Marcus theory

The Marcus theory (named after Rudolph Arthur Marcus ) occurs in redox reactions without bond formation or bond breaking in the place of the Eyring theory. Both theories lead to rate equations of the same exponential form. However, whereas the binding changes of the reactants are determinative in the Eyring theory during the reaction, the solvent ( outer sphere ) plays the central role in these redox reactions ( Einelektronenaustauschreaktionen ). The Marcus theory shows this and allows the calculation of the Gibbs free energy of activation of the polarization properties of the solvent, the size and spacing of the reactants in the electron transfer and the free enthalpy of the redox reaction.

Hinführung

Chemical reactions result in changes to material. They may e.g. lead to the substitution of a group in the molecule or a ligand in the complex, the elimination of a molecule or part of a ligand or to a rearrangement. However, it can also just change the charge state of the reactants, and these redox reactions appear to be particularly simple in inorganic chemistry with ions and complexes. To observe such reactions in ions of the transition metals are often of color changes, but also organic molecules may become discolored in electron uptake and release as the herbicide paraquat (1,1 ' -dimethyl-4, 4' -bipyridinium ), the blue for the absorption of electrons will. Hence the alternative name comes methyl viologen. For a type of redox reactions without structural changes Marcus has developed his theory. For the mathematical reasoning, the original papers should be consulted.

In a redox reaction, one partner acts as an electron donor D, A. The partners can react other than an electron, it must diffuse together. Means that they form the so-called Precusorkomplex, usually a kinetic unstable solvated collision complex, which passes through the electron transfer in a Successorkomplex which diffuses his hand apart. For the single-electron transfer, this yields the equation

(D and A can even carry loads ). Here, k12, k21 and k30 diffusion constants, k23 and k32 rate constants of activated partial reactions. The overall reaction may be controlled by diffusion, if the electron transfer is faster than the diffusion. Each shock leads to a reaction. If the reaction is activation- controlled be the one the association equilibrium, the electron transfer, the slower pace and the separation of the Sucessorkomplexes is the faster step.

Redox reactions usually take place in polar solvents, donor and acceptor centers then wear a solvation shell, and even precursors and Sucessorkomplex are solvated. The inner molecules of the solvation shell, which are very tightly bound in complexes of the ligands are referred to as inner sphere. Redox reactions in which this sphere is involved, it is called " inner-sphere " reactions. The outer sphere consists of the free solvent molecules. In the " outer-sphere " reactions, the inner sphere does not change, it will be broken or formed any bonds.

R. A. Marcus has with the nature and size of the outdoor activation in redox reactions, accurate one-electron transfer reactions of the outer-sphere type, busy and recognized the central role of the solvent. He has published two works first. Their results are often referred to as Marcus theory, although Marcus's work go much further later.

The problem

In outer-sphere redox reactions no bonds are formed or broken. There is only one electron transfer ( ET). A simple example is the Fe2 / Fe3 redox reaction. In the self- exchange reaction in an aqueous solution, for example, contains both FeSO4 and Fe2 (SO4 ) 3, takes place in both directions with the same measurable gross rate is the thermodynamic free standard enthalpy of reaction ( ΔG0 ) is equal to zero.

For example from the temperature dependence of the rate a reaction such as SN2 displacement reaction of the hydrolysis of an alkyl halide, an activation energy is mostly determined, and this again characterized as the energy of the transition state in the reaction diagram. The latter is drawn according to Arrhenius and Eyring as an energy diagram of the reaction coordinate as the abscissa, which describes the lowest energy path from the reactants to the products. The points of the coordinates of a reaction sequence of combinations of distances and angles between, and in the reactants during the formation and / or the breakage of bonds. The maximum in the energy diagram of the transition state, is characterized by a particular configuration of the atomic nuclei. In theory, the transition state of a particular core Eyring also coordinate change is responsible for the crossing of the maximum, and therefore the vibration is covered by this co-ordinate direction to the Eyring theory translation.

In the outer-sphere redox reactions can not give this reaction pathway. Yet one observes an activation energy. The rate constant for the activation of controlled reaction has the same form as the equation Eyringsche

In A acts mainly the transition probability and is the Gibbs free energy for the formation of the transition state.

The model of Marcus

When electron exchange is changing the charge distribution and this has a major impact on the solvent environment, because the solvent molecules rearrange in the charges from box ( this is called orientation polarization ), and the atoms and electrons in the solvent molecules are slightly displaced (atom or electron polarization ). This solvent polarization influences the magnitude of the activation energy and the reaction rate in redox reactions.

Substitution, elimination, and isomerization reactions are different from the outer-sphere redox reactions not only by structural changes, but also the fact that all individual core and charge transfers ( charge transfer, charge transfer, CT) continuously in the reactants on the energetically favored pathway and synchronously run, the core configurations are thus always in the "equilibrium". Consider the SN2 substitution of the hydrolysis of an alkyl halide in which the backside attack of the OH - ion displaces halide ion and where one must assume a transition state with fünfbindigem carbon atom. The Reaktantensysteme are strongly coupled in the course of the reaction, that they constitute a single entity, and finally an activated complex. In contrast, the solvent has a small but non-negligible influence.

In outer-sphere redox reactions the nuclear displacements in the reactants are very small, on the other hand, the solvent is determinative. The coupling of donor and acceptor is weak, both retain their identity throughout the reaction. Why can "jump" the electron as elementary only as a whole ( electron transfer ET). The electrons jump when it happens at all, is very much faster than the solvent molecules can move. The consequence is: so that the electron can jump, the core positions of the two reactants and all solvent molecules have before and after the fast electrons jump be the same (Franck -Condon principle ); and also the energy must not change when electrons jump.

Solvent arrangement is dependent on the load conditions. If, after the electron jump which should be the same before and then every correct arrangement for the Selbstaustauchreaktion already for reasons of symmetry would be realized by a solvent configuration, which ceased during transmission half elementary charge. At the same time presursor and Sucessorkomplex would have the same energy in the solvent environment. In this arrangement, the solvent conditions for the electron jump would be fulfilled.

Because the electron as elementary particles but can not be shared, it must be located on either the source or target atom or molecule. This means that the " transition state " solvent configuration is not ( equivalent to the transfer of a half charge ) and charge distribution ( charge on one of the partners ) in the " equilibrium". Nevertheless, this transition state must be reached before the electrons jump, and this can be done in the solvent by thermal fluctuations. The generation of the proper solvent configuration and the electron jump are effectively decoupled and happen out of sync. Actually, they have nothing to do with each other. The energy of the transition state is thus largely a polarization energy of the solvent.

The Marcus theory

The macroscopic system: two conducting spheres

Based on these considerations Rudolph A. Marcus its has developed a classical theory. Your goal is to calculate the Polarisationeenergie the mentioned non-equilibrium state. From thermodynamics we know that the energy of such a can be calculated if one finds a reversible way there. This is Marcus succeeded.

Four elements are constitutive model for the underlying theory: (1) Marcus initially used a classic, purely electrostatic model, in which the charge (very many elementary charges ) can be transferred to any portions between two bodies. ( 2) Marcus separates the fast electron polarization Pe of the solvent and the slow atomic and orientation polarization Pu due to the orders of magnitude different time constants with which they set themselves. ( 3) Marcus separates inner sphere ( reactant fixed solvated solvent shell in complexes: ligand ) and the outer sphere ( solvent -free outside ). ( 4) Marcus calculates only the outer-sphere energy of nonequilibrium polarization of the " transition state " in a solvent, the ( the Debye- Hückel theory of electrochemistry see ) is much bigger because of the long-range electrostatic forces mostly as the contribution of the inner sphere. To this end, he chooses a path with two reversible up or Umladungsschritte.

The tool provides the theory of dielectric polarization in solutions. Marcus solve the problem in general for a charge transfer between two bodies of any shape with a certain volume and surface charges. For the case of self-exchange redox reaction redox couple (eg, Fe ( H2O) 63 / Fe (H2O) 62 ) is replaced by two macroscopic conducting spheres particular charge state at a certain distance, between which a certain amount of charge to be reversibly exchanged.

In the first step, the energy state of WI is calculated in which both balls bear one half of the charge to be exchanged. This condition can be achieved by the reversible transfer of the half amount of charge from one to another through the transfer of the ball charge of the exchange of the donor ball into the vacuum and from there to the acceptor ball. The so charged spheres produce a certain electric field, in which the total solvent polarization Pe Pu set in the solvent. On the other hand, the polarization of the solvent now generates a field component, which acts back on charge and polarization.

In the second step, the energy is the reversible WII determined (re) transmission of the half exchanged charge back over the vacuum, on the first ball. In this case, however, the atomic and orientation polarization Pu is retained, only the electron polarization can be set in the new charge distribution and the now solid Pu field. Thereafter, the system is in the desired condition with an electron polarization corresponding to the initial state of the oxidation-reduction reaction, and an atomic polarization and orientation which correspond to the activated complex. The energy WI WII of this state is thermodynamically a free enthalpy G.

Of course in this classic model, the charge transfer is possible not only for the half amount of charge, but also for other portions DELTA.e. So you can sense the energy as a function of the distribution of the charge on the two spheres and thus the solvent polarization. Marcus has this elegant, the coordinates of all solvent molecules in a single Polarisationskoordinate Ap, which is determined by the charge transfer DELTA.e, summarized and thus a simplification of the energy representation in only two dimensions obtained: G = f ( DELTA.e ). The result for two conducting spheres in a solvent is the Marcus formula

Wherein r1 and r2 are the radii of the two spheres, R is the distance between them and εs and εop static and high frequency (optical ) dielectric constant of the solvent is. DELTA.e is the amount of charge transferred, the graph G vs. DELTA.e is a parabola ( Fig. 1). In the Marcus theory, the energy of the transfer of an entire charge corresponds ( DELTA.e = 1 ) is called the (outer sphere ) reorganization energy? O, ie the energy of the two- ball system in which the polarization corresponds to that of an entire charge transferred, the charge distribution, but the state before the transfer. The system is symmetrical with respect to the direction of exchange.

The microscopic system: the redox couple

For reduction of the classical two - sphere system to self-exchange reaction to get to certain quantum redox pair in which the charge can no longer be transmitted in any portions, but only as a whole elementary charge. The solvent polarization but can still be treated classically, ie it is not quantized, because it is determined jointly by many solvent molecules. Therefore, one can calculate the reorganization energy for a hypothetical transfer and return a partial elementary charge by the Marcus formula. The reorganization energy is also responsible for chemical redox systems a parabola (Fig. 2. ), And it corresponds to a free energy or free enthalpy. The energy that would have the system in the hypothetical transfer of a half- electron charge ( DELTA.e = 0.5 ), the activation of the self-exchange reaction? G ( 0) =? O ‡ / 4 (see Fig.1 and Fig.2, the intersection of the parabolas i and f and f ( 0 ))).

Up to this point everything was pure physics, chemistry now something happens. The self-exchange reaction is unique among the redox reactions. Most redox reactions take place between different partners, eg

And therefore also show a positive ( endergonic ) or negative ( exergonic ) free reaction enthalpy ΔG0.

Since the Marcus calculations are based solely on the electrostatic conditions in the solvent ( outer-sphere ), are ΔG0? O and independent of each other and simply additive, ie the Marcus parabolas of systems with different ΔG0 are vs in G. DELTA.e - diagram merely shifted up or down ( Fig. 2). A variation of ΔG0 can be experimentally achieved for example, that one offers a different donor acceptors.

From a simple calculation with the parabolas i ( equation y = x2 ), f ( 0) ( y = (x -d) 2 ), and f1 to f3 ( equation y = (x -d) 2 c) is obtained for the free energy of activation

Perhaps one should emphasize again that the intersection of the parabolas representing the polarization energy, not the energy of a particular configuration of all nuclei of the reactants such as in the above-mentioned substitution reaction. While ideally in this geometry of the transition state in each reactant pair is the same redox couples can meet with many different environments, the polarization energy condition. For this reason the use of the free energy of activation as a thermodynamic quantity is appropriate.

The formula of Marcus ( 2 ) shows a quadratic relationship between free reaction enthalpy and free energy of activation. It is an experience from the large experimental material chemistry: reactions usually proceed faster, the more negative ΔG0. In many cases, a linear free energy relationship (LFE ) is detected. Even after the Marcus formula to take the reaction rates when the reactions are exergonic, but only as long as ΔG0 is positive or moderately negative range of values. It is surprising that the activation energy for oxidation-reduction reactions in solution according to the formula ( 3) with very strong negative free reaction enthalpy should be back bigger, namely when ΔG0 is negative and greater in absolute terms than? O. This area of ​​the free reaction is " Marcus - inverted " called area. In Figure 2 you can see that in a further drop of ΔG0 the intersection of the i -and f- parabolas to the top left moves, which means an increase in the free energy of activation and decrease in speed. The representation of ln k vs. ΔG0 should therefore be a maximum curve.

The maximum reaction rate is expected at? G ‡ = 0. Here also DELTA.e = 0 and q = 0 ( Fig. 2). This means that the electron can jump in the equilibrium solvent polarization of the reactants, which is naturally realized much more likely than a thermally excited: the reaction is then complete barrier. In the inverted region, the polarization corresponds to the thinking difficult idea of a hypothetical charge distribution on the reactants, in which added the donor charge and the acceptor would have made charge. In reality, of course, does not happen, because not a real charge transfer generates this polarization, but thermal fluctuations in the solvent. The polarization that is necessary for the inverted region can be adjusted with a certain probability by the thermal fluctuation as well any other of equal energy. The electron waits a certain extent on the correct polarization, until it jumps.

Experimental results

Marcus has his theory published in 1956. For many years, the inverted area searched, but the experiments showed at reaction rows becomes smaller ΔG0 only an increase of k up to the diffusion-controlled value, that to the value at which each collision of the reactants leads to the reaction, and this limit was also in very strong negative ΔG0 values ​​obtained ( Rehm -Weller behavior). It took about 30 years to the inverted region by Miller, Calcaterra and Closs was clearly demonstrated and were indeed held in the intramolecular electron transfer in a molecule in which the donor and acceptor by a rigid bridge at a fixed distance (Fig. 3 ).

Ex post, one can assume that in freely diffusing reactants of the electron jump at the distance R is carried out in which? O = - ΔG0 is so? G # = 0. For? O depends on R, is? O at larger R increases, the opening of the parabola is smaller, and it is always possible formally the parabolas of Figure 2 to narrow so that the f - parabola passes through the vertex of the i- parabola. This means that more? G ‡ = 0 and the rate constant k has a maximum value that the diffusion- induced boundary, that for all redox reactions with very negative reaction enthalpy ΔG0 .. But there is also the view that a decrease in the rate for very fast electron meetings beyond the experimentally achievable very negative ΔG0 values ​​lie. The involvement of excited electronic states has been discussed.

R. A. Marcus and co-workers have refined various aspects of this theory in the following the first publication of the theory years. They have, inter alia, statistical considerations and quantum effects into account, they expanded to Chemiluminszenzsysteme and electrode reactions. R. A. Marcus was awarded for his work in 1992 the Nobel Prize in Chemistry, his Nobel lecture provides a comprehensive overview of his work.

Bills and Notes

• Kinetics (chemistry)