Outer-sphere formation constants, calculation

The first step is diffusion controlled while the second represents the fast formation of the outer sphere complex. The final step involves the conversion of the outer to the inner sphere complex. This is the rate determining step and is dependent on the equilibrium concentration of the outer sphere complex. Consequently, calculations of rate constants by the Eigen model involves estimation of the formation constant of the outer sphere species. 

Fuoss (15) proposed an equation which has been used frequently to calculate outer sphere formation constants. The equation has the form ... 

Under these conditions, the formation rate constant, k, can be estimated from the product of the outer sphere stability constant, Kos, and the water loss rate constant, h2o, (equation (28) Table 2). The outer sphere stability constant can be estimated from the free energy of electrostatic interaction between M(H20)q+ and L and the ionic strength of the medium [5,164,172,173]. Consequently, Kos does not depend on the chemical nature of the ligand. A similar mechanism will also apply to a coordination complex with polydentate ligands, if the rate-limiting step is the formation of the first metal-ligand bond [5]. Values for the dissociation rate constants, k, are usually estimated from the thermodynamic equilibrium constant, using calculated values of kf ... 

The dashed line in the complex in (4.21) and (4.22) indicates an outer-sphere (o.s.) surface complex, Kos stands for the outer-sphere complex formation constant and kads [M 1 s 1] refers to the intrinsic adsorption rate constant at zero surface charge (Wehrli et al., 1990). Kos can be calculated with the help of a relation from Gouy Chapman theory (Appendix Chapter 3). 

We can now calculate the outer-sphere complex formation constant according to Eq. (4.24) ... 

The overall rate const kf is measured by the relaxation methods. Assuming that the outer-sphere complex formation is faster than the water substitution by ligand L, the interchange rate constant k can be calculated using the equation... 

The rate constants (ket) of electron transfer from Fc to [(TPP)M] agree well with those evaluated in light of the Marcus equations [91] for outer-sphere electron transfer (Eq. 4) [215]. Such agreement clearly demonstrates that electron transfer from Fc to [(TPP)M]+ in Scheme 15 proceeds via an outer-sphere pathway. In contrast to this, the ket value of the acid-catalyzed electron transfer from (TPP)Co to O2 is 10 -fold larger than that expected from an outer-sphere electron transfer [215]. Such huge enhancement of the observed rate relative to that calculated for outer-sphere electron transfer indicates the strong inner-sphere nature of acid-catalyzed electron transfer from (TPP)Co to O2 this should result in formation of the hydroperoxo complex, [(TPP)Co02H]+ (Scheme 15, M = Co). Other metalloporphyrins (M = Fe and Mn) can also act as efficient catalysts of the reduction of... 

This expression relates the second-order rate constant, k, for an outer-sphere electron transfer reaction to the free energy of reaction, AG°, with one adjustable parameter, X, known as the reorganization energy. Wis the electrostatic work term for the coulombic interaction of the two reactants, which can be calculated from the collision distance, the dielectric constant, and a factor describing the influence of ionic strength. If one of the reactants is uncharged, Wis zero. In exact calculations, AG should be corrected for electrostatic work. The other terms in equation 46 can be treated as constants (Eberson, 1987) the diffusion-limited reaction rate constant, k, can be taken to be 10 M" is the equilibrium constant for precursor complex formation and Z is the universal collision frequency factor (see Eberson, 1987). 

The results of three ultrasonic investigations on lanthanide salts have been reported. The studies on erbium(iii) perchlorate in aqueous methanol suggest that inner-sphere perchlorate complexes occur at water mole fractions of less than 0.9. On that basis, the rate constant for the formation of the inner-sphere complex from the outer-sphere complex at 25 °C is 1.2 x 10 s. The case of erbium(m) nitrate in aqueous methanol is more complicated and it is suggested that the mechanism involves the existence of two forms of the solvated lanthanide ion, differing in coordination number, in equilibrium with the outer- and inner-sphere complexes. The results for aqueous yttrium nitrate, on the other hand, represent a simplification over those of previous ultrasonic studies on the lanthanides. The authors reject the normal multistep mechanism in favour of a single diffusion-controlled process. Unfortunately, the computed value for the formation rate constant kt of 1.0 x 10 1 mol s is at least two orders of magnitude lower than the value calculated on the Debye-Smoluchowski approach, but the discrepancy is attributed to steric effects. 

The mixed-valence complex [(bipy)2ClRu(III)Ph2PCH2PPh2Ru(n)Cl(bipy)2] is formed in mixtures of the corresponding [Ru(III),(III)] and [Ru(II),(II)] species and the rate of intramolecular electron transfer, 8.8 x 10 s has been calculated from an IT band at 1245 nm in acetonitrile solution. Comparison with the [Ru(bipy)2pyCl] self-exchange rate in acetonitrile is made possible by use of an estimated precursor complex formation constant of 0.6 mol liter" from which the outer-sphere intramolecular rate constant of 8 x 10 s" was evaluated. There is reasonable agreement especially when differences in the distance between redox sites (6.8 and 13.2 A) are taken into account. 

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