Marcus equation Page: 1 of 11
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September 21, 1998
In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating
the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are
reactions in which an electron is transferred from a donor to an acceptor without
any chemical bonds being made or broken. (Electron-transfer reactions in which
bonds are made or broken are referred to as inner-sphere reactions.) Marcus
derived several very useful expressions, one of which has come to be known as the
Marcus cross-relation or, more simply, as the Marcus equation. It is widely used
for correlating and predicting electron-transfer rates. For his contributions to the
understanding of electron-transfer reactions, Marcus received the 1992 Nobel
Prize in Chemistry.
In common with ordinary chemical reactions, an electron-transfer
reaction can be described in terms of the motion of the system on an energy
surface. As the reaction proceeds the system moves from the reactant minimum
(initial state) to the product minimum (final state). The nuclear configurations of
the reactants and products and the configuration of the surrounding solvent are
constantly changing as a consequence of thermal motion. Marcus showed that,
subject to certain assumptions, these fluctuations can be described in terms of
displacements on harmonic free-energy curves that are a function of a single
reaction coordinate. Two harmonic free-energy curves are needed to describe the
reaction - one refers to the reactants plus surrounding medium and the other to
the products plus surrounding medium. The two free-energy curves have
identical force constants and the reaction coordinate is the difference between the
reactant and product free energies at a particular nuclear configuration.
The free energy of the close-contact reactants plus surrounding medium
(Curve R) and the free energy of the close-contact products plus surrounding
medium (Curve P) are plotted vs. the reaction coordinate in Figure 1. The plot is
for an electron-transfer reaction with zero standard free-energy change (an
electron self-exchange reaction). The free-energy curves intersect where the
reactants plus surrounding solvent and the products plus surrounding solvent
have the same nuclear configurations and energies. This intersection defines the
transition state for the reaction: the energy required to reach the intersection is
the free energy of activation for the reaction. Also shown is X, the reorganization
parameter, which is the vertical difference between the free energies of the
(noninteracting) reactants and products of a self-exchange reaction at the
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Marcus equation, report, November 1, 1998; Upton, New York. (https://digital.library.unt.edu/ark:/67531/metadc711686/m1/1/: accessed March 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.