Molecular Imaging. Markus Rudin

in addition to Eq. (3.66), the result that:

      When this and the again given by Eq. (3.66), are introduced into the expression for ΔG^r, Eq. (3.67) follows again but with λ₀ replaced by λ₀ + λ_i, λ₀ being given by Eq. (3.69) and λ_i by Eq. (3.48), with similar remarks for the electrode reaction case.

      3.12.Additive Properties of the Reorganization Term λ

      For the reorganization term λ, there is an additivity property which is important to establish for subsequent correlations. Moreover, the λ’s for homogeneous and electrochemical systems are also related with each other.

      Let us consider the electron exchange reaction:

      and let us indicate its λ₀ as

      where a₁ and a₂ are the radii of the oxidized and of the reduced form respectively.

      Defining as and likewise we have that

      This equation is more simply written as:

      If the distance R between the ions is very large, so large that the force field from one reactant does not influence the other, the fluctuations around each reactant are independent [18] and one can write:

       and will be simply indicated in the following as λ₁ and λ₂.

      If, moreover, and we indicate with a their mean, we have:

      Considering now the reaction:

      with parallel reasoning and symbols we get:

      Considering finally the cross-reaction of the earlier electron exchange reactions:

      always in the hypotheses we have:

      For R large:

      so that finally a cross relation follows among the λ’s of reactions (3.76), (3.79), and (3.80):

      At finite distance, we have the approximate relation

      with R = a + b.

      For the electrochemical reaction:

      λ is given by Eq. (3.35) so that:

      More simply:

      and we see that for large R:

      where λ_ex and λ_el are the λ’s for the exchange and electrode reactions, respectively.

      The equation agrees with the fact that “in the electrochemical case there is only a contribution” to λ “from one ion” [18].

       3.13.A Synopsis of Equations for Chemical and Electrochemical ET Reactions

       3.14.Correlation between the Rate Constants of Two Electron Exchange Reactions and That of Their Cross-Reaction

      It follows from Eqs. (3.1, 3.78, 3.25) that when condition (3.26) is fulfilled, the forward rate constant k₁₂ of the cross-reaction (3.80) is correlated to the rate constants k₁₁ and k₂₂ of the isotopic exchange reactions (3.76)