Molecular Imaging. Markus Rudin. Читать онлайн. Newlib. NEWLIB.NET

Автор: Markus Rudin
Издательство: Ingram
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Жанр произведения: Медицина
Год издания: 0
isbn: 9781786346865
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the preceding discussion, we may write for the mechanism of reaction (4.19) the following scheme, analogous to that in Ref. [4]” and here reported in Chapter 2:

      “Here, a central ion just outside the double layer is denoted, in its initial electronic state, by A and in its final state by B. As noted earlier, step (4.20) involves a suitable reorganization of the solvent and ionic atmosphere and (if necessary) a suitable penetration of the electrical double layer. Step (4.21) is the actual ET itself, and step (4.22) involves a reversion of configuration of solvent and atmosphere to one in equilibrium with the new charge on the central ion. Step (4.22) also involves a motion away from the electrode. As in the redox theory, the reverse of Eq. (4.22) occurs but it needs not be considered in the computation of kf . Steady-state considerations for cX* and cX” (where the c’s are concentrations) “lead to the relation [4, p. 969]:

image

      As discussed later, when the probability of ET in the lifetime of the intermediate state X* (10−13 sec) is large, kf is about half of k1. k1 depends on the free energy of formation of X* from the initial state in reaction (4.20). We proceed first to the calculation of this free energy change and later to a discussion of the evaluation of the rate constants.”

      The treatment given here by Marcus in 1957 is today considered by him only as a formalism that helped him to get started, just a step in an evolution, “a thing that may be interesting from the point of view of how thoughts developed but not for presentation of a way of thinking. That was history, I’m not thinking anymore in those terms.” (M). I shall then not discuss the rate constants of the elementary steps and I refer the reader to Chapter 6 for the modern treatment.

      4.12.Free Energy of Formation of State X*

      “The electrostatic contribution, ΔF*, to the free energy of formation of state X* from state A will be different from zero because of

      (1)The work which may be required to transport the central ion from a point just outside the double layer to some particular point in it (if necessary)

      (2)The work required to reorient the solvent molecules and the ionic atmosphere to a nonequilibrium configuration, for this position of the central ion

      Let image and image denote the electrostatic free energy of the electrode system when the system is in the states A, X*, X, and B, respectively (throughout, asterisks will be used to designate the properties of X* and will be omitted in designating the properties of X). We have then for ΔF*:

image

      image is given by the general expression (4.24) and Fe is given by a similar equation, minus the asterisks:

      where image = electric field at point r exerted in a vacuum by all the ionic charges in the state X* and by those electrode charges which they would induce in a vacuum.

      E*(r) = electric field in state X*. It equals −∇φ*.

      φ*(r) = inner potential in state X* (cf. Eq. (4.1))

      αe = E-type polarizability = (Dop − 1)/4π

      αu = U-type polarizability = (DsDop)/4π

      ci (r) = concentration of ions of type i in states X* and X

      image = average concentration of ions of type i. It equals their number

      ni in the solution divided by the latter’s volume V

      Pu(r) = U-type polarization in states X* and X

      χ = potential drop at electrode–solution interface due to an oriented solvent dipolar layer”

      Note that ci(r) and Pu(r) in state X* are the same as those in state X because of the constant atomic configuration restriction.

      image is obtained from the general Eq. (4.24) introducing in it the equilibrium expressions (4.14) and (4.15) for Pu and ci.

      image is given by a similar equation, but the functions in the integrand now refer to state B.

      Equation (4.24) and subsequent equations treat χ as being independent of the average electrode charge density.” However, M. shows that “the final equations” (4.31)–(4.33) in the following “are unchanged even ifχ were a function of this quantity” [2, pp. 186–187].

       4.13.Constraint Imposed by the Constant Energy — Constant Atomic Configuration Restriction

      “The free energy of formation of X* from A consists of the electrostatic contribution ΔF* and of a term, described later, associated with the localization of the center of gravity of the central ion in a narrow region near the electrode. (3) The free energy of formation of X from B contains an electrostatic term Скачать книгу