Outer sphere rate constant Marcus Theory

How do we obtain kis and kos from such a complicated rate expression As indicated in equations (5.13) and (5.14), we can determine kso experimentally by carrying out the reaction between [Co(ox)2(en)] and [Co(en)3]2+ under pseudo-first-order conditions with varying excess [Co(en)3]2+. A plot of kobs vs [Co(en)3]2+ will give the second-order rate constant (kso) f°r the reaction. Using equation (5.15), and given the value for the inner-sphere rate constant, kis, which has been measured independently, the outer-sphere rate constant, kos, will be determined. In Experiment 5.6, Marcus theory will be used to model this reaction and the calculated vs the observed outer-sphere rate constants will be compared. 

EXPERIMENT 5.6 MARCUS THEORY THEORETICAL CALCULATION OF THE OUTER-SPHERE RATE CONSTANT, kos, FOR THE REACTION BETWEEN [Co(ox)2(EN)] AND [Co(en)3]2+11... 

According to the Marcus theory [64] for outer-sphere reactions, there is good correlation between the heterogeneous (electrode) and homogeneous (solution) rate constants. This is the theoretical basis for the proposed use of hydrated-electron rate constants (ke) as a criterion for the reactivity of an electrolyte component towards lithium or any electrode at lithium potential. Table 1 shows rate-constant values for selected materials that are relevant to SE1 formation and to lithium batteries. Although many important materials are missing (such as PC, EC, diethyl carbonate (DEC), LiPF6, etc.), much can be learned from a careful study of this table (and its sources). 

The value of E° was hence determined by the reaction of R4M with Fe3+ complexes as outer-sphere SET oxidizers. Using five complexes with a range of different E° values, from 1.15 to 1.42 V, the rate constants were determined193. This was followed up by Eberson who, by application of the Marcus theory, was able to determine from the E° values (shown in Table 18) standard potentials and reorganization energies. Most compounds... 

There are three points of significance of this result. One is that it provides strong support for the 10-step mechanism originally proposed for reaction 1. Another is that it facilitates a more robust fitting of the mechanism to the kinetic data obtained for that reaction. Thirdly, it confirms that reaction 2 has a rate constant that is four orders of magnitude greater than predicted by Marcus theory. It is concluded that reaction 2 is poorly modeled as an outer-sphere process and is better described as... 

The Marcus classical free energy of activation is AG , the adiabatic preexponential factor A may be taken from Eyring s Transition State Theory as (kg T /h), and Kel is a dimensionless transmission coefficient (0 < k l < 1) which includes the entire efiFect of electronic interactions between the donor and acceptor, and which becomes crucial at long range. With Kel set to unity the rate expression has only nuclear factors and in particular the inner sphere and outer sphere reorganization energies mentioned in the introduction are dominant parameters controlling AG and hence the rate. It is assumed here that the rate constant may be taken as a unimolecular rate constant, and if needed the associated bimolecular rate constant may be constructed by incorporation of diffusional processes as ... 

A model has been considered for Sn2 reactions, based on two interacting states. Relevant bond energies, standard electrode potentials, solvent contribntions (nonequi-librinm polarization), and steric effects are included. Applications of the theory are made to the cross-relation between rate constants of cross- and identity reactions, experimental entropies and energies of activation, the relative rates of Sn2 and ET reactions, and the possible expediting of an outer sphere ET reaction by an incipient SN2-type interaction (Marcus, 1997). 

Marcus attempted to calculate the minimum energy reaction coordinate or reaction trajectory needed for electron transfer to occur. The reaction coordinate includes contributions from all of the trapping vibrations of the system including the solvent and is not simply the normal coordinate illustrated in Figure 1. In general, the reaction coordinate is a complex function of the coordinates of the series of normal modes that are involved in electron trapping. In this approach to the theory of electron transfer the rate constant for outer-sphere electron transfer is given by equation (18). 

The Marcus theory provides an appropriate formalism for calculating the rate constant of an outer-sphere redox reaction from a set of non-kinetic parameters[347 350]. The simplest possible process is a self-exchange reaction, where AG = 0. In an outer-sphere electron self-exchange reaction the electron is transferred within the precursor complex (Eq. 11.5). 

These driving forces are exergonic and considerably more favorable than those involved in the electron-transfer reactions of the simple, monosub-stituted carbonylmanganese cations Mn(CO)5L+ and anions Mn(CO)4P-(where L and P are both monodentate phosphines and phosphites). Nonetheless, the rate constants for cis- and ra -Mn(CO)2( DPPE )2+ with Mn(CO)2(DPPE)2 are considerably slower than those qualitatively observed between Mn(CO)5L+ and Mn(CO)4P- (67). Such large rate differences that belie thermodynamics can be attributed to steric hindrance in the tetrasubstituted carbonylmanganese cations and the anion which are absent in the simpler ions. Such structural effects, even in these apparently outer-sphere electron transfers, merit a further quantitative evaluation as in the application of Marcus theory (83). 

Kinetic data have been reported for reduction of //-superoxo complexes by Fe2+,7 1 Mov,702 Co11703 and Ru11 complexes,704 and V2+, Cr2+ and Eu2+.705 These processes involve outer-sphere electron transfer and in some cases703,706 the Marcus theory has been applied to the rate constants obtained. Electron transfer quenching of the excited state of [Ru(bipy)3]2+ by various -superoxo cobalt(III) complexes leads to production of [Ru(bipy)3]3+ and the corresponding /z-peroxo species.706... 

The outer sphere character of these reactions has encouraged some workers to apply Marcus theory to the rate constants obtained " . Given the uncertainty in the values of the electrode potentials and the considerable electrostatic work function involved in the formation of the precursor complex, the significance of the intrinsic rate parameters obtained is not clear. 

Marcus theory showed a good correlation between experimental and calculated rate constants using Eq. (5). The 22 value was set at 10 M sec for this purpose and is considered as an upper limit for selfexchange of the diethyldithiocarbamate radical/anion pair. From the oxidation rates it was also estimated that E (edtc /edtc ) = 0.43(3)V vs SCE. A free-energy analysis for the oxidation of diethyldithiocarbamate (edtc ) by [FeiCNlgT also showed that the initial outer-sphere oxidation of the thiolate anion to its thio radical (Eq. 36) is the main energy barrier to be crossed along the reaction coordinate. 

Therefore, if the Marcus theory describes properly the effect of solvents of k, a linear correlation between In and ( op -fis ) should be observed in the experimental results. Before turning to the experimental studies, the (Sop - s ) parameter for various solvents used in electrochemical work is presented in Table 1. Inspection of these data reveals that the largest difference of the (Cop -Ss ) parameter for the listed solvents amounts to 0.263. Thus, on the basis of the Marcus theory for the outer-sphere electrode reactions, the largest change of the reaction rate for different solvents should amount to exp (const 0.263). In this estimation any double-layer effect on the rate constant was neglected. 

Aj may be evaluated from x-ray and infrared (IR) data or from theoretical calculations. However, for organic outer sphere electron transfers, this contribution is usually much smaller than Ao. In our opinion one of the greatest merits of the Marcus [43] and Levich-Dogonadze [44] theories is that they allow rather correct predictions of Aq through simple equations. Thus for most outer sphere electron transfers, reasonably accurate values of the rate constants can be predicted. 

In this experiment, the electrochemistry of both [Co(en)3]3+/2+ and [Co(ox)3]3+/2+ will be investigated using cyclic voltammetry, and the standard reduction potential (E°, V) for the [Co(en)3]3+/2+ couple will be measured. For metal complex stability reasons discussed below, it is not possible to use this technique to obtain reduction potentials for the mixed ligand cobalt systems an exercise at the end of this experiment helps to estimate these. The E° values obtained will be important for experiment 5.6, in which outer-sphere electron transfer rate constants between [Co(en)3)]2+ and [Co(en)2)(ox)]+ will be mathematically modeled using Marcus theory. 

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