7.11: Kinetics of Electron Transfer Reactions

Introduction

A large fraction of energetic reactions of importance to our technology and living organisms can be classified as redox reactions (reactions involving electron transfer). These reactions involve steps where an electron is transferred between a donor and an acceptor. One example are the electron transfers that occur at the electrodes of electrochemical cells. Another example is photosynthesis where an electron freed by the absorption of a photon undergoes multiple transfers to generate a large physical separation between the original electron donor and the electron, thereby reducing the probability of recombination before the energy can be used in another process. We can gain some insight into these types of processes through the Marcus theory.

As the model involves a reaction between a donor (D) and an acceptor (A) the mechanism involves the two species coming together and then transferring the electron:

\[\ce{D + A <=>[k_e][k_{-e}]DA}\nonumber\]

\[\ce{DA <=>[k_{et}][k_{-et}]D^+ A^-}\nonumber\]

\[\ce{D^+ A^- <=>[k_d][k_{-d}]D^+ + A^-}\nonumber\]

where k_±e are the rate constants for formation of an encounter pair and its separation, k_±et are the forward and reverse electron transfer rates, and k_±d are the forward and reverse dissociation rates for the charged species. In many cases these reactions are diffusion limited, the second two processes proceed rapidly compared to the rate the two initial species come together. However, in systems where the reaction is not diffusion limited the kinetics of the second reaction become important. Slow electron transfer and donor acceptor pairs held at fixed distances in inorganic complexes, proteins and extended biochemical structures are some examples.

In activation limited cases with slow electron transfer, the first reaction can be treated as a pre-equilibrium and the overall rate (assuming negligible back reactions in steps 2 and 3) will be:

\[\text{rate} \approx k_{et}\frac{K^{\ddagger}_e} {c^o}[A][D]\]

which implies an observed bimolecular rate of:

\[k_{obs} \approx k_{et}\frac{K^{\ddagger}_e} {c^o}\]

Marcus Theory¹

Marcus theory models the rate constant of the forward electron transfer (k_et). This theory was developed in the later half of the 1950s primarily by Rudi A. Marcus, who received the Nobel prize in chemistry for this work in 1992. The model is grounded in Transition State Theory. Because the complex is already formed we have a unimolecular reaction:

\[\ce{DA <=>DA^{\ddagger} ->[k_t] D^+A^-}\nonumber\]

with an equilibrium constant for the formation of the activated complex (indicated by \(\ddagger\)), \(K^\ddagger = e^{-\Delta G^\ddagger /R/T}\). The observed rate constant of electron transfer is then the rate constant of the second step (k_t) multiplied by the equilibrium constant for formation of the transition state.

\[ k_{et} = k_t e^{-\Delta G^\ddagger /R/T} \nonumber \]

The rate constant of the transfer (k_t) once in the activated complex (transition state) is dictated by the rate (probability) that the electron will jump the distance between the donor (D) location and the acceptor (A) location. This is a quantum tunneling process which is dependent on the separation distance. From quantum mechanical calculations and observations using scanning tunneling microscopes this distance dependence is known to drop off exponentially with distance. Calling the edge-to-edge jump (tunneling) distance r and \(\beta\) the distance decay rate for tunneling probability the rate of electron transfer becomes:

\[ k_{et} = e^{-\beta r}e^{-\Delta G^\ddagger /R/T} \label{ket1}\]

Because the geometry of DA may be different than D⁺A^- the transition state geometry is distorted from that of just D in contact with A. Marcus theory is able to account for this by representing the \(\Delta G^\ddagger\) as having two components; \(\Delta G^o\) the standard Gibbs energy of electron transfer (overall reaction); and \(\lambda\) the reorganization energy. The way they contribute to \(\Delta G^\ddagger\) is approximated using the assumption that the reaction potential energy curve can be modeled by overlapping parabolas (see reference 1 for a more detailed discussion):

\[\Delta G^\ddagger\ = \frac{(\Delta G^o +\lambda)^2}{4\lambda}\label{DGdd}\]

Combining this with equation \(\ref{ket1}\) we get:

\[ k_{et} = e^{-\beta r}\exp \left(-\frac{(\Delta G^o +\lambda)^2}{4\lambda R T}\right) \label{ket2}\]

If the same donor and acceptor are fixed at different distances from each other in a protein or by some other covalent linkages, \(\Delta G^\ddagger\) will be constant. In that case we can take the log of equation \(\ref{ket1}\) to get:

\[ ln k_{et} = -\beta r -\frac{\Delta G^\ddagger}{ RT} = -\beta r - \text{constant}\label{ket3}\]

so a plot of ln k_et versus r gives a straight line with a slope of \(-\beta\). In vacuum it is found that 28 nm^-1<β< 35 nm^-1 while in proteins 9.0 nm^-1 < β< 16 nm^-1. The smaller values for proteins indicated that the probability of a jump decays less rapidly with distance. This agrees with the observation that large electron transfer distances on the order of 2 nm are observed for donor acceptor pairs in proteins.

If the distance is held constant and the donor acceptor pair is changed the variation in k_et can be interpreted in terms of variation in \(\Delta G^o\) and \(\lambda\). Because the sum of \(\Delta G^o\) and \(\lambda\) is squared in equations \(\ref{DGdd}\) and \(\ref{ket2}\) this means the numerator, \((\Delta G^o + \lambda)^2 \ge 0\), making \(\Delta G^\ddagger \ge 0\). With the negative sign in the exponential this means that the maximum for k_et will be reached when \(\Delta G^o + \lambda = 0\). This has been verified in experiments.²

Marcus Cross Relation

The Marcus cross relation provides a way of estimating k_et from self exchange rates, the rate of electron exchange between identical species:

\[\ce{A + A ->[k_{AA}] A^+ + A^-} \quad \text{and} \quad \ce{D + D ->[k_{DD}] D^+ + D^-}\quad\nonumber\]

The final estimate when \(|\Delta G^o | \ll \lambda\) is (see below for a justification of this approximate relationship):

\[k_{et} \approx \sqrt{k_{DD}k_{AA}K_{DA}}\label{Mxrel}\]

where K_DA is the equilibrium constant for \(\ce{D + A <=> D^+ + A^-}\). K_DA is easily obtained from redox potentials since \(\Delta G^o = -n_e FE^o\), where n_e = number moles electrons transferred in the reaction, F = Faraday's constant and E^o = the redox potential of the reaction. Thus:

\[K_{DA} = \exp\left(\frac{-\Delta G^o}{RT}\right) = \exp\left(\frac{n_e FE^o}{RT}\right)\nonumber\]

Time resolved measurements of the self exchange rate constants can thus be combined with the known redox potential of a reaction to estimate the electron transfer rates as long as the reorganization energy is significant compared to the redox free energy change using equation \(\ref{Mxrel}\). When this is not true an additional factor, which is difficult to estimate must be included.

Justification of approximate Marcus Cross Relation

Equation \(\ref{ket2}\) can be expanded to:

\[ \begin{align} k_{et} &= e^{-\beta r}\exp \left(-\frac{ \Delta {G^o}^2 +2 \lambda \Delta G^o + \lambda^2}{4 \lambda R T}\right)
\\[4pt] &= e^{-\beta r}\exp \left(-\frac{1}{4RT} \left(\frac{ \Delta {G^o}^2}{ \lambda} + \frac{2 \lambda \Delta G^o}{4\lambda} + \frac{\lambda^2}{\lambda}\right)\right)\label{ket4} \end{align}\]

When \(|\Delta G^o| \ll \lambda\) the \({\Delta G^o}^2 /\lambda\) term is negligible compared to the other two and can be dropped leading to:

\[ \begin{align} k_{et} &\approx e^{-\beta r}\exp \left(-\frac{1}{4RT} \left(\frac{2 \lambda \Delta G^o}{\lambda} + \frac{\lambda^2}{\lambda}\right)\right) \\[4pt] &\approx e^{-\beta r}\exp \left(-\frac{\Delta G^o}{2RT} - \frac{\lambda}{4RT}\right) \\[4pt] &\approx e^{-\beta r}\exp\left(- \frac{\lambda}{4RT}\right)\exp \left(-\frac{\Delta G^o}{2RT}\right) \label{ket5} \end{align}\]

The last exponential is the square root of the overall equilibrium constant (\(K_{DA}^{1/2}\)), leading to:

\[ k_{et} \approx e^{-\beta r}\exp\left(- \frac{\lambda}{4RT}\right)\sqrt{K_{DA}}\label{ket6} \]

If we assume that the exchange energy \(\lambda\) and distances (\(r\)) can be approximated as the averages of the values in the self exchange reactions this becomes:

\[ \begin{align} k_{et} &\approx \exp\left({-\beta \frac{r_{DD}+r_{AA}}{2}}\right)\exp\left(- \frac{\lambda_{DD} + \lambda_{AA}}{8RT}\right)\sqrt{K_{DA}} \\[4pt] &\approx
{\color{blue}{\exp\left(-\beta \frac{r_{DD}}{2}\right)\exp\left(- \frac{\lambda_{DD}}{8RT}\right)}} {\color{red}\exp\left(-\beta\frac{r_{AA}}{2}\right) \exp\left( -\frac{\lambda_{AA}}{8RT}\right)}\sqrt{K_{DA}}\label{ket7} \end{align}\]

The blue pair of exponentials are \(\sqrt{k_{DD}}\) and the red pair are \(\sqrt{k_{AA}}\) because \(\Delta G^o\) for the self exchange reactions is 0 (see equation \(\ref{ket2}\)). This recovers equation \(\ref{Mxrel}\).

References

1. P.W. Atkins, J. de Paula, Physical Chemistry, 10th Ed. (W.H. Freeman and Co. New York, 2014) Pp. 1114-1021.
2. J. R. Miller et al. J. Am. Chem Soc. 106, 3047 (1984).