The Marcus model

During the 1950's chemists were able to establish that reactions consisting of a single electron transfer, have a well-defined rate. From one reaction to another the difference may be as great as that between a snail and an express train. Why? Rudolph Marcus solved this riddle by considering all the details of the reaction. Although no bonds are broken during the reaction, there are still small changes in structure when electrons are added or removed. The lengths of the chemical bonds are changed and the molecules of the solvent are thrown about. Such structural changes require the reorganization energy l.
    In fig 1, l is the energy difference between the bottom of the left parabola (representing the equilibrium positions before the reaction) and a point vertically above on the upper part of the other curve (representing the reaction products A/A^- before any atomic positions have been changed).
    Marcus was the first to calculate l. He also realized that the leap of the electron only requires the energy l/4 via the crossing point of the parabolas. The activation energy is thus E_a = l/4. One may then, according to well-known principles, derive that:

Reaction rate = n exp(-E_a/kT)

where k is the Boltzmann constant and T the absolute temperature. The preexponential factor (n) is the vibration frequency of the atoms around their equilibrium positions. The reorganization energy l, and hence E_a, varies greatly from one reaction to another. In this way one may explain the great variations in reaction rate.
     A reaction where the energy E is liberated corresponds in the Marcus model to moving the energy curve of the products (marked D⁺/A^-) down by the amount E. As is evident from fig 2 the activation energy (E_a) is decreased and is instead E_a = (E-l)²/4l. The dependence of the reaction rate on E is then parabolic, and this theoretical prediction agrees very well with experimental results.
    If the liberated energy E is equal to l, then E_a will be equal to zero and the reaction is very fast.
     For E>l a so called inverted region is reached (see fig 3), where the activation energy E_a increases with E. The reaction is slower the greater the energy liberated. This rather remarkable result has been experimentally confirmed.
    An important part of the reorganization energy, l, is dependent on the solvent. The higher the dielectric constant of the latter the larger the value obtained for l. Water with a high dielectric constant is thus a poor solvent if fast electron transfer reactions are desired.
__________

n nu    l lambda

Fig 1
Self-exchange reaction
A^- + A A + A^- e.g.
*MnO₄^2- + MnO₄^-
*MnO₄^- + MnO₄^2-

Fig 2
A reaction D + A D⁺ + A^-
where the energy E is liberated e.g.
Fe²⁺ + Ce⁴⁺Fe³⁺ + Ce³⁺

Fig 3
The inverted region where E >l

Safety light

When the rod is bent a glass ampoule is broken, the liquids from the ampoule and the rod are mixed and 'cold light' is produced. The phenomenon is an example of chemiluminescence, a complicated process involving electron transfer steps. Due to the inverted region the total reaction results in an excited state from which light is emitted. Safety lights of this kind, which are non-flammable and weatherproof are used by seamen and divers in emergency.