Molecular Imaging. Markus Rudin

and ΔS^‡ defined by the Eyring relation:

At room temperature, the term kT/h equals 10¹³ sec⁻¹ while Z, the collision number in solution, is about equal to 10¹¹ liter mole⁻¹ sec⁻¹ [16]. M. will later use the more precise value 10¹² [16a].

      Marcus calls ΔS^* and ΔF^* excess entropy and excess free energy of activation. The excess for ΔF^* is the free energy of formation of the intermediate state from the reactants in excess of what it would be if the state were simply the usual transient collision complex at the thermally averaged geometry of two neutral, nonreactive (i.e., not forming or breaking chemical bonds) molecules in a solution. ΔS^* and ΔS^‡, ΔΔF^* and ΔF^‡ are related in Note 13 [1, p. 868, M19, Part II of the Theory].

      Equations (3.1) and (3.7) express the rate constant of the redox step in terms of ΔF^* and ΔS^*, and Eqs. (3.4) and (3.6) relate ΔF^* and ΔS^* to ΔF⁰, the standard free energy change of the redox step, and to the polarizing radii (vide infra) of the reactants a₁ and a₂. Defining now and as:

      from Eqs. (3.4) and (3.6) we get:

      In Eq. (3.9), the last term is generally very small and we then have as an approximate equation for ΔS^*:

      3.2.Frequency Factors, Activation Energy, and Transition Probability Factor κ

      Equation (3.1) is also written [4, p. 156] as:

      and the usual Arrhenius expression for k is:

      The preexponential coefficient A is generally known as the frequency factor. The name derives from the interpretation of the expression in terms of the energy barrier to reaction: the exponential states the probability of surmounting the barrier E_a while A is related to the frequency of attempts [17, p. 87]. Writing E_a as E_a = ΔF^* + T ΔS^* we see that:

      The calculated and experimental frequency factors provide the most direct measure of the probability factors κ [4, p.160]. M. found a good agreement between experimental and calculated frequency factors for the and for the systems, supporting — at least for these systems — a probability of adiabatic reaction of the order of one (vide infra).

      3.3.The Dependence of ΔF^* on ΔF0 for a Series of Compounds at Constant λ

      From Eqs. (3.2) and (3.4), a simple dependence of ΔF^* on ΔF⁰ can be deduced for the reaction of a reagent with a series of compounds chemically similar to each other in the sense that the charged part of the molecule remains the same throughout the series and the molecules differ only in some substituent.

      Two such series will be discussed later. The polarizing radius is that of the charged part of the molecule which causes the electrical polarization of the medium. For each member of a series, it will be taken to be the same, since each has the same charged group with the same radius which is then a constant for all members of the series. It will therefore be seen from Eq. (3.2) that λ is the same for each member of the series. Accordingly, we are interested in the dependence of ΔF^* on ΔF⁰ at a given λ, that is, in (∂ ΔF^*/∂ ΔF⁰)_λ. We find, from Eqs. (3.3) and (3.4) that:

      This predicts that when is very small, a plot of ΔF^* versus ΔF⁰ should be a straight line with a slope of When is large, a plot of ΔF^* versus ΔF⁰ will be approximately linear for small changes in ΔF⁰; the slope will be about unity if is as large as unity.

       3.4.Theoretical Equations for Isotopic Exchange Reactions

      For such reactions, the expression for ΔF^* simplifies because and

      Moreover, “since, in the present case, reacting particle 1 as a product is the same as 2 as a reactant, and conversely, we have a₁ = a₂ and will denote these by a. Since R is taken to equal a₁ + a₂ it therefore equals 2a.” The expression for ΔF^* simplifies then to:

      We immediately see that the reorganization term increases as a decreases and Δe increases, as expected from the discussion in Chapter 2.

      As