Molecular Imaging. Markus Rudin

The remaining three terms of Eq. (4.35) should be added to Eqs. (4.17) and (4.18)” [2, pp. 203–204].

      M. demonstrates that “because of a cancellation of the newly added terms” the equation for ΔF* remains unchanged “except for the new significance of φ_s” [2, p. 203].

       4.17.Summary and Final Remarks

      At the end of this chapter, it may be useful to highlight some pivotal points of Marcus’ treatment.

      M. begins with an extension to the electrode system of his general equation for the electrostatic free energy F_e of systems with nonequilibrium dielectric polarization. In the electrode system, he explicitly considers that the central ions have ionic atmospheres. Moreover, a potential drop χ is present at the electrode–solution interface. The expression for F_e is obtained by the two stages charging process that has been described in detail, for a simplified model, in Chapter 3. The expression for F_e in a nonequilibrium polarization system is then obtained from by a variational calculation considering that F_e obtains from when, because of fluctuations (variations) δP_u and δc_i, F_e departs from its equilibrium value. M. found an expression for in which P_u = α_uE—as expected for a system in electrostatic equilibrium—and an expression for c_i(r) is given as function of φ(r). M. considers here also explicitly for the first time the hypothetical equilibrium system, called equivalent equilibrium system, whose P_u is the same as that in X* and X but in electrostatic equilibrium with fictitious charges on the ions. The treatment of the equivalent equilibrium distribution in Ref. [20] will later extend this concept.

      The ionic charge on the electrode is described in terms of the electrical images of the ions in the body of the solution and at the electrical double layer. With a second variational calculation, Marcus finds the expression for the minimum value of ΔF* corresponding to the activation free energy for the ET process.

      Formulas (4.31)–(4.33) for ΔF*, λ, and m are very similar to those for the redox systems and were discussed in detail in Chapter 3. A unifying point of view of the formulas for redox and electrode systems consists in reducing the ET in the electrode case to that between an ion and its electrical image in the electrode.

      It is finally demonstrated that the presence of χ can be neglected in deriving the final equations because of cancellations. For the same reason, the presence of adsorbed ions in the double layer can also be overlooked.

      The contribution of the ionic atmosphere to the free energy of activation has also been demonstrated to be small. In Chapter 6, this last point will be taken up again and an expression for the contribution of to ΔF* will be given.

       NOTES

      1.M: “The charge on the electrode is an image charge, it is not uniformly distributed over the electrode. It is on the surface of the electrode but it behaves as though is situated a certain distance into the electrode, that’s the image.”

      2.M: “If you have a certain species of ions you may change a local concentration keeping the total number of ions of that species constant.”

      3.M: “We have, in the case of a reaction at an electrode, a localization analogous to that considered by Sutin when he considers the reactants to come together and to form a complex.”

       References

1.R. A. Marcus, “Theory and Applications of Electron Transfers at Electrodes and in Solution,” in Special Topics in Electrochemistry, p. 161, P. A. Rock, Editor, Elsevier, New York (1977).

2.R. A. Marcus, “Electrostatic Free Energy and Other Properties of States Having Nonequilibrium Polarization. II. Electrode Systems. (ONR Report No. 11, dated 1957),” in Special Topics In Electrochemistry, p. 210, P. A. Rock, Editor, Elsevier, New York (1977); (a) R. A. Marcus, “On the Theory of Overvoltage for Electrode Processes Possessing Electron Transfer Mechanisms. I. (ONR Report No. 12 dated 1957),” in Special Topics in Electrochemistry, p. 180, P. A. Rock, Editor, Elsevier, New York (1977).

3.A. J. Bard, L. R. Faulkner, Electrochemical Methods, Fundamentals and Applications, John Wiley & Sons, Inc., New York, Chichester, Brisbane (1980).

4.R. A. Marcus, J. Chem. Phys. 24, 966–978, (1956).

5.R. Parsons, in Modern Aspects of Electrochemistry, Chap. 3, J. O’M. Bockris, Editor, Academic Press, Inc., New York (1954).

6.Compare Reference 5 for bibliography.

7.M. Mason, W. Weaver, The Electromagnetic Field, Dover Publications, Inc., New York (1929).

8.R. G. Sachs, D. L. Dexter, J. Appl. Phys. 21, 1304, (1950).

      9.J. O’M. Bockris, A. K. N. Reddy, Modern Electrochemistry 1, 2nd Edition, Plenum Press, New York and London (1998).

10.R. A. Marcus, J. Chem. Phys. 24, 979–989, (1956).

11.J. E. B. Randles, Trans. Faraday Soc. 48, 828, (1952).

12.A. A. Vlcek, Coll. Czech. Comm. 20, 894 (1955).

13.R. A. Marcus, Trans. N.Y. Acad. Sci. 19, 423, (1957).

14.D. C. Grahame, Chem. Rev. 41, 441, (1947).

      15.Compare reviews by (a) J. O’M. Bockris, Modern Aspects of Electrochemistry, Chap. 4, Academic Press, Inc., New York, 1954; (b) P. Delahay, New Instrumental Methods in Electrochemistry, Interscience Publishers, Inc., New York, 1954; (c) A. E. Remick, J. Chem. Ed. 33, 564, (1956).

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