Communications to the Editor (4) (5) (6) (7) (8)
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S.Nagase and C. W. Kern, J. Am. Chem. SOC.. 101,2544 (1979): S.Nagase and C. W. Kern, ibid., submitted for publication. R. Ditchfield, W. J. Hehre, and J. A. Popie, J. Chem. Phys., 54, 724 (1971). We used the IMS ab initio package (K. Morokuma, S.Kato, K. Kitaura, I. Ohmine and S.Sakai, unpublishedresults) consisting of the GAUSSIAN 70, HONDO, own gradient and optimization, and force constant programs. J. A. Eyre, T. Hikida, andL. M. Dorfman. J. Chem. Phys., 53, 1281 (1970); T. Hikida. J. A. Eyre, and L. M. Dorfman, ibid., 54, 3422 (1971). M. J. Kurylo, N. C. Peterson, and W. Braun, J. Chem. Phys., 53, 2776 (1970). J. V. Michael, D. T. Osborne. and G. N. Suess. J. Chem. Phys., 58, 2800 (19731. - -, D. Mihelcic, V. Schubert, F. Hofler, and P. Potzinger, Ber. Bunsenges. Phys. Chem.. 79. 1230 11975). -, Y. Ishikawa, M. Yamabe, A. Noda, and S.Sato, Bull. Chem. SOC.Jpn., 51, 2488 (1978). J. H. Lee, J. V. Michael, W. A. Payne, and L. J. Stief, J. Chem. Phys., 68, 1817 (1978). J. V. Michael and G. N. Suess, J. Chem. Phys., 58,2807 (1973). W. Braun and M. Lenzi, Disc. Faraday SOC.,44, 252 (1967). A. F. Dodonov, G. K. Lavrovskaya, and V. L. Tal'roze, Kinet. Katal., 10, 22 (1969). P. 0.Penzhorm and B. de B. Darwent, J. Chem. Phys., 55, 1508 (1971). Y. lshikawa and S.Sato, Bull. Chem. SOC.Jpn., 52, 984 (1979). C2H4D1for example, €0 = 2.08 k c a l h o l . For the reaction D C2H4 €0 = 2.33 and 2.34 kcal/mol for the H addition toward the cis and trans isomers, respectively. P. I. Abell in "Comprehensive Chemical Kinetics", Vol. 18. C. H.Bamford and C. F. H. Tipper, Eds., Elsevier, Amsterdam, 1976, p 111. For a recent review, see J. M. Tedder and J. C. Walton, Adv. Phys. Org. Chem.. 16. 51 (1978). J. E. Bennett and B.'Miie, J. Chem. SOC., Faraday Trans. 7, 69, 1398 (1973). \
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Shigeru Nagase, Takayuki Fueno* Department of Chemistry Faculty of Engineering Science Osaka Unicersity, Toyonaka, Osaka 560, Japan
Keiji Morokuma Institute f o r Molecular Science Myodaiji, Okazaki 444, Japan Received May 21 I979 ~
the Correlation of the Kinetics of Electron-Transfer Processes with the Change in Reaction Free Energy. Application to Electron-Exchange-Initiated Chemiluminescence Reactions Sir: Single-electron-transfer reactions to generate radical-ion intermediates have been shown to be of crucial importance in the chemistry of organic and organometallic electronically excited states,' singlet oxygen,2 radical^,^ and organic peroxi d e ~The . ~ radical ions so produced have been observed to return to reactants (fluorescence quenching), to undergo further electron-transfer reactions that lead eventually to products, to combine to generate products, and to annihilate to form electronically excited states (chemiluminescence). Often the evidence for the involvement of radical-ion intermediates in these processes is a correlation of the observed rate constant for reaction with the one-electron oxidation or reduction potential of the r e a ~ t a n tOur . ~ recent work on chemically initiated electron-exchange luminescence (CIEEL) has produced several such c ~ r r e l a t i o n s . ~We ~ - ~report herein an interpretation of the kinetics of these apparently irreversible, endergonic, electron-transfer reactions based upon the model proposed by Weller and co-workers.6 Equation 1 is an adapted version of the general reaction scheme presented by Rehm and Weller:6b
D+A
*
+ D-A k21
2 2D+.-2A-. k32
k30
P
(1)
The subscripts on the rate constants have been kept the same for consistency but the reactants have been generalized as 0002-7863/79/l501-5851$01 .OO/O
donor ( D ) and acceptor ( A ) and the electronic states of these are left unspecified. The products (P) of the reaction may be different electronic states of the reactants or new substances. The first equilibrium defines the diffusion of the reactants, the second the electron-transfer process. It should be noted also that the final step (rate constant k30) is irreversible and may correspond to diffusion apart of the ions or to chemical reaction. Application of the usual steady-state approximation to the concentration of encounter complex ( D - A ) and radical-ion pair (2D+.-2A-.)leads to
k3ok23k 1 2 (2) (k32 + k30)(k21 + k23) - k32k23 (analogous to eq 2 of Rehm and Weller6b) which defines the rate constant for formation of P ( k , ) . It is the correlation of this rate constant with the energetics of the electron transfer (AG23)' that can provide the kinetic evidence for an electron transfer in a reaction scheme. The relationship among the magnitudes of the various rate constants of eq 2 apparently determines the nature of this correlation. Weller and coworkers,6 for example, have observed diffusion limited electron transfer for which k , is independent of AC23 (case i, below) and endergonic electron transfer for which k , depends strongly on AG23 (case ii, below). Our o b s e r v a t i ~ n s ~suggest ~-l a third form for the correlation of k , with AG23 (case iii, below) which occurs when the electron-transfer step is both endergonic and irreversible. Case i. The electron-transfer step (k23)is exergonic (AC23 < 0) and irreversible. The two conditions specified above are not unrelated. The reverse electron transfer (k32) must be endergonic and therefore activated. Diffusion of the radical ions or chemical reaction (k30) could easily be orders of magnitude larger than k3+ (k32 > k21. When these inequalities are applied to eq 2 , they lead to the conclusion that, for this case, there should be no correlation between AC23 and k,. The slope of a semilog plot of k , against AG23 should be equal to zero, as has been seen experimentally.6a.b Case ii. The electron-transfer step k23 is endergonic (AC23 > 0) and reversible. As in the above case, these two conditions are related. Back electron transfer (k32) is now the exergonic direction and since exergonic electron transfer competes effectively with diffusion k32 >> k30. By analogy, k23 0) and irreversible. As in case ii, k23
