Chemistry 1956

On the Theory of Oxidation–Reduction Reactions Involving Electron Transfer

Rudolph A. Marcus

Before an electron can jump, the solvent must reorganize — and that sets the speed.

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In depth · the introduction

Make a reaction more favourable and it speeds up — usually. Marcus found the strange exception, and a whole theory of how electrons move was hiding inside it.

The idea, unpacked

A huge class of chemistry comes down to a single electron jumping from one molecule or ion to another — the heart of batteries, rust, respiration, and the first step of photosynthesis. Rudolph Marcus asked a deceptively simple question: what sets the speed of that jump?

His answer: the electron can't jump until its surroundings are ready. An ion in water is wrapped in a shell of solvent molecules arranged to suit its present charge. Move the electron and that arrangement is suddenly wrong. So nothing happens until ordinary thermal jiggling momentarily twists the solvent — and the molecule's own bonds — into a shape that fits the 'before' and the 'after' equally well. Only then does the electron slip across, costing no energy. The work needed to reach that in-between shape, the reorganization energy, together with how downhill the reaction is, sets the speed.

Where it came from

In the early 1950s, working at the Polytechnic Institute of Brooklyn, Marcus was puzzling over why some electron-exchange reactions in solution are lightning fast and others sluggish, when in each case no chemical bond is made or broken. The reigning theory of reaction rates, built for reactions that rearrange atoms, simply did not apply. In a 1956 paper in the Journal of Chemical Physics — the first of a long series — he found the missing picture by following what the surrounding solvent had to do.

The theory made one prediction so strange that many chemists refused to believe it: beyond a certain point, making a reaction more favourable should make it slower. It took until 1984 for an experiment to catch this 'inverted region' in the act. Eight years later, in 1992, Marcus received the Nobel Prize in Chemistry.

Why it mattered

For the first time, the rate of an electron transfer could be calculated rather than just measured, from two understandable quantities. The same equation works for a rusting nail, a redox enzyme, a battery electrode, and a solar cell. And the inverted region turned out to be useful, not just curious: it is part of why the charge-separated state in photosynthesis — and in well-designed solar cells — survives long enough to do work instead of instantly collapsing back.

An everyday analogy

Picture a crowd in a stadium, all leaning toward one side to watch a play. To move the action to the far side (to transfer the electron), the crowd must first shift its lean so neither side is favoured — an awkward, effortful in-between. Only from that balanced moment can the play jump across without anyone lurching. Now the twist: if the far side is much more exciting (a big driving force), you might expect an instant switch — but reaching the balanced lean actually takes more contortion, so the change can come slower, not faster. Slide the controls below and watch the barrier shrink, vanish, then grow again.

Where it sits

Marcus's theory completed a story begun by transition-state theory (Eyring, 1935), which had explained rates for reactions that rearrange atoms but not for bare electron jumps. It draws on the Franck–Condon idea from spectroscopy — that electrons move far faster than nuclei — and connects to the quantum chemistry of the bond (Pauling) on one side and the redox machinery of life on the other. Closely related results were found by Hush, and a quantum version by Levich and Dogonadze.

The original document

Original source text

R. A. Marcus · J. Chem. Phys. 24(5), 966–978 · May 1956 · Polytechnic Institute of Brooklyn

The problem

A large class of reactions in solution does nothing but pass an electron between two species — the exchange Fe²⁺ + Fe³⁺ → Fe³⁺ + Fe²⁺ is the type case — with no bonds made or broken. Transition-state theory, built for reactions that rearrange atoms along a single coordinate, gave no handle on them: why is one such electron exchange fast and another slow, when chemically nothing seems to happen?

A mechanism for electron transfer reactions is described, in which there is very little spatial overlap of the electronic orbitals of the two reacting molecules in the activated complex.

The slight-overlap mechanism

Because the electronic coupling is weak, the electron cannot simply hop whenever the partners collide: the solvent, polarized around the old charge distribution, would be left out of equilibrium with the new one, and energy would not be conserved (the Franck–Condon principle, applied to the nuclei). Marcus required instead that thermal fluctuations of the solvent polarization — and of the reactants' own bond lengths — first carry the system to a nuclear configuration in which the reactant and product electronic states have equal energy. Only there can the electron move at constant energy; the solvent then relaxes about the new charges.

Two parabolas and the reorganization energy

Treating the solvent as a dielectric continuum, Marcus showed that the free energy of the reactant state and of the product state are each, to a good approximation, a parabola in a collective reaction coordinate measuring the nonequilibrium polarization. The reaction proceeds through their intersection. The height of that intersection — the activation free energy — is fixed by just two quantities: the driving force ΔG° and the reorganization energy λ, the energy it would take to distort the reactants' nuclei and surrounding solvent into the products' equilibrium arrangement without moving the electron.

The result is compact — ΔG‡ = (λ + ΔG°)² / 4λ — and for the solvent contribution Marcus gave λ as a continuum expression in the reactant radii, their separation, and the optical and static dielectric constants of the medium.

[ … ]

The inverted region

The formula carries a startling consequence. As the reaction is made more favourable (−ΔG° rising from zero) the barrier falls — until, at −ΔG° = λ, it vanishes and the rate is greatest. Push further and the barrier returns: the rate now decreases as the driving force grows. This 'inverted region' was thought so implausible that it was doubted for some twenty-five years, until rigid donor–acceptor molecules confirmed it in 1984.

What followed

This was the first of a celebrated series running through the 1950s and 1960s, in which Marcus added the cross-relation linking a reaction's rate to its self-exchange rates. N. S. Hush reached closely related results; Levich and Dogonadze recast the theory quantum-mechanically; the experimental foundation came from Taube, Sutin, and others. The work brought Marcus the 1992 Nobel Prize in Chemistry.

Polytechnic Institute of Brooklyn · 1956