly exponential decrease of the first-order electron transfer rate k(r) in the normal region, dominated by electronic factors, but they calculate a maximum curve for k(r) in the inverted region caused by the opposing effects of electronic and reorganizational factors. The rates in the inverted region are higher at large transfer distances than those in the normal region even at smaller distances. In a system with diffusion the pair distribution function is time dependent: separations with high k(r) will dominate the whole process and their fast depletion is prevented by diffusion. The experimental results of our system exactly reflect Sutin's predictions. 4.3. The Quenching Reaction in EtOH/H,O. The diffusionlimited quenching constant for unmodified Ru(bpy),2+ has been calculated according to eq la. The values in Table I show that quenching is faster than expected from the Smoluchowski/Debye diffusion equation. In order to obtain the experimental value of k , = 2.7 X lo8 M-' s-l from the calculations we have to use a transfer distance of 1.9 nm which certainly is not the contact distance of the reaction partners. Bridging this distance requires solvent participation. The difference in redox potentials of the C H 3 C N / H 2 0and E t O H / H 2 0 is demonstrated in our experiments by the fact that excited MV2+oxidizes the alcoholic but not the acetonitrile solvent. Therefore participation of EtOH but not of CH3CN in the electron transfer of the quenching reaction seems likely. But if the transfer distance is so large one does not understand the dependence of k, on rD (Figure 3) which is less expressed than that in the acetonitrile solvent. One would expect no dependence at all. At the moment we have no explanation. 4.4. Marcus and Rehm-Weller Behavior. The results of this paper might contribute to the understanding of the puzzle of the Marcus inverted region of electron transfer. Figure 6a represents the classical Marcus representation which certainly is much simplified but allows an easy mathematical derivation of the Marcus equation

[+

AG* = AG*(O) 1

2

4A;'(O)]

(3)

The point of intersection of the initial and final state potential energy curves (assumed to be like parabolas) moves down and up again (Marcus inverted region) when the free energy of the final state is lowered. But note, in Figure 6a the nuclear coordinates are kept constant when AG is changed. Indeed, if this requirement is met, the Marcus inverted region is observed. The ~ ~geminal lightD-Sp-A molecules of Miller and C ~ O SorSthe created ion pairs of Farid39 represent such systems. In systems with variable transfer distances where solvent molecules are involved at highly negative AG, new vibrational modes are introduced and consequently the nuclear configuration space is widened. Figure 6b represents this case in the same level (35) Personal communication. (36) Brunschwig, D. S.; Ehrenson, S.; Sutin, h'.J . Am. Chem. SOC.1984, 106, 6858. (37) Ulstrup, J. "Charge Transfer Processes in Condensed Media" in Lecture Notes in Chemislry No. I O ; Springer Verlag: Heidelberg, 1979. (38) Kuznetsov, A. M.; Ulstrup, J. J . Chem. Phys. 1981, 75, 2047. (39) Farid, S., personal communication.

The Journal of Physical Chemistry, Vol. 90, No. 1 I , 1986 2481

Dependence of Electron Transfer on R and AG

MARCUS

finds the optimum conditions for electron transfer. This, however, resulted in Rehm-Weller behavior, and no Marcus inverted region should be expected in normal diffusional systems. On a much higher level of sophistication Marcus and Siders2*have reached a similar conclusion "that the r dependence of the solvent reorganization energy increases the predicted rate constant in the inverted region". Figure 6b represents Rehm-Weller behavior and it can be shown that the Marcus eq 3 and the Rehm-Weller equationI2

+

are related. If [ l '/2(AG/2)2/(2AC*(0)2)] is considered the then the Marcus expansion of a square root (1 + g)'/2 = 1 '/g, equation can be transformed to the Rehm-Weller equation with a factor of in the second term of the root. However, the condition for the expansion of the root to be valid is (AC/2)2 < 2AG*(0)2. Therefore this factor 1 / 2 is not important. In the usual Marcus picture of Figure 6a, the condition can only be met.in a very narrow region of small AG values as AG*(O) is constant. If a displacement along the coordinate q in Figure 6b is admitted, AG*(O) may increase and the condition can be met approximate]~.~'

REHM-WELLER

5. Conclusion

Gi

We have investigated systems in which free donor and acceptor molecules have to diffuse together in order to react by electron transfer. In these systems in which electron transfer is dominated by diffusional processes the transfer distance and the free enthalpy change are interrelated: at high negative AG the transfer is over large distances. Our experiments are in perfect agreement with calculations by It seems that electronic and nuclear factors are adjusted to an optimum for the reaction by the diffusional process. In systems in which the transfer distance is variable with AG we expect to observe Rehm-Weller behavior; in systems with constant and uniform transfer distances we expect to observe Marcus behavior. Systematic studies are in progress on the influence of different solvent mixtures and solvents on the electron-transfer reaction. These experiments will provide a broader basis for the discussion of the influence of the dielectric constant, viscosity, and specific solvent properties. Variation of AG is planned by changing the acceptor of the quenching reaction.

Figure 6. Potential energy diagrams for Marcus and Rehm-Weller

+

behavior. q is a nuclear coordinate of the total system donor solvent acceptor. i is the initial state before reaction; fare the final stata after electron transfer. a is the intrinsic free enthalpy of activation (reorganization) AG*(O), b is the free enthalpy of activation AG*, and c is the free reaction enthalpy AG. (a) Marcus behavior: d = qr- qi and AG*(O) are independent of AG. AG' increases at highly negative AG. (b) Rehm-Weller behavior: d and AG*(O) increase at highly negative AG, AG* is near zero.

+

+

of sophistication as that of Figure 6a. Now the displacement of the parabola of the final state compensates partly for the effect of the lowering of AG on the movement of the point of intersection.40 One might suspect that a real system with diffusion (40) If reduced nuclear coordinates are used a change in the width of the parabola corresponds to a shift in this coordinate representation (Rau, H.; Steiner, U., manuscript in preparation).

Acknowledgment. We gratefully acknowledge helpful, sometimes controversial, discussions with Prof. U. Steiner, Konstanz, Prof. A. Weller, Gottingen, Dr. J. R. Miller, Argonne, Prof. G . L. Closs, Chicago, and Dr. S . Farid, Rochester and the work of Mrs. A. Woll in our laboratory. Our work is supported by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie, Frankfurt. Registry No. [ R ~ ( b p y ) ~ ] ~ '15 , 158-62-0; [ R ~ [ ( C H ~ ) , b p y ] ~ l ~ + , 3288 1-03-1; [Ru[(C5Hl I ) ~ ~ P Y ] ~101 I ~ '5,19-46-4; [Ru[ (C7H15)2bp~]3]~', 101 5 19-47-5; [ R u [ (Cl IH13)2bp~]3] 10 15 1 9 - 4 8 - 6 ; [ RU[(C13H27)2bpy]3] 1015 19-49-7; MV2+, 4685- 14-7.

'+,

'+,

(41) If the lengths of ul and c3 are taken then the ratio (c3 2)*/2u,2 = 0.47. The third term in the development of the square root - 1 / 8 i w o u l d be -0.03. The approximation is good.