The energy of reorganization

There is a small complication in that the frequency to is different for the reduced and oxidized states so that one has to take an average frequency. Marcus has suggested taking u av = 2woxu red/(wox + a reci)-When several inner-sphere modes are reorganized, one simply sums over the various contributions. The matter becomes complicated if the complex is severely distorted during the reaction, and the two states have different normal coordinates. While the theory can be suitably modified to account for this case, the mathematics are cumbersome. 

To obtain an estimate for the energy of reorganization of the outer sphere, we start from the Born model, in which the solvation of an ion is viewed as resulting from the Coulomb interaction of the ionic charge with the polarization of the solvent. This polarization contains two contributions one is from the electronic polarizability of the solvent molecules the other is caused by the orientation and distortion of the 

In a static field both components of the polarization contribute, and the static value es of the dielectric constant must be used in Eq. (6.25). The slow polarization is obtained by subtracting Pf, which gives  

The reorganization of the solvent molecules can be expressed through the change in the slow polarization. Consider a small volume element AC of the solvent in the vicinity of the reactant it has a dipole moment m = Ps AC caused by the slow polarization, and its energy of interaction with the external field Eex caused by the reacting ion is —Ps Eex AC = —Ps D AC/eo, since Eex = D/eo- We take the polarization Ps as the relevant outer-sphere coordinate, and require an expression for the contribution AU of the volume element to the potential energy of the system. In the harmonic approximation this must be a second-order polynomial in Ps, and the linear term is the interaction with the external field, so that the equilibrium values of Ps in the absence of a field vanishes  

During the reaction the dielectric displacement changes from Dox to Dred (or vice versa), and the equilibrium value from Dox/2aeo to Drec[/2a eo. From Eq. (6.5) the contribution of the volume element AV to the energy of reorganization of the outer sphere is  

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Thus, the energy of reorganization A from Marcus theory is replaced by the sum A + D, and the activation energy is significantly enhanced. 

As demonstrated in Section 2.2, the energy of activation of simple electron transfer reactions is determined by the energy of reorganization of the solvent, which is typically about 0.5-1 eV. Thus, these reactions are typically much faster than bondbreaking reactions, and do not require catalysis by a J-band. However, before considering the catalysis of bond breaking in detail, it is instructive to apply the ideas of the preceding section to simple electron transfer, and see what effects the abandomnent of the wide band approximation has. 

Calculate the intersection point of these two parabolas and the energy of reorganization. Prove Eqs. (6.6) and (6.7) for this system. 

From Eq. (6.31) calculate the energy of reorganization of a single spherical reactant in the bulk of a solution. Derive Eq. (6.32) for a reactant in front of a metal electrode. 

We identify Aj = mjUtjAj/2 as the contribution of the mode j to the energy of reorganization [see Eq. (6.5)]. The thermal averaging is simplified by the fact that the expression does not depend on the nuclear momenta, which dropped out when the two Hamilton functions were subtracted. Explicitly we have ... 

The part that remains is the interaction with the solvent. Up to now, we have treated this in a comparatively simple manner, assuming that the energy of reorganization drops by a factor of 2 as the reactant approaches the surfaces [56]. This approximation is sufficient, as long as we focus on the catalytic properties of the electrode, comparing the electronic properties of various materials, as we do here. However, if we want to compare different processes, we need a better approximation. We shall return to this point toward the end of this chapter. In the next sections, we focus on hydrogen evolution/oxidation. 

The contribution from iimer-sphere modes are easily obtained within the harmonic approximation. Let Aq be the change in the bond length of a mode with frequency co and effective mass m then its contribution to the energy of reorganization is... 

The energy of reorganization is a molecular concept, and an equation such as (10), which is based on macroscopic electrostatics, can only be a rough approximation. Several other expressions, on the basis... 

When the electronic interaction A is large and the energy of reorganization X is small, there is only one stationary state. This corresponds to an adsorbed state, which sits in a single potential well on the electrode surface. 

When the electronic interaction is weak and the energy of reorganization is large, there are three stationary solutions two minima separated by a barrier. The two minima correspond to two different... 

Fig. 3 Adiabatic potential-energy curves for various values of the energy broadening A full line A = 0.01 eV long dashes A = 0.05 eV short dashes A = 0.1 eV. The energy of reorganization was taken as 6 ksT.

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