Diffusion constants, electron-transfer

Elegant evidence that free electrons can be transferred from an organic donor to a diazonium ion was found by Becker et al. (1975, 1977a see also Becker, 1978). These authors observed that diazonium salts quench the fluorescence of pyrene (and other arenes) at a rate k = 2.5 x 1010 m-1 s-1. The pyrene radical cation and the aryldiazenyl radical would appear to be the likely products of electron transfer. However, pyrene is a weak nucleophile the concentration of its covalent product with the diazonium ion is estimated to lie below 0.019o at equilibrium. If electron transfer were to proceed via this proposed intermediate present in such a low concentration, then the measured rate constant could not be so large. Nevertheless, dynamic fluorescence quenching in the excited state of the electron donor-acceptor complex preferred at equilibrium would fit the facts. Evidence supporting a diffusion-controlled electron transfer (k = 1.8 x 1010 to 2.5 X 1010 s-1) was provided by pulse radiolysis. 

At slow ionization and fast diffusion the electron transfer is expected to be under kinetic control, and its rate constant klt defined in Eq. (3.37) is diffusion-independent. Moreover, if a sharp exponential function (3.53) is a good model for W(r), the kinetic rate constant may be approximately estimated as follows ... 

Marcus theory deals with the first-order electron transfer rate constant for a donor-acceptor system presuming fixed separation. However, there are often systems that do not undergo Marcus behavior and have to be treated using the Rehm-Weller formalism.Here, the second-order diffusion-mediated electron transfer rates are calculated using Eq. 4, where AG =free energy of activation for... 

Table 2 Second order rate constants of completely diffusion controlled electron transfer reactions measured with the CFMIO at 296 K.

The topic has received considerable attention of late, stimulated in part by the interest in solar energy conversion. Recent papers have examined the theory of electron transfer rate constant ket(r), which in nondiffusing systfims decreases exponentially with distance, and the influence of energetics and diffusion on electron transfer rates (24b). Electron transfer can occur over distances of tens of Angstroms. 

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

The explicit mathematical treatment for such stationary-state situations at certain ion-selective membranes was performed by Iljuschenko and Mirkin 106). As the publication is in Russian and in a not widely distributed journal, their work will be cited in the appendix. The authors obtain an equation (s. (34) on page 28) similar to the one developed by Eisenman et al. 6) for glass membranes using the three-segment potential approach. However, the mobilities used in the stationary-state treatment are those which describe the ion migration in an electric field through a diffusion layer at the phase boundary. A diffusion process through the entire membrane with constant ion mobilities does not have to be assumed. The non-Nernstian behavior of extremely thin layers (i.e., ISFET) can therefore also be described, as well as the role of an electron transfer at solid-state membranes. 

The electron transfer from a methanol molecule to the activated diazonium ion is obviously a diffusion-controlled reaction. The rate constant is of the same order... 

Tl(III) < Pb(IV), and this conclusion has been confirmed recently with reference to the oxythallation of olefins 124) and the cleavage of cyclopropanes 127). It is also predictable that oxidations of unsaturated systems by Tl(III) will exhibit characteristics commonly associated with analogous oxidations by Hg(II) and Pb(IV). There is, however, one important difference between Pb(IV) and Tl(III) redox reactions, namely that in the latter case reduction of the metal ion is believed to proceed only by a direct two-electron transfer mechanism (70). Thallium(II) has been detected by y-irradiation 10), pulse radiolysis 17, 107), and flash photolysis 144a) studies, butis completely unstable with respect to Tl(III) and T1(I) the rate constant for the process 2T1(II) Tl(III) + T1(I), 2.3 x 10 liter mole sec , is in fact close to diffusion control of the reaction 17). 

It is clear that one of the major limitations of this analysis is the assumption of constant excited-state coverage. Deviations from the behavior described by Eq. (45) in the low frequency range have been observed at photocurrent densities higher than 10 Acm [50]. These deviations are expected to be connected to excited-state diffusion profiles similar to those considered by Dryfe et al. [see Eq. (38)] [127]. A more general expression for IMPS responses is undoubtedly required for a better understanding of the dynamics involved in back electron transfer as well as separation of the photoproducts. 

In agreement with the theory of electrolysis, treated in Sections 3.1 and 3.2, the parts of the residual current and the limiting current are clearly shown by the nature of the polarographic waves because for the cathodic reduction of Cd2+ and Zn2+ at the dme we have to deal with rapid electron transfer and limited diffusion of the cations from the solution towards the electrode surface and of the metal amalgam formed thereon towards the inside of the Hg drop, we may conclude that the half-wave potential, Eh, is constant [cf., Fig. 3.13 (a ] and agrees with the redox potential of the amalgam, i.e., -0.3521V for Cd2+ + 2e - Cd(Hg) and -0.7628 V for Zn2+ + 2e -> Zn(Hg) (ref. 10). The Nernst equation is... 

Very often, the electrode-solution interface can be represented by an equivalent circuit, as shown in Fig. 5.10, where Rs denotes the ohmic resistance of the electrolyte solution, Cdl, the double layer capacitance, Rct the charge (or electron) transfer resistance that exists if a redox probe is present in the electrolyte solution, and Zw the Warburg impedance arising from the diffusion of redox probe ions from the bulk electrolyte to the electrode interface. Note that both Rs and Zw represent bulk properties and are not expected to be affected by an immunocomplex structure on an electrode surface. On the other hand, Cdl and Rct depend on the dielectric and insulating properties of the electrode-electrolyte solution interface. For example, for an electrode surface immobilized with an immunocomplex, the double layer capacitance would consist of a constant capacitance of the bare electrode (Cbare) and a variable capacitance arising from the immunocomplex structure (Cimmun), expressed as in Eq. (4). 

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