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d 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 5 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 edit 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 k 12 k 21 and k 30 are diffusion constants k 23 and k 32 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 problem edit in outer sphere redox reactions no bonds are formed or broken only an electron transfer et takes place a quite simple example is the fe 2 fe 3 redox reaction the self exchange reaction which is known to be always occurring in an aqueous solution containing the aquo complexes fe h 2 o 6 2 and fe h 2 o 6 3 redox occurs with gibbs free reaction energy δ g 0 displaystyle delta g circ 0 from the reaction rate s temperature dependence an activation energy is determined and this activation energy is interpreted as the energy of the transition state in a reaction diagram the latter is drawn according to arrhenius and eyring as an energy diagram with the reaction coordinate as the abscissa the reaction coordinate describes the minimum energy path from the reactants to the products and the points of this coordinate are combinations of distances and angles between and in the reactants in the course of the formation and or cleavage of bonds the maximum of the energy diagram the transition state is characterized by a specific configuration of the atoms moreover in eyring s tst 3 4 a quite specific change of the nuclear coordinates is responsible for crossing the maximum point a vibration in this direction is consequently treated as a translation for outer sphere redox reactions there cannot be such a reaction path but nevertheless one does observe an activation energy the rate constant for activation controlled reactions has the same exponential form as the eyring equation k act a e δ g r t displaystyle k_ text act a cdot e frac delta g ddagger rt δ g displaystyle delta g ddagger is the gibbs free energy of the formation of the transition state the exponential term represents the probability of its formation a contains the probability of crossing from precursor to successor complex marcus model edit the consequence of an electron transfer is the rearrangement of charges and this greatly influences the solvent environment for the dipolar solvent molecules rearrange in the direction of the field of the charges this is called orientation polarisation and also the atoms and electrons in the solvent molecules are slightly displaced atomic and electron polarization respectively it is this solvent polarization which determines the free energy of activation and thus the reaction rate substitution elimination and isomerization reactions differ from the outer sphere redox reaction not only in the structural changes outlined above but also in the fact that the movements of the nuclei and the shift of charges charge transfer ct on the reactions path take place in a continuous and concerted way nuclear configurations and charge distribution are always in equilibrium this is illustrated by the s n 2 substitution of the alkaline hydrolysis of an alkyl halide where the rear side attack of the oh ion pushes out a halide ion and where a transition state with a five coordinated carbon atom must be visualized the system of the reactants becomes coupled so tightly during the reaction that they form the activated complex as an integral entity the solvent here has a minor effect by contrast in outer sphere redox reactions the displacement of nuclei in the reactants are small here the solvent has the dominant role donor acceptor coupling is weak both keep their identity during the reaction therefore the electron being an elementary particle can only jump as a whole electron transfer et if the electron jumps the transfer is much faster than the movement of the large solvent molecules with the consequence that the nuclear positions of the reaction partners and the solvent molecules are the same before and after the electron jump franck condon principle 6 the jump of the electron is governed by quantum mechanical rules it is only possible if also the energy of the et system does not change during the jump the arrangement of solvent molecules depends on the charge distribution on the reactants if the solvent configuration must be the same before and after the jump and the energy may not change then the solvent cannot be in the solvation state of the precursor nor in that of the successor complex as they are different it has to be somewhere in between for the self exchange reaction for symmetry reasons an arrangement of the solvent molecules exactly in the middle of those of precursor and successor complex would meet the conditions this means that the solvent arrangement with half of the electron on both donor and acceptor would be the correct environment for jumping also in this state the energy of precursor and successor in their solvent environment would be the same however the electron as an elementary particle cannot be divided it resides either on the donor or the acceptor and arranges the solvent molecules accordingly in an equilibrium the transition state on the other hand requires a solvent configuration which would result from the transfer of half an electron which is impossible this means that real charge distribution and required solvent polarization are not in an equilibrium yet it is possible that the solvent takes a configuration corresponding to the transition state even if the electron sits on the donor or acceptor this however requires energy this energy may be provided by the thermal energy of the solvent and thermal fluctuations can produce the correct polarization state once this has been reached the electron can jump the creation of the correct solvent arrangement and the electron jump are decoupled and do not happen in a synchronous process thus the energy of the transition state is mostly polarization energy of the solvent theory description edit macroscopic system two conducting spheres edit on the basis of his reasoning r a marcus developed a classical theory with the aim of calculating the polarization energy of the said non equilibrium state from thermodynamics it is well known that the energy of such a state can be determined if a reversible path to that state is found marcus was successful in finding such a path via two reversible charging steps for the preparation of the transition state from the precursor complex four elements are essential for the model on which the theory is based marcus employs a classical purely electrostatic model the charge many elementary charges may be transferred in any portion from one body to another marcus separates the fast electron polarisation p e and the slow atom and orientation polarisation p u of the solvent on grounds of their time constants differing several orders of magnitude marcus separates the inner sphere reactant tightly bound solvent molecules in complexes ligands and the outer sphere free solvent in this model marcus confines himself to calculating the outer sphere energy of the non equilibrium polarization of the transition state the outer sphere energy is often much larger than the inner sphere contribution because of the far reaching electrostatic forces compare the debye hückel theory of electrochemistry marcus tool is the theory of dielectric polarization in solvents he solved the problem in a general way for a transfer of charge between two bodies of arbitrary shape with arbitrary surface and volume charge for the self exchange reaction the redox pair e g fe h 2 o 6 3 fe h 2 o 6 2 is substituted by two macroscopic conducting spheres at a defined distance carrying specified charges between these spheres a certain amount of charge is reversibly exchanged in the first step the energy w i of the transfer of a specific amount of charge is calculated e g for the system in a state when both spheres carry half of the amount of charge which is to be transferred this state of the system can be reached by transferring the respective charge from the donor sphere to the vacuum and then back to the acceptor sphere note 2 then the spheres in this state of charge give rise to a defined electric field in the solvent which creates the total solvent polarization p u p e by the same token this polarization of the solvent interacts with the charges in a second step the energy w ii of the reversible back transfer of the charge to the first sphere again via the vacuum is calculated however the atom and orientation polarization p u is kept fixed only the electron polarization p e may adjust to the field of the new charge distribution and the fixed p u after this second step the system is in the desired state with an electron polarization corresponding to the starting point of the redox reaction and an atom and orientation polarization corresponding to the transition state the energy w i w ii of this state is thermodynamically speaking a gibbs free energy g fig 1 the parabolas of outer sphere reorganisation energy of the system two spheres in a solvent parabola i the charge on the first transfer to the second parabola f the charge on the second transfer to the first the abscissa is the transferred amount of charge δ e or the induced polarization p the ordinate the gibbs free energy δg 0 λ o 4 is the reorganization energy at δe 0 5 it corresponds to the activation energy of the self exchange reaction of course in this classical model the transfer of any arbitrary amount of charge δ e is possible so the energy of the non equilibrium state and consequently of the polarization energy of the solvent can be probed as a function of δ e thus marcus has lumped together in a very elegant way the coordinates of all solvent molecules into a single coordinate of solvent polarization δ p which is determined by the amount of transferred charge δ e so he reached a simplification of the energy representation to only two dimensions g f δ e the result for two conducting spheres in a solvent is the formula of marcus g 1 2 r 1 1 2 r 2 1 r 1 ϵ opt 1 ϵ s δ e 2 displaystyle g left frac 1 2r_ 1 frac 1 2r_ 2 frac 1 r right cdot left frac 1 epsilon _ text opt frac 1 epsilon _ text s right cdot delta e 2 where r 1 and r 2 are the radii of the spheres and r is their separation ε s and ε opt are the static and high frequency optical dielectric constants of the solvent δ e the amount of charge transferred the graph of g vs δ e is a parabola fig 1 in marcus theory the energy belonging to the transfer of a unit charge δ e 1 is called the outer sphere reorganization energy λ o i e the energy of a state where the polarization would correspond to the transfer of a unit amount of charge but the real charge distribution is that before the transfer note 3 in terms of exchange direction the system is symmetric microscopic system the donor acceptor pair edit shrinking the two sphere model to the molecular level creates the problem that in the self exchange reaction the charge can no longer be transferred in arbitrary amounts but only as a single electron however the polarization still is determined by the total ensemble of the solvent molecules and therefore can still be treated classically i e the polarization energy is not subject to quantum limitations therefore the energy of solvent reorganization can be calculated as being due to a hypothetical transfer and back transfer of a partial elementary charge according to the marcus formula thus the reorganization energy for chemical redox reactions which is a gibbs free energy is also a parabolic function of δ e of this hypothetical transfer for the self exchange reaction where for symmetry reasons δ e 0 5 the gibbs free energy of activation is δ g 0 λ o 4 see fig 1 and fig 2 intersection of the parabolas i and f f 0 respectively up to now all was physics now some chemistry enters the self exchange reaction is a very specific redox reaction most of the redox reactions are between different partners note 4 e g fe ii cn 6 4 ir iv cl 6 2 fe iii cn 6 3 ir iii cl 6 3 displaystyle ce fe ii cn 6 4 ir iv cl6 2 fe iii cn 6 3 ir iii cl6 3 and they have positive endergonic or negative exergonic gibbs free energies of reaction δ g displaystyle delta g circ as marcus calculations refer exclusively to the electrostatic properties in the solvent outer sphere δ g displaystyle delta g circ and λ 0 displaystyle lambda _ 0 are independent of one another and therefore can just be added up this means that the marcus parabolas in systems with different δ g displaystyle delta g circ are shifted just up or down in the g displaystyle g vs δ e displaystyle delta e diagram fig 2 variation of δ g displaystyle delta g circ can be affected in experiments by offering different acceptors to the same donor simple calculation of the intersection point between the parabolas i y x 2 displaystyle y x 2 and f i displaystyle f_ i y x d 2 c displaystyle y x d 2 c give the gibbs free energy of activation δ g λ 0 δ g 2 4 λ 0 displaystyle delta g ddagger frac lambda _ 0 delta g circ 2 4 lambda _ 0 where λ 0 displaystyle lambda _ 0 d 2 displaystyle d 2 and δ g displaystyle delta g circ c the intersection of those parabolas represents an activation energy and not the energy of ...