The Inverted Region

The Inverted Region.— The expression (11) for the activation energy as a function of the standard free-energy change is well known. It predicts that when — 

In the theory of Fischer and Van Duyne, the different vibrational states w of the precursor and of the successor complex are treated as different reactants and products, with rate constants k ,n given by equation (12), in which LZ is the binary 

Experimental data supporting the inverted rate correlation have been put forward by Creutz and Sutin, using excited ruthenium(ii) species as reductants. Fluorescence 

Science. Physical Chemistry Series Two, Vol. 9, Butterworths, London, 1976, pp. 213—263. 

According to Eq. (30), the rate should maximize at AGe, = —X therefore, the onset of the inverted region should be related to the parameters which control X. This fact can be exploited when it is desired to change the rate of an electron transfer by increasing X [53]. The practical implications are discussed in a later section. 

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The reaction exothermicities ( —AG°) for forward and back ET in polar media were approximately estimated to be 1.39 and 2.18 eV, respectively [120], Since the back ET is highly exothermic, the relatively small kb-1 values for the compartmentalized system may be ascribed to the combined effect of the inverted region [97-99] and the loose ion-pair state. 

To examine the shape that this equation enables us to predict for log k or AG as a function of AG, we substitute the parameter for a specific case. The value of kfc will be taken as 7.4 x 109 L mol-1 s l, that being the value in water at 298 K. Values of k calculated from Eq. (10-66) are shown in Fig. 10-10 as a function of AG. Values of AG are also depicted. The value A = 80 kJ mor1 was used and Z was taken from TST as 6.21 x 1012 s l at 298 K. The effect of introducing the diffusion-controlled limit is that the plot is shaped like a truncated parabola. This figure was drawn with K = k /k-Ac = 0.2 L mol-1. The left side of each of the diagrams shows the inverted region where k decreases and AG increases as AG becomes more negative. 

Electron transfer corresponds to the transition from U, to at the transitional confignration P. AF is the free-energy of the transition. The lower free-energy snrface of the final state corresponds to the inverted region. 

Equation (34.32) is remarkable in the relation that it shows that (1) the observable symmetry factor is determined by occupation of the electron energy level in the metal, giving the major contribution to the current, and (2) that the observable symmetry factor does not leave the interval of values between 0 and 1. The latter means that one cannot observe the inverted region in a traditional electrochemical experiment. Equation (34.32) shows that in the normal region (where a bs is close to ) the energy levels near the Fermi level provide the main contribution to the current, whereas in the activationless (a bs 0) and barrierless (a bs 1) regions, the energy levels below and above the Fermi level, respectively, play the major role. 

FIGURE 34.5 Scheme explaining the absence of the inverted region in the electrochemical processes. The position of the free-energy surfaces 7 and U corresponds to the inverted region. However, the major contribution to the current is due to the transition from to U(, which are in activationless configuration. 

One striking prediction of the energy gap law and eq. 11 and 14 is that in the inverted region, the electron transfer rate constant (kjjj. = ket) should decrease as the reaction becomes more favorable (lnknr -AE). Some evidence has been obtained for a fall-off in rate constants with increasing -AE (or -AG) for intermolecular reactions (21). Perhaps most notable is the pulse radiolysis data of Beitz and Miller (22). Nonetheless, the applicability of the energy gap law to intermolecular electron transfer in a detailed way has yet to be proven. 

Schmickler W, Tao N (1997) Measuring the inverted region of an electron transfer reaction with a scanning tunneling microscope. Electrochim Acta 42 2809-2815... 

Most photoinduced CS processes take place within the Marcus normal region. However, chaige recombination processes take place within the inverted region. Thus, the small reorganization eneigy of CHI will slow down CR processes. 

One of the most important consequences of taking all electrode electronic states into account is the disappearance of the inverted region that is predicted by the simplified treatment. Equation (1.32) indeed entails that the forward rate constant should increase as E = EP becomes more and more negative, reach its maximal value for E — E° = —1, and decrease further on (Figure 1.16a). Similarly, the backward rate constant should increase as = ° becomes more and more positive, reach its maximal value for E — = 1, and decrease... 

The inverted region is so-called because the rate of the electron-transfer process decreases with increasing thermodynamic driving force. 

The inverted region in electron transfer reactions is studied for the reaction of electronically-excited ruthenium(II) tris-bipyridyl ions with various metal(III) tris-bipyridyl complexes. Numerical calculations for the diffusion-reaction equation are summarized for the case where electron transfer occurs over a range of distances. Comparison is made with the experimental data and with a simple approximation. The analysis reveals some of the factors which can cause a flattening of the In k versus AG curve in the inverted... 

To search for the inverted region, experiments were designed so that the donor-acceptor distances were kept fixed by attaching them to a covalent network of rigid spacers, frozen media, electrostatic com-plexation, and protein frameworks. In all of these cases, the... 

The best way to search for the existence of an inverted region (if any) would be to use a single electrochemical electron transfer reaction in one solvent medium at a particular electrode and determine the effect of high overpotential on the reaction rate or the current density. Many experiments were carried out at organic spacer-covered ( 2.0 nm thick) electrodes to search for the inverted region for the outer-sphere ET reactions however, no inverted region was observed." ... 

It was suggested that the absence of an inverted region for the ET reactions at spacer-covered metal electrodes is due to the availabihty of a continuum of electronic states in metal electrodes below the Fermi level. For the same reason, the inverted region is also not expected to be seen for the homogeneous intermolecular ET reactions because a continuum of electronic states are also available below and above the respective ground states of acceptor and donor ions in solutions involved in homogeneous ET reactions. 

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