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Marcus theory

Marcus theory is a chemistry topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Marcus theory rather than just read about it. In short: In theoretical chemistry, Marcus theory is a theory originally developed by Rudolph A. Marcus, starting in 1956, to explain the rates of electron transfer reactions – the rate at which an electron can move or jump from one chemical species (called the electron donor) to another (called the electron acceptor).

Marcus theory — main illustration
Marcus theory — illustration

Key takeaways

  • Marcus theory belongs to chemistry; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Marcus theory to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Marcus theory from memory before moving on to harder problems.

Reference excerpt

In theoretical chemistry, Marcus theory is a theory originally developed by Rudolph A. Marcus, starting in 1956, to explain the rates of electron transfer reactions – the rate at which an electron can move or jump from one chemical species (called the electron donor) to another (called the electron acceptor). It was originally formulated to address outer sphere electron transfer reactions, in which the two chemical species only change in their charge with an electron jumping (e.g. the oxidation of an ion like Fe2+/Fe3+), but do not undergo large structural changes. It was extended to include inner sphere electron transfer contributions, in which a change of distances or geometry in the solvation or coordination shells of the two chemical species is taken into account (the Fe-O distances in Fe(H2O)2+ and Fe(H2O)3+ are different). For electron transfer reactions without making or breaking bonds Marcus theory takes the place of Eyring's transition state theory which has been derived for reactions with structural changes. Both theories lead to rate equations of the same exponential form. However, whereas in Eyring theory the reaction partners become strongly coupled in the course of the reaction to form a structurally defined activated complex, in Marcus theory they are weakly coupled and retain their individuality. It is the thermally induced reorganization of the surroundings, the solvent (outer sphere) and the solvent sheath or the ligands (inner sphere) which create the geometrically favourable situation prior to and independent of the electron jump. The original classical Marcus theory for outer sphere electron transfer reactions demonstrates the importance of the solvent and leads the way to the calculation of the Gibbs free energy of activation, using the polarization properties of the solvent, the size of the reactants, the transfer distance and the Gibbs free energy Δ G ∘ {\displaystyle \Delta G^{\circ }} of the redox reaction. The most startling result of Marcus' theory was the "inverted region": whereas the reaction rates usually become higher with increasing exergonicity of the reaction, electron transfer should, according to Marcus theory, become slower in the very negative Δ G ∘ {\displaystyle \Delta G^{\circ }} domain. Scientists searched the inverted region for proof of a slower electron transfer rate for 30 years until it was unequivocally verified experimentally in 1984. R. A. Marcus received the Nobel Prize in Chemistry in 1992 for this theory. Marcus theory is used to describe a number of important processes in chemistry and biology, including photosynthesis, corrosion, certain types of chemiluminescence, charge separation in some types of solar cells and more. Besides the inner and outer sphere applications, Marcus theory has been extended to address heterogeneous electron transfer.

Outer vs inner electron transfer In a redox reaction an electron donor D must diffuse to the acceptor A, forming a precursor complex, which is labile but allows electron transfer (ET) to give successor complex. The pair then dissociates. For a one electron transfer the reaction is

D + A ⇌ k 21 k 12 [ D ⋯ A ] ⇌ k 32 k 23 [ D + ⋯ A − ] → k 30 D + + A − {\displaystyle {\ce {{D}+A<=>[k_{12}][k_{21}][D{\dotsm }A]<=>[k_{23}][k_{32}][D+{\dotsm }A^{-}]->[k_{30}]{D+}+{A^{-}}}}}

(D and A may already carry charges). Here k12, k21 and k30 are diffusion constants, k23 and k32 are rate constants of activated reactions. The total reaction may be diffusion controlled (the electron transfer step is faster than diffusion, every encounter leads to reaction) or activation controlled (the "equilibrium of association" is reached, the electron transfer step is slow, the separation of the successor complex is fast). The ligand shells around A and D are retained. This process is called outer sphere electron transfer. Outer sphere ET is the main focus of traditional Marcus Theory. The other kind or redox reactions is inner sphere where A and D are covalently linked by a bridging ligand. Rates for such ET reactions depend on ligand exchange rates.

… excerpt ends here. Continue reading the full article.

Illustrations

Marcus theory: Fig. 2 Marcus-Parabolas for different redox reactions: f1 for positive Δ G ∘ {\displaystyle \Delta G^{\circ }} , f ( 0 ) {\displaystyle f(0)} for the self-exchange reaction with Δ G ∘ = 0 {\displaystyle \Delta G^{\circ }=0} (broken line), f 2 {\displaystyle f_{2}} for moderately negative Δ G ∘ {\displaystyle \Delta G^{\circ }} with Δ G ‡ = 0 {\displaystyle \Delta G^{\ddagger }=0} and f 3 {\displaystyle f_{3}} for strongly negative Δ G ∘ {\displaystyle \Delta G^{\circ }} . The free energy of activation Δ G ‡ {\displaystyle \Delta G^{\ddagger }} decreases from f 1 {\displaystyle f_{1}} ( b 1 {\displaystyle b_{1}} ) via f ( 0 ) {\displaystyle f(0)} (a) to f 2 {\displaystyle f_{2}} (zero) and increases again for f 3 {\displaystyle f_{3}} ("Marcus inverted region").
Fig. 2 Marcus-Parabolas for different redox reactions: f1 for positive Δ G ∘ {\displaystyle \Delta G^{\circ }} , f ( 0 ) {\displaystyle f(0)} for the self-exchange reaction with Δ G ∘ = 0 {\displaystyle \Delta G^{\circ }=0} (broken line), f 2 {\displaystyle f_{2}} for moderately negative Δ G ∘ {\displaystyle \Delta G^{\circ }} with Δ G ‡ = 0 {\displaystyle \Delta G^{\ddagger }=0} and f 3 {\displaystyle f_{3}} for strongly negative Δ G ∘ {\displaystyle \Delta G^{\circ }} . The free energy of activation Δ G ‡ {\displaystyle \Delta G^{\ddagger }} decreases from f 1 {\displaystyle f_{1}} ( b 1 {\displaystyle b_{1}} ) via f ( 0 ) {\displaystyle f(0)} (a) to f 2 {\displaystyle f_{2}} (zero) and increases again for f 3 {\displaystyle f_{3}} ("Marcus inverted region").
Marcus theory: Fig. 3 Energy diagram for Electron Transfer including inner and outer sphere reorganization and electronic coupling: The vertical axis is the free energy, and the horizontal axis is the "reaction coordinate" – a simplified axis representing the motion of all the atomic nuclei (including solvent reorganization)
Fig. 3 Energy diagram for Electron Transfer including inner and outer sphere reorganization and electronic coupling: The vertical axis is the free energy, and the horizontal axis is the "reaction coordinate" – a simplified axis representing the motion of all the atomic nuclei (including solvent reorganization)
Marcus theory: Fig.4. Marcus behaviour in a molecule, which is composed of a biphenyl entity, whose anion (produced by means of pulse radiolysis) acts as a donor, a steroid entity, which is a rigid spacer and different aromatic hydrocarbons and quinones, which are the acceptors (A).
Fig.4. Marcus behaviour in a molecule, which is composed of a biphenyl entity, whose anion (produced by means of pulse radiolysis) acts as a donor, a steroid entity, which is a rigid spacer and different aromatic hydrocarbons and quinones, which are the acceptors (A).

Worked examples

Example 1 — a first encounter with Marcus theory

Start with the simplest possible case. Write down what Marcus theory claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In chemistry, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Marcus theory before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Marcus theory ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Marcus theory

In research
Marcus theory appears in chemistry research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Marcus theory in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Marcus theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Physical chemistry, Physical organic chemistry, so understanding it makes those chapters shorter.
In everyday life
Look for Marcus theory outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Marcus theory in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Marcus theory means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Marcus theory out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Marcus theory in simple terms?

In theoretical chemistry, Marcus theory is a theory originally developed by Rudolph A. Marcus, starting in 1956, to explain the rates of electron transfer reactions – the rate at which an electron can move or jump from one chemical species (called the electron donor) to another (called the electron…

Why does Marcus theory matter?

Because it connects several chemistry ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Marcus theory?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Marcus theory.

Tags

  • Physical chemistry
  • Physical organic chemistry

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