Coulombic work

Since the electronic interaction of the two reactants becomes more favorable with decreasing separation, the most favorable configuration for electron transfer is generally one in which the two reactants are in close proximity. Opposing this is the coulombic work required to bring similarly-charged reactants together, and ultimately the electron-electron... 

Step 1 To calculate the quantity of electricity in coulombs, work in amperes and seconds. 

The Tafel slope can be obtained from eqn. (153) by differentiation and substitution from eqn. (154) as a Br nsted slope. It should be noted that a is an empirical parameter defined in Sect. 2.1 and is not part of the theory. From eqns. (153) and (154), the transfer coefficient at constant Coulombic work terms is given by... 

In the absence of ion pairing and rate limitation by solvent dynamics, the volume of activation for adiabatic outer-sphere electron transfer in couples of the type j (z+i)+/z ju principle, be calculated as in equation 2 from an adaptation of Marcus-Hush theory. In equation 2, the subscripts refer respectively to volume contributions from internal (primarily M-L bond length) and solvent reorganization that are prerequisites for electron transfer, medium (Debye-Huckel) effects, the Coulombic work of bringing the reactants together, and the formation of the precursor complex. 

Since ionization potentials of anionic donors and electron affinities of cationic acceptors are not readily available, Mulliken correlations for charge-transfer ion pairs are generally presented in a modified form using electrochemical oxidation or reduction potentials, respectively. A typical example of such a modified Mulliken plot with unit slope is shown in Figure 2 for the CT ion pairs of TpMo(CO)3 [Tp = hydrido-trM-(3,5-dimethylpyrazolyl)borate] as the donor and various pyr-idinium acceptors [127]. Similar (modified) Mulliken correlations with unit slopes have been found for numerous other ion pairs with pyridinium acceptors and Mn(CO)s [126], Co(CO)4 [118], or V(CO)e [118] as donors. It is important that the Coulombic work term (co) in Eq. 8 is explicitly included in all Mulliken evaluations of ion pairs with different structures since co reflects the electrostatic energy of the (ground-state) ion pair which strongly depends on the inter-ionic distance [125]. 

The reason for the higher activation energy of surface conductivity could first of all be the strong coulombic work term AG for proton motion along the array of S OIT groups. This contribution is essential, if the proton is transferred from an SC)7 group to an intermediate water molecule in the rate-limiting step. 

AeGp and AcGr are the Coulombic work terms for the reactants and products, respectively. They are discussed in more detail below in the section dealing with formation of the precursor complex, k r and K p are the force constants of the yth vibrational coordinate in a species when it participates as a reactant and product, respectively, and Alj is the associated change in bond length. The latter quantities allow one to calculate the work associated with inner sphere reorganization. 

Estimate the outer sphere reorganization energy for the homonuclear electron transfer between Fe(H20)g and Fe(H20)g assuming that the radii of the reactants are 353 and 337 pm, respectively, and that the Coulombic work terms AcGr and AcGp are zero. In the transition state the reactants are assumed to be in contact, so that a is equal to + Tb-... 

For a homonuclear reaction AG° is zero. Since the Coulombic work terms are neglected in this calculation, the factor m is equal to 1/2. It follows from equation (7.8.27) that the outer sphere contribution to the Gibbs activation energy is equal to... 

The first term in Equation 1.4 was retained from Debye s colliding sphere model the electron-donor and electron-acceptor species were viewed as spheres of radii r-i and r2 that possessed charges of z, and z2, respectively. This term is associated with the electrostatic energy (Coulombic work) required to bring the two spheres from an infinite distance to the center-to-center separation distance, ri2 G + G, which is also known as the distance of closest approach (formation of the precursor complex [Dn—Am] in Scheme 1.1). The magnitude of the Coulombic term is modified by a factor exp(—yr12), which accounts for the effects of the dielectric medium (of dielectric constant D) and of the total ionic strength /a. 

Here, AG is defined by Equation 1.4, k is the transmission coefficient, and Z is the collision frequency in units of M 1 s 1. The transmission coefficient is discussed above. In practice, k is often set equal to 1. Although this gives reasonable results in numerous cases, this is one of the many assumptions embedded within the familiar, classical form of the Marcus equation (Equation 1.4). Expansion of Equation 1.4 gives Equation 1.12, in which the Coulombic work term (i.e., the first term on the right-hand side of Equation 1.4) is abbreviated as W(r). 

As mentioned above, Z is the pre-exponential factor and Wy are Coulombic work terms (Equation 1.4) associated with all four combinations of the reacting species. If k22 is known, one can use kx2 and Equations 1.24—1.26 to calculate k, which is related by Equation 1.19 to its reorganization energy In. The C 2 and/12 terms often approach unity for molecules that possess small charges.4,22 For reactions of charged inorganic complexes, however, these terms can be important. 

Here, k is the electronic transmission coefficient (k = 1 for adiabatic electron transfer) and v x the nuclear frequency factor, whereas is the equilibrium constant for assembly of a precursor state and effectively includes any coulombic work and medium (Debye-Hiickel) terms [4, 5]. Following the approach taken by Stranks [7], the observed volume of activation AV for a simple, adiabatic, outer-sphere, bimolecular electron transfer reaction can be represented as... 

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