Electron-transfer rate

Mediators are meant to carry electrons and transfer them to another species, which could be either another redox-active atom or an electrode. The rate of electron transfer at an interface is typically described by the Butler-Volmer equation, which is included in Equation 9.7. The two exponential terms describe anodic and cathodic charge transfer reactions from a species in solution to the electrode. Electron transfer between species in solution is given by Equation 9.9, where the rate et is the product of two exponential terms [7]  

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At low currents, the rate of change of die electrode potential with current is associated with the limiting rate of electron transfer across the phase boundary between the electronically conducting electrode and the ionically conducting solution, and is temied the electron transfer overpotential. The electron transfer rate at a given overpotential has been found to depend on the nature of the species participating in the reaction, and the properties of the electrolyte and the electrode itself (such as, for example, the chemical nature of the metal). 

At higher current densities, the primary electron transfer rate is usually no longer limiting instead, limitations arise tluough the slow transport of reactants from the solution to the electrode surface or, conversely, the slow transport of the product away from the electrode (diffusion overpotential) or tluough the inability of chemical reactions coupled to the electron transfer step to keep pace (reaction overpotential). 

Chemical reactions can be studied at the single-molecule level by measuring the fluorescence lifetime of an excited state that can undergo reaction in competition with fluorescence. Reactions involving electron transfer (section C3.2) are among the most accessible via such teclmiques, and are particularly attractive candidates for study as a means of testing relationships between charge-transfer optical spectra and electron-transfer rates. If the physical parameters that detennine the reaction probability, such as overlap between the donor and acceptor orbitals. 

Figure C1.5.12.(A) Fluorescence decay of a single molecule of cresyl violet on an indium tin oxide (ITO) surface measured by time-correlated single photon counting. The solid line is tire fitted decay, a single exponential of 480 5 ps convolved witli tire instmment response function of 160 ps fwiim. The decay, which is considerably faster tlian tire natural fluorescence lifetime of cresyl violet, is due to electron transfer from tire excited cresyl violet (D ) to tire conduction band or energetically accessible surface electronic states of ITO. (B) Distribution of lifetimes for 40 different single molecules showing a broad distribution of electron transfer rates. Reprinted witli pennission from Lu andXie [1381. Copyright 1997 American Chemical Society.

Figure C3.2.10.(a) Dependence of electron transfer rate upon reaction free energy for ET between biphenyl radical anions and various organic acceptors. Experiments were perfonned with the donors and acceptors frozen into...

In Debye solvents, x is tire longitudinal relaxation time. The prediction tliat solvent polarization dynamics would limit intramolecular electron transfer rates was stated tlieoretically [40] and observed experimentally [41]. 

This lineshape analysis also implies tliat electron-transfer rates should be vibrational-state dependent, which has been observed experimentally [44]- Spin-orbit relaxation has also been identified as an important factor in controlling tire identity of botli electron and vibrational-state distributions in radiationless ET reactions. 

Early studies showed tliat tire rates of ET are limited by solvation rates for certain barrierless electron transfer reactions. However, more recent studies showed tliat electron-transfer rates can far exceed tire rates of diffusional solvation, which indicate critical roles for intramolecular (high frequency) vibrational mode couplings and inertial solvation. The interiDlay between inter- and intramolecular degrees of freedom is particularly significant in tire Marcus inverted regime [45] (figure C3.2.12)). 

Beratan D N, Betts J N and Onuchic J N 1991 Protein electron transfer rates set by the bridging secondary and tertiary structure Science 252 1285-8... 

Spears K G, Wen X and Zhang R 1996 Electron transfer rates from vibrational quantum states J. Phys. Chem. 100 10 206-9... 

The electron transfer rates in biological systems differ from those between small transition metal complexes in solution because the electron transfer is generally long-range, often greater than 10 A [1]. For long-range transfer (the nonadiabatic limit), the rate constant is... 

M Tachiya. Relation between the electron-transfer rate and the free energy change of reaction. J Phys Chem 93 7050-7052, 1989. 

The first type of interaction, associated with the overlap of wavefunctions localized at different centers in the initial and final states, determines the electron-transfer rate constant. The other two are crucial for vibronic relaxation of excited electronic states. The rate constant in the first order of the perturbation theory in the unaccounted interaction is described by the statistically averaged Fermi golden-rule formula... 

In the strong-coupling limit at high temperatures the electron transfer rate constant is given by the Marcus formula [Marcus 1964]... 

As an illustration of these considerations, the Arrhenius plot of the electron-transfer rate constant, observed by DeVault and Chance [1966] (see also DeVault [1984]), is shown in fig. 13. 

The exponent in this formula is readily obtained by calculating the difference of quasiclassical actions between the turning and crossing points for each term. The most remarkable difference between (2.65) and (2.66) is that the electron-transfer rate constant grows with increasing AE, while the RLT rate constant decreases. This exponential dependence k AE) [Siebrand 1967] known as the energy gap law, is exemplified in fig. 14 for ST conversion. 

S.2.2 Carbon Electrodes Solid electrodes based on carbon are currently in widespread use in electroanalysis, primarily because of their broad potential window, low background current, rich surface chemistry, low cost, chemical inertness, and suitability for various sensing and detection applications. In contrast, electron-transfer rates observed at carbon surfaces are often slower than those observed at metal electrodes. The electron-transfer reactivity is strongly affected by the origin... 

Further improvements can be achieved by replacing the oxygen with a non-physiological (synthetic) electron acceptor, which is able to shuttle electrons from the flavin redox center of the enzyme to the surface of the working electrode. Glucose oxidase (and other oxidoreductase enzymes) do not directly transfer electrons to conventional electrodes because their redox center is surroimded by a thick protein layer. This insulating shell introduces a spatial separation of the electron donor-acceptor pair, and hence an intrinsic barrier to direct electron transfer, in accordance with the distance dependence of the electron transfer rate (11) ... 

Both Marcus27 and Hush28 have addressed electron transfer rates, and have given detailed mathematical developments. Marcus s approach has resulted in an important equation that bears his name. It is an expression for the rate constant of a net electron transfer (ET) expressed in terms of the electron exchange (EE) rate constants of the two partners. The k for ET is designated kAS, and the two k s for EE are kAA and bb- We write the three reactions as follows ... 

The reason for the exponential increase in the electron transfer rate with increasing electrode potential at the ZnO/electrolyte interface must be further explored. A possible explanation is provided in a recent study on water photoelectrolysis which describes the mechanism of water oxidation to molecular oxygen as one of strong molecular interaction with nonisoenergetic electron transfer subject to irreversible thermodynamics.48 Under such conditions, the rate of electron transfer will depend on the thermodynamic force in the semiconductor/electrolyte interface to... 

Figure 42. Scheme comparing expected potential-independent charge-transfer rates from Marcus-Gerischer theory of interfacia) electron transfer (left) with possible mechanisms for explaining the experimental observation of potential-dependent electron-transfer rates (right) a potential-dependent concentration of surface states, or a charge-transfer rate that depends on the thermodynamic force (electric potential difference) in the interface. 

Electron transfer rate and its exponential increase at zinc oxide-electrolyte interfaces, 512 Electronic conductivity... 

This apparent low efficiency for reducing N2 was observed under conditions of saturating N2 and is not due to restrictions in electron transfer rate. Furthermore, the low efficiency is mirrored in bacterial cultures at 30°C (185). 

EPR studies on electron transfer systems where neighboring centers are coupled by spin-spin interactions can yield useful data for analyzing the electron transfer kinetics. In the framework of the Condon approximation, the electron transfer rate constant predicted by electron transfer theories can be expressed as the product of an electronic factor Tab by a nuclear factor that depends explicitly on temperature (258). On the one hand, since iron-sulfur clusters are spatially extended redox centers, the electronic factor strongly depends on how the various sites of the cluster are affected by the variation in the electronic structure between the oxidized and reduced forms. Theoret-... 

Figure 23. Arrhenius plot of the electron transfer rate. The electronic coupling strength is TIad = 0.0001 a.u. Solid line-Bixon-Jortner perturbation theory Ref. [109]. FuU-circle present results of Eq. (26 ). Dashed line-results of Marcus s high temperature theory [Eq.(129)]. Taken from Ref. [28].

Figure 25. Electron-transfer rate the electronic coupling strength at T = 500 K for the asymmetric reaction (AG = —3ffl2, oh = 749 cm ). Solid line-present full dimensional results with use of the ZN formulas. Dotted line-full dimensional results obtained from the Bixon-Jortner formula. Filled dotts-effective ID results of the quantum mechanical flux-flux correlation function. Dashed line-effective ID results with use of the ZN formulas. Taken from Ref. [28].

Figure 26. Electron-transfer rates for 2,7-dinitronaphthalene as a function of temperature for three solvents. Symbols are the experimental results by Nelsen et al. [112]. Lines are the present theoreical results. Solid line and square CH3CN. Dot-dash line and PrCN. Dashed line and circle N, A-dimethylforamide (DMF). Taken from Ref. [113].

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