physical inorganic chemistry

Rate constants for thesereactions vary in a nonlinear fashion as functions of ionic strength, and the models areintimately tied to contemporary and later developments in electrolyte theo

PHYSICAL INORGANIC CHEMISTRY

Copyright  2010 by John Wiley & Sons, Inc All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Physical inorganic chemistry : reactions, processes and applications / [edited by]

10 9 8 7 6 5 4 3 2 1

To Jojika

Ophir Snir and Ira A Weinstock

2 Proton-Coupled Electron Transfer in Hydrogen

Shunichi Fukuzumi

Mahdi M Abu-Omar

4 Mechanisms of Oxygen Binding and Activation

Elena V Rybak-Akimova

Gregory J Kubas and Dennis Michael Heinekey

10 Organometallic Radicals: Thermodynamics, Kinetics,

Tamas Kegl, George C Fortman, Manuel Temprado, and Carl D Hoff

vii

11 Metal-Mediated Carbon–Hydrogen Bond Activation 495

Thomas Brent Gunnoe

12 Solar Photochemistry with Transition Metal Compounds Anchored

Gerald J Meyer

viii CONTENTS

This book is a natural extension of “Physical Inorganic Chemistry: Principles,Methods, and Models,” a 10-chapter volume describing the methods, techniques,and capabilities of physical inorganic chemistry as seen through the eyes of amechanistic chemist This book provides an insight into a number of reactions thatplay critical roles in areas such as solar energy, hydrogen energy, biorenewables,catalysis, environment, atmosphere, and human health None of the reaction typesdescribed here is exclusive to any particular area of chemistry, but it seems thatmechanistic inorganic chemists have studied, expanded, and utilized these reactionsmore consistently and heavily than any other group The topics include electrontransfer (Weinstock and Snir), hydrogen atom and proton-coupled electron transfer(Fukuzumi), oxygen atom transfer (Abu-Omar), ligand substitution at metal centers(Swaddle), inorganic radicals (Stanbury), organometallic radicals (Kegl, Fortman,Temprado, and Hoff), and activation of oxygen (Rybak-Akimova), hydrogen (Kubasand Heinekey), carbon dioxide (Joo´), and nitrogen monoxide (Olabe) Finally, thelatest developments in carbon–hydrogen bond activation and in solar photochemistryare presented in the respective chapters by Gunnoe and Meyer

I am grateful to this group of dedicated scientists for their hard work andprofessionalism as we worked together to bring this difficult project to a successfulconclusion I am also thankful to my family, friends, and colleagues who providedinvaluable support and encouragement throughout the project, and to my editor,Anita Lekhwani, who has been a source of ideas and professional advice through theentire publishing process

ix

Engineering, Osaka University, Suita, Osaka, Japan

Charlottesville, VA, USA

Seattle, WA, USA

CARLD HOFF, Department of Chemistry, University of Miami, Coral Gables, FL,USA

University of Debrecen and Research Group of Homogeneous Catalysis,Debrecen, Hungary

TAMA ´ S KE ´ GL, Department of Organic Chemistry, University of Pannonia,Veszprem, Hungary

THOMASW SWADDLE, Department of Chemistry, University of Calgary, Calgary,Canada

FL, USA

Beer Sheva, Israel

xii CONTRIBUTORS

1 Electron Transfer Reactions

OPHIR SNIR and IRA A WEINSTOCK

In that sense, it is written from a “reaction chemist’s”1perspective In 1987, Ebersonpublished an excellent monograph that provides considerable guidance in the context

of organic reactions.2In addition to a greater focus on inorganic reactions, this chaptercovers electrolyte theory and ion pairing in more detail, and worked examples arepresented in a step-by-step fashion to guide the reader from theory to application.The chapter begins with an introduction to Marcus’ theoretical treatment of outer-sphere electron transfer The emphasis is on communicating the main features of thetheory and on bridging the gap between theory and practically useful classical models.The chapter then includes an introduction to models for collision rates betweencharged species in solution, and the effects on these of salts and ionic strength, whichall predated the Marcus model, but upon which it is an extension Collision rate andelectrolyte models, such as those of Smoluchowski, Debye, H€uckel, and others, apply

in ideal cases rarely met in practice The assumptions of the models will be defined,and the common situations in which real reacting systems fail to comply with themwill be highlighted These models will be referred to extensively in the second half ofthe chapter, where the conditions that must be met in order to use the Marcus modelproperly, to avoid common pitfalls, and to evaluate situations where calculated valuesfail to agree with experimental ones will be clarified

Those who have taught themselves how to apply the Marcus model and relation correctly will appreciate the gap between the familiar “formulas” published

cross-in numerous articles and texts and the assumptions, definitions of terms, and physical

Physical Inorganic Chemistry: Reactions, Processes, and Applications Edited by Andreja Bakac Copyright  2010 by John Wiley & Sons, Inc.

1

constants needed to apply them This chapter will fill that gap in the service of thoseinterested in applying the model to their own chemistry In addition, the task ofchoosing compatible units for physical constants and experimental variables will besimplified through worked examples that include dimensional analysis.

The many thousands of articles on outer-sphere electron transfer reactions involvingmetal ions and their complexes cannot be properly reviewed in a single chapter Fromthis substantial literature, however, instructive examples will be selected Importantly,they will be explaining in more detail than is typically found in review articles ortreatises on outer-sphere electron transfer In fact, the analyses provided here are quitedifferent from those typically found in the primary articles themselves There, in nearlyall cases, the objective is to present and discuss calculated results Here, the goal is toenable readers to carry out the calculations that lead to publishable results, so that theycan confidently apply the Marcus model to their own data and research

The importance of Marcus’ theoretical work on electron transfer reactions wasrecognized with a Nobel Prize in Chemistry in 1992, and its historical development

is outlined in his Nobel Lecture.3The aspects of his theoretical work most widelyused by experimentalists concern outer-sphere electron transfer reactions These arecharacterized by weak electronic interactions between electron donors and acceptorsalong the reaction coordinate and are distinct from inner-sphere electron transferprocesses that proceed through the formation of chemical bonds between reactingspecies Marcus’ theoretical work includes intermolecular (often bimolecular) reac-tions, intramolecular electron transfer, and heterogeneous (electrode) reactions Thebackground and models presented here are intended to serve as an introduction tobimolecular processes

The intent here is not to provide a rigorous and comprehensive treatment of thetheory, but rather to help researchers understand basic principles, classical modelsderived from the theory, and the assumptions upon which they are based This focus isconsistent with the goal of this chapter, which is to enable those new to this area toapply the classical forms of the Marcus model to their own science

For further reading, many excellent review articles and books provide more in-depthinformation about the theory and more comprehensive coverage of its applications tochemistry, biology, and nanoscience Several recommended items (among many) are

a highly cited review article by Marcus and Sutin,4excellent reviews by Endicott,5Creutz and Brunschwig,6and Stanbury,7a five-volume treatise edited by Balzano,8andthe abovementioned monograph by Eberson2that provides an accessible introduction

to theory and practice in the context of organic electron transfer reactions

1.2.1 Collision Rates Between Hard Spheres in Solution

In 1942, Debye extended Smoluchowski’s method for evaluating fundamentalfrequency factors, which pertained to collision rates between neutral particles D

2 ELECTRON TRANSFER REACTIONS

and A, randomly diffusing in solution, to include the electrostatic effects of chargedreacting species in dielectric media containing dissolved electrolytes.9–11Debye’scolliding sphere model was derived assuming that collisions between Dn(electrondonors with a charge of n) and Am(electron acceptors of charge m) resulted in thetransient formation of short-lived complexes, Dn Am

Rate constants for thesereactions vary in a nonlinear fashion as functions of ionic strength, and the models areintimately tied to contemporary and later developments in electrolyte theory.Marcus12and others13extended this model to include reactions in which electrontransfer occurred during collisions between the “donor” and “acceptor” species, that

is, between the short-lived DnAmcomplexes In this context, electron transfer withintransient “precursor” complexes ([Dn Am] in Scheme 1.1) resulted in the formation

of short-lived “successor” complexes ([D(nþ 1) A(m1)] in Scheme 1.1) TheDebye–Smoluchowski description of the diffusion-controlled collision frequencybetween Dnand Amwas retained This has important implications for application ofthe Marcus model, particularly where—as is common in inorganic electron transferreactions—charged donors or acceptors are involved In these cases, use of the Marcusmodel to evaluate such reactions is only defensible if the collision rates between thereactants vary with ionic strength as required by the Debye–Smoluchowski model.The requirements of that model, and how electrolyte theory can be used to verifywhether a reaction is a defensible candidate for evaluation using the Marcus model,are presented at the end of this section

After electron transfer (transition along the reaction coordinate from Dn Am

to

D(nþ 1) –A(m1)in Scheme 1.1), the successor complex dissociates to give the finalproducts of the electron transfer, D(nþ 1)and A(m1) The distinction between thesuccessor complex and final products is important because, as will be shown, theMarcus model describes rate constants as a function of the difference in energybetween precursor and successor complexes, rather than between initial and finalproducts

1.2.2 Potential Energy Surfaces

As noted above, outer-sphere electron transfer reactions are characterized by theabsence of strong electronic interaction (e.g., bond formation) between atomic ormolecular orbitals populated, in the donor and acceptor, by the transferred electron.Nonetheless, as can be appreciated intuitively, outer-sphere reactions must requiresome type of electronic “communication” between donor and acceptor atomic ormolecular orbitals This is referred to in the literature as “coupling,” “electronicinteraction,” or “electronic overlap” and is usually less than 1 kcal/mol Inner-sphere electron transfer reactions, by contrast, frequently involve covalent bond

SCHEME 1.1THEORETICAL BACKGROUND AND USEFUL MODELS 3

formation between the reactants and are often characterized by ligand exchange oratom transfer (e.g., of O, H, hydride, chloride, or others).

The two-dimensional representation of the intersection of two N-dimensionalpotential energy surfaces is depicted in Figure 1.1.4The curves represent the energiesand spatial locations of reactants and products in a many-dimensional (N-dimen-sional) configuration space, and the x-axis corresponds to the motions of all atomicnuclei The two-dimensional profile of the reactants plus the surrounding medium isrepresented by curve R, and the products plus surrounding medium by curve P Theminima in each curve, that is, points A and B, represent the equilibrium nuclearconfigurations, and associated energies, of the precursor and successor complexesindicated in Scheme 1.1, rather than of separated reactants or separated products As

a consequence, the difference in energy between reactants and products (i.e., thedifference in energy between A and B) is not the Gibbs free energy for the overallreaction,DG, but rather the “corrected” Gibbs free energy,DG0 For reactions ofcharged species, the difference betweenDG andDG0can be substantial

The intersection of the two surfaces forms a new surface at point S in Figure 1.1.This (N 1)-dimensional surface has one less degree of freedom than the energysurfaces depicted by curves R and P Weak electronic interaction between thereactants results in the indicated splitting of the potential surfaces This gives rise

to electronic coupling (resonance energy arising from orbital mixing) of the reactants’electronic state with the products’, described by the electronic matrix element, HAB.This is equal to one-half the separation of the curves at the intersection of the R and

energy surface of reactants plus surrounding medium is labeled R and that of the products plussurrounding medium is labeled P Dotted lines indicate splitting due to electronic interactionbetween the reactants Labels A and B indicates the nuclear coordinates for equilibriumconfigurations of the reactants and products, respectively, and S indicates the nuclear config-uration of the intersection of the two potential energy surfaces

4 ELECTRON TRANSFER REACTIONS

P surfaces The dotted lines represent the approach of two reactants with no electronicinteraction at all.

This diagram can be used to appreciate the main difference between inner- andouter-sphere processes The former are associated with a much larger splitting of thesurfaces, due to the stronger electronic interaction necessary for the “bonded”transition state A classical example of this was that recognized by Henry Taube,recipient of the 1983 Nobel Prize in Chemistry for his work on inorganic reactionmechanisms In a famous experiment, he studied electron transfer from CrII(H2O)62þ(labile, high spin, d4) to the nonlabile complex (NH3)5CoIIICl2þ(low spin, d6) underacidic conditions in water Electron transfer was accompanied by a change in color ofthe solution from a mixture of sky blue CrII(H2O)6 þand purple (NH3)5CoIIICl2þtothe deep green color of the nonlabile complex (H2O)5CrIIICl2þ (d3) and labile

CoII(H2O)6 þ (high spin, d7) (Equation 1.1).14,15

CrIIIproduct

1.2.3 Franck–Condon Principle and Outer-Sphere Electron Transfer

The mass of the transferred electron is very small relative to that of the atomic nuclei

As a result, electron transfer is much more rapid than nuclear motion, such that nuclearcoordinates are effectively unchanged during the electron transfer event This is theFranck–Condon principle

Now, for electron transfer reactions to obey the Franck–Condon principle, whilealso complying with the first law of thermodynamics (conservation of energy),electron transfer can occur only at nuclear coordinates for which the total potentialenergy of the reactants and surrounding medium equals that of the products andsurrounding medium The intersection of the two surfaces, S, is the only location inFigure 1.1 at which both these conditions are satisfied The quantum mechanicaltreatment allows for additional options such as “nuclear tunneling,” which isdiscussed below

1.2.4 Adiabatic Electron Transfer

The classical form of the Marcus equation requires that the electron transfer beadiabatic This means that the system passes the intersection slowly enough for thetransfer to take place and that the probability of electron transfer per passage is large(near unity) This probability is known as the transmission coefficient,k, defined later

in this section In this quantum mechanical context, the term “adiabatic” indicates that

THEORETICAL BACKGROUND AND USEFUL MODELS 5

nuclear coordinates change sufficiently slowly that the system (effectively) remains

at equilibrium as it progresses along the reaction coordinate The initial eigenstate ofthe system is modified in a continuous manner to a final eigenstate according to theSchr€odinger equation, as shown in Equation 1.2 At the adiabatic limit, the timerequired for the system to go from initial to final states approaches infinity (i.e.,[tf ti]! ¥)

yðx; tfÞ

When the system passes the intersection at a high velocity, that is, the abovecondition is not met even approximately, it will usually “jump” from the lower Rsurface (before S along the reaction coordinate) to the upper R surface (after S) That

is, the system behaves in a “nonadiabatic” (or diabatic) fashion, and the probabilityper passage of electron transfer occurring is small (i.e., k  1) The nuclearcoordinates of the system change so rapidly that it cannot remain at equilibrium

At the nonadiabatic limit, the time interval for passage between the two states atpoint S approaches zero, that is, (tf ti) ! 0 (infinitely rapid), and the probabilitydensity distribution functions that describe the initial and final states remainunchanged:

The “fast” and “slow” changes described here, which refer to “velocities” ofpassage through the intersection, S, correspond to “high” and “low” frequencies ofnuclear motions Hence, “nuclear frequencies” play an important role in quantummechanical treatments of electron transfer

1.2.5 The Marcus Equation

In his theoretical treatment of outer-sphere electron transfer reactions, Marcus relatedthe free energy of activation,DGz, to the corrected Gibbs free energy of the reaction,

DG0, via a quadratic equation (Equation 1.4).2,4,13

The termsDG0andl in Equation 1.4 are represented schematically in Figure 1.1, and

w is the reciprocal Debye radius (Equation 1.5).11,16

w ¼ 4pe

2DkT

Xi

an infinite distance to the center-to-center separation distance, r12¼ r1þ r2, which isalso known as the distance of closest approach (formation of the precursor complex[DnAm

] in Scheme 1.1) The magnitude of the Coulombic term is modified by

a factor exp(wr12), which accounts for the effects of the dielectric medium (ofdielectric constant D) and of the total ionic strengthm

The corrected Gibbs free energy,DG0, in Equation 1.4 is the difference in freeenergy between the successor and precursor complexes of Scheme 1.1 as shown inFigure 1.1 The more familiar, Gibbs free energy,DG, is the difference in free energybetween separated reactants and separated products in the prevailing medium Thecorrected free energy,DG0, is a function of the charges of the reactants and products

It is calculated using Equation 1.6, where z2is the charge of the electron donor and z1

is the charge of the electron acceptor

a reaction For example, the heteropolyanion, CoIIIW12O405(E¼ þ 1.0 V), canoxidize organic substrates with standard potentials as large asþ 2.2 V This is becausethe attraction between the donor and acceptor in the successor complex, generated byelectron transfer, leads to a favorable attraction between the negative heteropolyanionand the oxidized (now positively charged) donor This attraction makes the correctedfree energy more favorable, the activation energy smaller, and the electron transferreaction kinetically possible.2,17

THEORETICAL BACKGROUND AND USEFUL MODELS 7

In Equation 1.6, the electrostatic correction to DG vanishes when z1 z2¼ 1(e.g., when z1and z2are equal, respectively, to 3 and 2, 2 and 1, 1 and 0, 0 and1, 1and 2, etc.).2 In these cases, the difference in Gibbs free energy between thesuccessor and precursor complexes is not significantly different from that between theindividual (separated) reactants and final (separated) products (Scheme 1.1).The relation betweenDG and the standard reduction potential of the donor andacceptor, E, is given by

where n is the number of electrons transferred and F is the Faraday constant This,combined with Equation 1.6, is often used to calculateDG0from electrochemical data.1.2.5.1 Reorganization Energy Thel term in Equation 1.4 is the reorganizationenergy associated with electron transfer, and more specifically, with the transitionfrom precursor to successor complexes As noted above, there are two different andseparable phenomena, termed “inner-sphere” and “outer-sphere” reorganizationenergies, commonly indicated by the subscripts “in” and “out.” The total reorganiza-tion energy is the sum of the inner- and outer-sphere components (Equation 1.8)

The inner-sphere reorganization energy refers to changes in bond lengths andangles (in-plane and torsional) of the donor and acceptor molecules or complexes.Due to electron transfer, the electronic properties and charge distribution of thesuccessor complex are different from those of the precursor complex This causesreorientation or other subtle changes of the solvent molecules in the reaction mediumnear the reacting pair, and the energetic cost associated with this is the outer-sphere(solvent) reorganization energy

The inner-sphere reorganization energy can be calculated by treating bonds withinthe reactants as harmonic oscillators, according to Equation 1.9

of the dielectric polarization on charging parameters is quadratic Marcus then used

a two-step thermodynamic cycle to calculate lout.19,20 This treatment allows theindividual solvent dipoles to move anharmonically, as indeed they do in the liquidstate The forms of the relationships that describeloutdepend on the geometrical modelchosen to represent the charge distribution For spherical reactants,l is given by

8 ELECTRON TRANSFER REACTIONS

Equation 1.10.

lout¼ ðDeÞ2 1

2r1 þ 12r2 1

Here,De is the charge transferred from one reactant to the other, r1and r2are the radii

of the two (spherical) reactants, r12is, as before, the center-to-center distance, oftenapproximated18as the sum of r1þ r2, and Dsand Dopare the static and optical (square

of refractive index) dielectric constants of the solvent, respectively This model for

louttreats both the reactants as hard spheres (i.e., the “hard sphere” model) For othershapes, more complex models are needed, which are rarely used by reactionchemists.21

1.2.6 Useful Forms of the Marcus Model

1.2.6.1 The Eyring Equation and Linear Free Energy Relationships In ple, one could use nonlinear regression to fit the Marcus equation (Equation 1.4) to aplot ofDGzversusDG0values for a series of reactions, withl as an adjustable param-eter To obtain a reasonably good fit, the shapes, sizes, and charges of reactants andproducts, and theirl values, must be similar to one another A good fit between calculatedand experimental curves would be evidence for a common outer-sphere electron transfermechanism, and the fitted value ofl is an approximate value for this parameter Inpractice,DGzvalues cannot be measured directly However, bimolecular rate constants,

princi-k, can be These are related toDGzby the Eyring equation (Equation 1.11)

IfjDG0j  l, the lastterminEquation 1.12can beignored,and the Marcusequationcan be approximated by a linear free energy relationship (LFER) (Equation 1.14).

If the equilibrium constants for a series of reactions can be measured or calculated,one can plot ln k versus ln K (Equation 1.15) A linear result with a slope of 0.5 isindicative of a common outer-sphere electron transfer mechanism

ln k¼ ln ZWðrÞ

RT  l4RT þ 0:5 ln K þ ln A ð1:15ÞAnother useful linear relationship is based on electrochemical data and is obtained

by recourse to the fact that DG¼ nFE For a series of outer-sphere electrontransfer reactions that meet the criteria discussed in context with Equation 1.14, a plot

of ln k versus Ewill have a slope of 0.5(nF/RT), and a plot of log k versus Ewill have

a slope of 0.5(nF)/2.303RT or 8.5 V1for n¼ 1 at 25C.5All the above methods can

be used to obtain a common (approximate) value ofl for a series of similar reactions.For single reactions of interest, however,l values can often be measured directly byelectron self-exchange

1.2.6.2 Electron Self-Exchange In many cases, DG can be easily measured(usually electrochemically), while l is more difficult to determine The methodsdiscussed above require data for a series of similar reactions This information isnot always accessible, or of interest An alternative and more direct method is todeterminel values from rate constants for electron self-exchange This requires that

a kinetic method be available for measuring the rate of electron exchange betweenone-electron oxidized and reduced forms of a complex or molecule One requirementfor this is that the oxidation or reduction involved does not lead to rapid, irreversiblefurther reactions of either partner In this sense, self-exchanging pairs whosel valuescan be measured kinetically are often reversible or quasi-reversible redox couples

In self-exchange reactions, such as that between Amand Amþ 1(Equation 1.16),

BecauseDGzis not directly measurable,l11can be calculated from the observedrate constant k for the self-exchange reaction by using Equation 1.18 This is obtained

by substituting Equation 1.17 into Equation 1.11 and assuming thatk ¼ 1

For reactions in solution, Z is often on the order of 1011M1s1 (values of

Z¼ 6  1011M1s1are also used).7To calculate W(r), one must know the chargesand radii of the reactants, the dielectric constant of the solvent, and the ionic strength

of the solution The reorganization energyl11can then be calculated from k Workedexamples from the literature are included in Section 1.3

1.2.6.3 The Marcus Cross-Relation The rate constant, k12, for electron transferbetween two species, Amand Bn(Equation 1.20) that are not related to one another byoxidation or reduction, is referred to as the Marcus cross-relation (MCR)

Amþ Bn þ 1> Am þ 1þ Bn ð1:20Þ

It is called the “cross-relation” because it is algebraically derived from expressionsfor the two related electron self-exchange reactions shown in Equations 1.21 and 1.22.Associated with these reactions are two self-exchange rate constants k11and k22andreorganization energiesl11andl22

A

* mþ Am þ 1> A* m þ 1þ Am; rate constant ¼ k11 ð1:21ÞB

* mþ Bm þ 1> B* m þ 1þ Bm; rate constant ¼ k22 ð1:22ÞThe MCR is derived by first assuming that Equation 1.23 holds This means that thereorganization energy for the cross-reaction, l12, is equal to the mean of thereorganization energies,l11andl22, associated with the two related self-exchangereactions

l12ffi1

The averaging over the outer-sphere components ofl11andl22, that is,l11outand

l , is only valid if Amand Bnþ 1(Equation 1.20) are of the same size (i.e., r ¼ r)

THEORETICAL BACKGROUND AND USEFUL MODELS 11

As is demonstrated later in this chapter, differences in size between donors andacceptors can lead to large discrepancies between calculated and experimental values.The assumption in Equation 1.23 is used to derive the MCR (Equations 1.24–1.26)

k12¼ ðk11k22K12f12Þ1 =2C

where

ln f12¼14

ln K12þ ðw12w21Þ=RT

lnðk11k22=Z2Þ þ ðw11w22Þ=RT ð1:25Þand

1.2.7 Additional Aspects of the Theory

1.2.7.1 The Inverted Region The Marcus model predicts that as absolute values

ofDG0decrease (i.e., as electron transfer becomes more thermodynamically able), electron transfer rate constants should decrease Because energetically morefavorable reactions generally occur more rapidly, it is counterintuitive to expect theopposite to occur However, this is precisely the case in the inverted region wheremore thermodynamically favorable reactions occur more slowly The exothermicregion in which this occurs is therefore referred to as “inverted.” Marcus predicted thisbehavior in 1960,13,22and the first experimental evidence for it was provided morethan two decades later.23

favor-In a series of related reactions possessing similarl values but differing in DG0,

a plot of the activation free energy DGzversusDG0(from Equation 1.12) can beseparated into two regions In the first region,DGzdecreases, and rates increase, as

DG0decreases from zero to more negative values This is the “normal” region When

DG0becomes sufficiently negative such thatDG0¼ l, DGzbecomes zero (This istrue for reactions in which W(r)¼ 0; for cases where W(r) 6¼ 0, DGz¼ W(r).) In thenext region (the inverted region),DGzbegins to increase (and rates decrease), asDG0becomes even more negative thanl

The dependence of ln k onDG0is depicted in Figure 1.2 The rate first increases as

DG0becomes more negative (region I, the normal region) and reaches a maximum at

DG0¼ l (point II) Then, as DG0 becomes even more negative, ln k begins todecrease (region III, the inverted region)

12 ELECTRON TRANSFER REACTIONS

The reason for this can be understood by reference to Figure 1.1 The plots inFigure 1.1 show the locations of the R and P surfaces in the normal region To reach theinverted region,DG0must become more negative This corresponds to lowering the Psurface relative to the R surface in Figure 1.1 As this proceeds, the free energy barrier

DGzdecreases until it becomes zero atDG0¼ l At this point, the intersection of the

R and P surfaces occurs at the minimum of the R curve, and the reaction has noactivation barrier Further decrease inDG0then raises the energy at which the R and Psurfaces intersect This corresponds to an increase in the activation barrier,DGz, and

a decrease in rate This case, that is, the inverted region, is shown in Figure 1.3, whichdepicts the relative locations of the reactant and product energy surfaces in theinverted region

Indirect evidence for the inverted region was first provided by observations thatsome highly exothermic electron transfer reactions resulted in chemiluminescence,

an indication that electronically excited products had been formed (surface P inFigure 1.3) When the ground electronic state potential energy surface of the products,

P, intersects the R surface at a point high in energy on the R surface (intersection ofcurves R and P in the inverted region; Figure 1.3), the reaction is slow In this case, lessthermodynamically favorable electron transfer to a product excited state (P surface)can occur more rapidly than electron transfer to the ground electronic state of theproducts (P surface) Electron transfer to a P state, and chemiluminescence asso-ciated with a subsequent loss of energy to give the ground electronic state, wasobserved by Bard and coworkers.24Direct experimental confirmation of the invertedregion (electron transfer from the R to P surfaces in Figure 1.3) was provided a fewyears later.25,26

THEORETICAL BACKGROUND AND USEFUL MODELS 13

product potential energy surfaces That intersection, S in Figure 1.1, defines thenuclear coordinates and energy of the transition state for electron transfer Thissection deals with a quantum mechanical phenomenon in which electron transferoccurs without the nuclear coordinates first reaching the intersection point.Graphically, this means that the system passes from curve R to curve P at

a lower (more negative) energy than that of the intersection, S This is referred

to as nuclear “tunneling” from surface R to surface P The material provided inthis section is designed to help the reader understand some basic aspects of thisphenomenon

When nuclear tunneling occurs, the system passes from the R surface to the Psurface by crossing horizontally from the first to the second of these surfaces This

is depicted schematically by the horizontal line that extends from “a” to “b” inFigure 1.4 In practice, at room temperature, and for reactions in the “normal” region,nuclear tunneling usually accounts for only minor contributions to rate constants Thecross-relation in the normal region is even less affected by nuclear tunneling, due topartial cancellation of the quantum correction in the ratio k12=ðk11k22K12Þ1 =2.When it is a viable pathway, nuclear tunneling tends to dominate at low tem-peratures, at which the probability of the system reaching the intersection point, S,

is low At the same time, nuclear tunneling rates are independent of temperature This

is because, in tunneling, electron transfer occurs at energies near the zero-point

exothermic reaction, the P surface has dropped in energy to such an extent that furtherdecreases in its energy result in larger activation energies and smaller rate constants The blackarrow indicates the intersection of the R and P surfaces (the splitting of these surfaces due to

more rapid transition from the R surface to an electronically excited state of one of the products(see discussion in the text)

14 ELECTRON TRANSFER REACTIONS

vibrational energy of the reactants and surrounding media (provided that the energy ofthe lowest point on the R surface equals or exceeds that of the P surface).

Temperature affects nuclear motion and thereby the Boltzmann probability ofattaining the nuclear configuration that corresponds to the intersection between the Rand P surfaces For this reason, temperature determines the contribution that nucleartunneling makes to overall reaction rates (HABis not directly affected by temperature,although it will vary somewhat with nuclear configuration.) Hence, the existence of

a temperature-dependent rate constant at high temperatures and an independent one atlow temperatures may be a manifestation of nuclear tunneling

The terms adiabatic and nonadiabatic that were discussed above can be engageddirectly to the phenomenon of tunneling If a system reacts via an adiabatic pathway,the system follows the R surface in the initial stages of the reaction, then remains

on the lower surface caused by electronic coupling at the intersection, and continues tothe P surface In a nonadiabatic reaction, the electronic coupling of the reactants is soweak, that is, 2HABis so small, that the probabilityk of going from the R to the Psurface when the system is in proximity to the intersection region in Figure 1.1 issmall In the majority of collisions that result in attaining the energy of the intersectionregion, the system will stay on the R surface, instead of going on to the P surface Forreactions with intermediatek values, expressions known as the Landau–Zener typeare available for calculatingk.13 In the “inverted region,” a reaction in which thesystem goes directly from the R to the P surface is necessarily nonadiabatic, and there

is no adiabatic path: the system must “jump” from one solid curve to the other in order

to form directly the ground-state products

Nuclear tunneling from the R to the P surfaces is represented by the horizontal linefrom “a” to “b” in Figure 1.5 Unlike in Figure 1.4, the slopes of the R and P potentialenergy surfaces have the same sign when approaching the intersection region.Semiclassical models of electron transfer show that in this case nuclear tunneling

is much more important

electron transfer via nuclear tunneling from point “a” on surface R to point “b” on surface P

THEORETICAL BACKGROUND AND USEFUL MODELS 15

1.2.9 Reactions of Charged Species and the Importance of Electrolyte Theory1.2.9.1 Background and Useful Models The Marcus equation is an extension ofearlier models from collision rate theory As such, compliance with collision ratemodels is a prerequisite to defensible use of the Marcus equation This is particularlyimportant for reactions of charged species, and therefore, for reactions of manyinorganic complexes In these cases, the key question is whether electron transfer rateconstants vary with ionic strength as dictated by electrolyte theory, on which thecollision rate models are based When they do not, differences between calculated andexperimental values can differ by many orders of magnitude.

The theory of electrolyte behavior in solution is indeed complex, and has beendebated since 1923, when Debye and H€uckel27,28described the behavior of electro-lyte solutions at the limit of very low concentration Subsequently, intense discussions

in the literature lasted into the 1980s, by which time a number of quite complexapproaches had been promulgated The latter do give excellent fits up to large ionicstrength values, but are generally not used by most kineticists

The effects of electrolyte concentrations on the rate constants depend on the nature

of the interaction between the reacting species If the reacting species repelone another, rate constants will increase with ionic strength This is because theelectrolyte ions attenuate electrostatic repulsion between the reacting ions If thereactants have opposite charges, and attract one another, electrolyte ions willattenuate this attraction, resulting in smaller rate constants

One convenient option is to use the Debye–H€uckel equation, also referred to as theDavies equation.29

exothermic (inverted) reaction

16 ELECTRON TRANSFER REACTIONS

log k¼ log k0þ2z1z2a ffiffiffiffi

mp

In Equation 1.27,a and b are the Debye–H€uckel constants, equal to 0.509 and0.329, respectively, at 25C in aqueous solutions k0 is the rate constant for thereaction at infinite dilution (m ¼ 0 M) In this form, a is dimensionless and b has units

Alternatively, one may use the Guggenheim equation (Equation 1.28), which isrigorously correct for solutions of mixed electrolytes In this equation, the specificinteraction parameters are moved from the denominator to a second term, in which b is

an adjustable parameter

log k¼ log k0þ2z1z2a ffiffiffiffi

mp

If one ignores the second term in Equation 1.28, one obtains the “truncated”Guggenheim equation (Equation 1.29), which is identical to the Davies equation, butwith br equal to unity The reader should be aware that many authors refer toEquation 1.29 as the Guggenheim equation

log k¼ log k0þ2z1z2a ffiffiffiffi

mp

Alternatively, one can use more elaborate models.35These can yield fine fits toextraordinarily high ionic strengths, but they do not generally provide much addi-tional insight

In practice, the most common approach36 is to use the truncated Guggenheimequation (Equation 1.29) and ionic strength no greater than 0.1 M This is alsothe method espoused by Espenson.37 If this fails, the Guggenheim equation(Equation 1.28) is sometimes used This has more adjustable parameters, and therefore

is more likely to produce a linear fit However, the slopes of the lines obtained oftendeviate from theoretical values, defined as a function of the charge product, z1z2, of thereacting species Nonetheless, if a good linear fit is obtained (even though the slope isnot correct), this might still be used as an argument against the presence of significantion pairing or other medium effects A good example of this is provided in an article byBrown and Sutin,38who fitted the same data set to a number of models before finallyobserving a linear relationship between rate constants and ionic strength

THEORETICAL BACKGROUND AND USEFUL MODELS 17

1.2.9.2 Graphical Demonstration of the Truncated Guggenheim Equation InFigure 1.6, the truncated Guggenheim equation (Equation 1.29) was used to calculaterate constants as a function of ionic strength Then, log k values were plotted as

a function of ionic strength This reveals the effects of ionic strength on rate constantsthat one might observe experimentally The curves shown are for reactions withcharge products of z1z2¼ 2, 1, 0, 1, and 2 Charge products of 2 and 1 indicaterepulsion between like-charged reactants and those of1 and 2 indicate attractionbetween oppositely charged reactants

In Figure 1.7, the same rate constants and ionic strengths shown in Figure 1.6 arenow plotted according to the truncated Guggenheim equation (Equation 1.29) Thehorizontal line corresponds to z1z2¼ 0 The slopes of the lines are equal to 2z1z2a andhave values of2.036, 1.018, 0, 1.018, and 2.036, respectively, for z1z2values of

2, 1, 0, 1, and 2

As discussed above, verification that a reaction that involves charged speciessatisfies the requirements of electrolyte theory is a necessary prerequisite to use of theMarcus model For this, a plot of log k versus ffiffiffiffi

m

p =ð1 þ ffiffiffiffipmÞ (truncated Guggenheimequation) should be linear with a slope of 2z1z2a, as shown in Figure 1.7 Deviationsfrom linearity, or, to a lesser extent, slopes that give incorrect charge products, z1z2,are indications that the system does not obey this model When this occurs, othermodels can be tried If these fail, ion pairing, other specific medium effects, cationcatalysis, or other reaction mechanisms are likely involved.21 In these cases, thereaction is not a defensible candidate for evaluation by the Marcus model for outer-sphere electron transfer

Finally, readers should be aware of a comment by Duncan A MacInnes (in

193939) that “There is no detail of the derivation of the equations of the

varied from 0.001 to 1 M

18 ELECTRON TRANSFER REACTIONS

Debye–H€uckel theory that has not been criticized.” From this perspective, lyte models are simply the best tools available to assess whether the dependence ofelectron transfer rate constants on ionic strength is sufficiently well behaved tojustify use of the Marcus model For this, and despite their shortcomings, they areindispensable.

This section is designed to fill the gap between the familiar “formulas” presentedabove and the assumptions and definitions of terms and physical constants needed toapply them Values for all physical constants and needed conversion factors areprovided, and dimensional analyses are included to show how the final results andtheir units are obtained This close focus on the details and units of the equationsthemselves is followed by worked examples from the chemical literature The goal is

to provide nearly everything the interested reader may need to evaluate his or her owndata, with reasonable confidence that he or she is doing so correctly

1.3.1 Compliance with Models for Collision Rates Between Charged Species

In this section, applications of the Davies and truncated Guggenheim equations aredemonstrated through worked examples from the literature

1.3.1.1 The Davies Equation The Davies equation (introduced earlier in thischapter and reproduced here for convenience in Equation 1.30) is one of severalclosely related models, derived from electrolyte theory, that describe the functional

m

GUIDE TO USE OF THE MARCUS MODEL 19

dependence of rate constants on ionic strength.

log k¼ log k0þ 2az1z2m1=2=ð1 þ br m1 =2Þ ð1:30Þ

In Equation 1.30, z1and z2are the (integer) charges of the reacting ions and r is thehard sphere collision distance (internuclear distance) The latter term, r, is approxi-mated as the sum of the radii of the reacting ions, r1þ r2.18The termm is the totalionic strength It is defined as m ¼1

in Equation 1.30 refers to log10k, rather than to the natural logarithm, ln k Thisdeserves mention because in many published reports, log k is (inappropriately) used

to refer to ln k

Typically, log k (i.e., log10k) is plotted (y-axis) as a function of m1/2/(1þ brm1/2

)(x-axis) If the result is a straight line, its slope should be equal to a simple function ofthe charge product, that is, 2z1z2a, and its y-intercept gives log k0, the log of the rateconstant at the zero ionic strength limit The constant k0 is the ionic strength-independent value of the rate constant and can be treated as a fundamental parameter

of an electron transfer reaction

1.3.1.2 Dimensional Analysis The constantb in Equation 1.31 is the “reciprocalDebye radius.”11,16Dimensional analysis of this term is instructive because it involves

a number of often needed constants and occurs frequently in a variety of contexts

e¼ electron charge (4.803  1010electrostatic units (esu) or StatC)

Ds¼ static dielectric constant (78.4) (water at 298K)

k¼ Boltzmann constant (1.3807  1016erg/K)

Evaluation ofb is as follows:

21000DskT

b ¼ 8pð6:022  10

23mol1Þð4:803  1010StatCÞ2

1000ð78:4Þð1:3807  1016erg=KÞ298K

ðerg cmÞStatC2

b cm ¼ 3:29  107 cm

3mol

 1 =2

ð1:38Þ

The units in Equation 1.38 can be viewed as (vol/mol)1/2and are cancelled (i.e.,reduced to unity) when multiplied by the units of m1/2(mol/vol)1/2 in the Daviesequation (Equation 1.30)

Note here that the denominator in Equation 1.33 (definition ofb) is multiplied by

a factor of 1000 This is needed to “scale up” from cm3to L, so that, for r in cm andm

in mol/L, the units associated with the product brm exactly cancel one another.Similarly, for r in units of A andm in units of mol/L, b ¼ 0:329 A1=2=mol1 =2.

In published articles,b is often presented as dimensionless (e.g., as 0.329), or withunits of cm1or A1 The units of cm1are obtained if the correction factor of 1000 inthe denominator of Equation 1.33 is assigned units of cm3 Once the final units are in

cm1, these can be converted to A1 These options can be disconcerting to those new

to the use of these models For practical purposes, however, one only needs to knowthat, form in units of molarity (M), b ¼ 0.329 for r in units of Aand 3.29 107for r inunits of cm

1.3.1.3 Literature Example: Reaction Betweena-PW12O404anda-PW12O403

In a detailed investigation of electron self-exchange between Keggin tungstate anions in water, Kozik and Baker used line broadening of31P NMR signals

heteropoly-to determine the rates of electron exchange betweena-PW12O403and one-electronreduceda-PW12O404(Figure 1.8) and betweena-PW12O404and the two-electronreduced aniona-PW12O40.531,40The W ions in the parent aniona-PW12O403are intheir highest þ 6 oxidation state (d0electron configuration)

Structurally,a-PW12O403, which is 1.12 nm in diameter,41may be viewed as atetrahedral phosphate anion, PVO4, encapsulated within a neutral, also tetrahedral,a-WVI12O360shell (“clathrate” model42–44) According to this model, the PVO4anion in the one-electron reduced anion, a-PW12O404, is located at the center of

a negatively charged W12O36shell,45which contains a single d (valence) electron.Moreover, the single valence electron is not localized at any single W atom, but is

GUIDE TO USE OF THE MARCUS MODEL 21

rapidly exchanged between all 12 chemically equivalent W centers The rate ofintramolecular exchange at 6K is108s1and considerably more rapid46,47 thanmost electron transfer reactions carried out at near-ambient temperatures between thereduced anion,a-PW12O404, and electron acceptors.

Kozik and Baker combined the acid forms of the two anions,a-H3PW12O40anda-H4PW12O40 (1.0 mM each), in water using a range of ionic strengths (m) from0.026 to 0.616 M.31Ionic strength values were adjusted by addition of HCl and NaCl(pH values ranged from 0.98 to 1.8) Under these conditions, the anions are present intheir fully deprotonated, “free anion” forms, a-PW12O403 and a-PW12O404.Observed rate constants, kobs, were fitted to the Davies equation (Equation 1.30)

An internuclear distance of r¼ 11.2 A, or twice the ionic radius of the Keggin anions,was used At 25C in water,a ¼ 0.509 (dimensionless) and b ¼ 3.29  107cm/mol1/2

a-Keggin anions are shown in coordination polyhedron notation Each anion is 1.12 nm in

22 ELECTRON TRANSFER REACTIONS

For the units to “work out,” r must be converted to 1.12 107cm UsingEquation 1.30, log k was plotted as a function ofm1/2/(1þ brm1/2) A linear relation-ship was observed (R2¼ 0.998), whose slope (equal to 2az1z2) gave a charge product

of z1z2¼ 14.3 (Figure 1.9) The theoretical charge product is 12 Linearity andcomparison to the theoretical charge product are both important considerations inevaluating whether an electron transfer reaction between charged species obeyselectrolyte theory to an extent sufficient for proceeding with use of the data in theMarcus model In the present case, few would argue that the system fails to complywith electrolyte theory

The linearity and close-to-theoretical slope in Figure 1.9 was surprising becauseEquation 1.30 was derived for univalent ions at low ionic strengths (up to 0.01 M).Agreement at much higher ionic strengths (greater than 0.5 M) was attributed to thefact that POM anions, “owing to the very pronounced inward polarization of theirexterior oxygen atoms, have extremely low solvation energies and very low van derWaals attractions for one another.”48

In many published examples of careful work, some models fail, while others(including those with empirical corrections) give better fits In those cases, somejudgment is required to assess whether the Marcus model might be used The mostoften encountered reasons for failure to comply with these models are the presence ofalternative mechanistic pathways and significant ion association between electrolyteions and the charged species involved in the electron transfer reaction Ion association

is discussed in more detail at the end of this chapter

1.3.2 Self-Exchange Rate Expression

For self-exchange reactions such as that in Figure 1.8, Equation 1.18 applies This isobtained by setting the free energy terms in Equation 1.12 equal to zero In most cases,the work term W(r) can be calculated This is the energy required to bring the reactantsfrom effectively infinite separation to within collision distance r, approximated as the

) (from the Davies equation,

GUIDE TO USE OF THE MARCUS MODEL 23

sum of the radii of the reacting ions, r1þ r2 The reorganization energyl is moredifficult to calculate This is because it requires detailed knowledge of all the changes

in bond lengths and angles required to reach the transition state for electron transfer(see Equation 1.9) and this information is usually not available In practice, therefore,self-exchange reactions are most commonly carried out as a means for determiningthe reorganization energies This fundamental parameter of the self-exchange reac-tion can then be used to determine the inherent reorganization barriers associatedwith other electron transfer reactions (cross-reactions) of interest In this sense, thespecies involved in the self-exchange reaction can be deployed as physicochemical

“probes.”17,32,33

The work term W(r), Equation 1.39, is as shown in Equation 1.4, but withthe subscript on the collision distance r modified to indicate a self-exchangereaction

A dramatic example of the effect of charge and W(r) on rate constants involvesself-exchange between a-AlW12O405 and the one-electron reduced aniona-AlW12O406 At an ionic strength,m, of 175 mM, the rate constant for this reaction

By taking the natural logarithm of each side of Equation 1.18, one obtainsEquation 1.42 (reproduced here from Section 1.2) The subscript on k is a pair of

24 ELECTRON TRANSFER REACTIONS

1’s, used to indicate that this rate constant is for a self-exchange reaction.

A well-known self-exchange reaction is that between RuIII(NH3)6 þ and

RuII(NH3)6 þ.38,49 At 25C and m ¼ 0.1 M, the experimentally determined exchange rate constant k11is 4 103

self-M1s1.For D¼ 78.4, the effective radius of the Ru complexes equal to 3.4  108cm (i.e.,

r¼ 6.8  108cm), and the other constants defined as shown above, W(r) is calculated

The ionic strengthm in units of mol/L is equivalent to units of mmol/cm3 Note that

b has units of cm1/2/mol1/2, such that the units of the productbrm cancel one anotherdue to the factor of 1000 in the denominator of b itself (i.e., as defined inEquation 1.33)

By solving Equation 1.42 forl, one obtains

In the above example, the reorganization energyl is the total reorganizationenergy and includes both inner-sphere and outer-sphere components, linandlout,discussed earlier in this chapter Once the total reorganization energy,ltotal, is known,

linandloutcan be calculated using Equation 1.45

self-exchange reactions, r1¼ r2and r12¼ 2 r1, and for a one-electron process,De¼ e.Thus, Equation 1.10 reduces to Equation 1.46.

lout¼ e2 1

r1

 1

lin¼ 15.9 kcal/mol, which left 29.6 kcal/mol for lout Next, they used Equation 1.46

to calculate the “effective” radius of O2 (i.e., r1/2) and obtained a value of 3 A.More recent work suggests that this seemingly large value might reflect orienta-tional restrictions imposed on collisions between the nonspherical reactants, O2and O2.51

1.3.3 The Marcus Cross-Relation

Rate constants for outer-sphere electron transfer reactions that involve net changes

in Gibbs free energy can be calculated using the Marcus cross-relation(Equations 1.24–1.26) It is referred to as a “cross-relation” because it is derivedfrom expressions for two different self-exchange reactions

1.3.3.1 Derivation of the Marcus Cross-Relation The cross-relation is derivedalgebraically by first assuming that the reorganization energy for the “cross” reaction

is the average of the reorganization energies associated with the two self-exchangereactions involved To clarify this, consider the two self-exchange reactions inEquations 1.49 and 1.50 These reactions are, respectively, assigned rate constants

26 ELECTRON TRANSFER REACTIONS