“Similarly the ratio:
for each member of the series oxidized or reduced by two reagents, a and b, should be constant. This result was found experimentally for the Co(NH3)X compounds reduced by and respectively, with X being NH3, H2O, and Cl− [70] (Table II of Ref. [6]).
The restriction to a given charge type will not be important if the work terms are relatively minor.
The comparison involving V+2 should be accepted with some reserve since the V(II) reaction is not necessarily an ‘outer sphere’ one, as Taube has pointed out.” [6, p. 855]
3.24.Early Comparisons of Theoretical and Experimental Results
(a) Excess Free Energy of Activation
The very first application of the theory consisted in comparisons of experimental and theoretical ΔF*’s [1, p. 870].
The experimental are known very accurately because they depend only on the logarithm of kbi and so a factor of 2 in kbi introduces an error of only 0.4kcal mole−1 in
is found measuring kbi while Eq. (3.17); (or, more simply, Eq. 3.4) is used to compute
In most instances, M. found an encouraging agreement between experimental and calculated ΔF*’s, considering that his theory is free from adjustable parameters. The agreement was in particular excellent for compounds of class I (“tightly knit covalently bound ions”), but was not as good, even if reasonable, for small hydrated cations [3, p. 426].
The reasons given to explain differences between experimental and calculated ΔF*’s are the following:
(i)It is possible that the innermost solvation layer of the ions is not completely dielectrically saturated as assumed. In this case, there would be an additional contribution to ΔF* arising from any changes which may have to occur in interatomic distances in the innermost solvation layer, that is, within the sphere of radius a, prior to the electronic jump, in order to form the activated state.
(ii)The electron tunneling factor may be somewhat less than unity. Note that this factor is not temperature dependent and therefore does not enter into any comparison between experimental and calculated values of the activation energy.
(iii)For reactions in which there is uncertainty of mechanism, say between two successive one-electron transfer and one two-electron transfer it is necessary to know the mechanism before calculating
From calculations applied to concrete ET cases one finds that the relative magnitude of the two contributions to ΔF* in Eq. (3.4), the Coulombic repulsion and the solvent reorganization free energy are generally of the same order of magnitude. From this, it may be inferred that no simple correlation between the reaction rate and the size of the Coulombic term would be expected.
(b) Excess Entropy of Activation
From the carefully studied temperature dependence of the ferrous–ferric reaction, the experimental value of the excess entropy of activation was computed with the aid of Eq. (3.7). It was −23cal mole−1 deg−1 at 0∘C. “The theoretical value is found from Eq. (3.15) to be −14cal mole−1 deg−1. Considering the experimental errors that always accompany a measurement of ΔS* and considering the assumptions of the theory, the experimental and calculated values agree reasonably well.” [1, p. 871]
3.25.Reaction Rates in Heavy Water—A Possible Criterion of Mechanism
The way in which the reaction rate changes when D2O is used as solvent instead of H2O is rather subtle. Any changes of the interatomic O−H distances in the innermost hydration layer needed to form the activated state are easier for the O−H bonds than for the O−D bonds since the former have a higher zero-point energy and so they span a larger range of interatomic distances. Note that the influence of D2O on ET can only happen through the differences in atomic polarization in the dielectrically saturated innermost solvation layer since the two solvents do not differ appreciably in their Ds or Dop and so the difference of rates in the two solvents cannot arise from the dielectrically unsaturated part of the medium.
M. proposed [1, p. 871] an experimental method to help distinguishing between a simple outer-sphere ET mechanism and an atom transfer mechanism in a series of reactions. If one measures the reaction rates for the reactions Fe+2–Fe+3, Fe+2–FeOH+2 and Fe+2–FeCl+2 in H2O and in D2O, one should find a comparable D2O–H2O isotope effect on the reaction rates if these reactions have a small-overlap electron transfer mechanism, that is, if there is but a small overlap of the electronic orbitals of the two reactants in the activated complex, a basic assumption of the theory. Since the Fe+2–Fe+3 and Fe+2–FeOH+2 rate constants were twice as great in H2O as they were in D2O, this effect would predict an effect of similar magnitude for the Fe+2–FeCl+2 reaction.
If a reaction has, on the other hand, an atom transfer mechanism, then a D2O–H2O effect would probably be expected only if the atom were hydrogen. Thus an isotope effect could occur for the Fe+2–Fe+3 and Fe+2–FeOH+2 reactions, but not for the Fe+2–FeCl+2 reaction if it involves a chlorine atom transfer. This expected absence of isotope effect in a chlorine atom transfer process can, of course, be tested by measuring the rate constant of the atom transfer Cr+2–CrCl+2 reaction in the two solvents (vide supra).
3.26.Two Case Studies in Detail (Verbatim with Glosses) [2]