Activation free energy rate constant

Having separated the dynamical from equilibrium (or, more accurately, quasi-equilibrium) effects, one can readily discover the origin of the activation free energy and define the concept of the potential of mean force by analysis of the expression for the TST rate constant, k in (A3.8.3). The latter can be written as [7]... 

Our problem now is to determine the functional form of this experimental free energy curve for the intrinsic rate constant ki for electron transfer. In addition to the Marcus eq 4, two other relationships are currently in use to relate the activation free energy to the free energy change in electron transfer reactions (15, JL6). 

The forward and backward rate constants are related to the corresponding activation free energies, AG and AGf, by equation (1.25) below, introducing koo (and kf ) as the maximal rate constants, reached when A Gf or A Gf vanish. The main laws and models describing the way in which the forward and backward rate constants, or the corresponding free energies of activation, vary with the driving force are discussed in Section 1.4.2. 

AGq is the standard activation free energy, also termed the intrinsic barrier, which may be defined as the common value of the forward and backward activation free energies when the driving force is zero (i.e., when the electrode potential equals the standard potential of the A/B couple). Expression of the forward and backward rate constants ensues ... 

The forward and backward activation free energies and the corresponding rate constants thus depend on an extrinsic factor, the standard free energy of the reaction, AG° = E — E°, and an intrinsic factor, the standard activation free energy, that reflects the solvent and internal reorganization energy, Aq and A [equation (1.31)]. 

If the interaction between the donor and acceptor in the encounter pair (D. .. A) is weak (Scheme 4.2), the rate constant kET can be estimated by the Marcus theory. This theory predicts a quadratic dependence of the activation free energy AG versus AG° (standard free energy of the reaction). 

The reorganization energy of a self-exchange reaction is denoted A(0) (from the fact that AG° = 0) and is an important quantity in the Marcus theory, where it can be shown that the activation free energy of a self-exchange reaction, AG(0), is equal to X.(0)/4. It is also possible to measure rate constants of self-exchange processes experimentally and thus get access to (0) via this relationship. 

It therefore seems reasonable that the deviations of the activation free energies for highly exoergic electrochemical and homogeneous reactions, illustrated in Figures 2 and 5, may arise partly from the same source, i.e., from values of for the oxidation half reactions that are unexpectedly small. That is not to say that other factors are not responsible, at least in part, for these discrepancies. Nonadiabaticity, work terms, specific solvation, and other environmental effects may all play important roles depending on the reactants. For example, there is evidence to suggest that the true rate constant for outer-3+/2+... 

Early theories of electron transfer attempted to give a theoretical foundation to the phenomenological rate expression (Eq. 15) and the constants A, a, and o. A brief summary of the history and references to recent reviews have recently been given by Marcus. " Marcus theory has been able to derive an expression similar to Eq. (15), in which the relationship between the activation free energy and the reaction free energy is given in terms of a new quantity called the reorganization free... 

The experimental kinetic data obtained with the butyl halides in DMF are shown in Fig. 13 in the form of a plot of the activation free energy, AG, against the standard potential of the aromatic anion radicals, Ep/Q. The electrochemical data are displayed in the same diagrams in the form of values of the free energies of activation at the cyclic voltammetry peak potential, E, for a 0.1 V s scan rate. Additional data have been recently obtained by pulse radiolysis for n-butyl iodide in the same solvent (Grim-shaw et al., 1988) that complete nicely the data obtained by indirect electrochemistry. In the latter case, indeed, the upper limit of obtainable rate constants was 10 m s", beyond which the overlap between the mediator wave and the direct reduction wave of n-BuI is too strong for a meaningful measurement to be carried out. This is about the lower limit of measurable... 

The final step of the convolution analysis is the determination of the transfer coefficient a. This coefficient, sometimes called the symmetry factor, describes how variations in the reaction free energy affect the activation free energy (equation 26). The value of a does not depend on whether the reaction is a heterogeneous or a homogeneous ET (or even a different type of reaction such as a proton transfer, where a is better known as the Bronsted coefficient). Since the ET rate constant may be described by equation (4), the experimental determination of a is carried out by derivatization of the ln/Chet-AG° and thus of the experimental Inkhei- plots (AG° = F E — E°)) (equation 27). 

The E° difference is a necessary but not a sufficient condition. The rate constant for either ET (in general, / et) may be described in a simple way by equation (4). The activation free energy AG is usually expressed as a quadratic function of AG°, no matter whether we deal with an outer-sphere ET or a dissociative ET. However, even if the condition (AG")c < (AG°)sj holds (hereafter, subscripts C and ST will be used to denote the parameters for the concerted and stepwise ETs, respectively), the kinetic requirements (intrinsic barriers and pre-exponential factors) of the two ETs have to be taken into account. While AGq depends only slightly on the ET mechanism, is dependent on it to a large extent. For a concerted dissociative ET, the Saveant model leads to AG j % BDE/4. Thus, (AGy )c is significantly larger than (AG )sj no matter how significant AGy, is in (AG( )gj (see, in particular. Section 4). In fact, within typical dissociative-type systems such as... 

Note that, with the minimized rate constant in hand, a generalized activation free energy can be defined as the difference between the free energy of the reactants and that for the point. s mm- Note also that for the computation of isotope effects, VTST proceeds exactly like conventional TST, except that there is no requirement at a given temperature that the value of. y that minimizes the rate constant for the light-atom-substituted system will be the same value of. y that minimizes the rate constant for the heavy-atom-substituted system. Each must be determined separately, at which point the ratio of rate constants for that temperature may be expressed. 

The formal rate constants kfh and kb h are potential dependent. At any given potential, in order for the transition from the oxidized form to occur, it will be necessary to pass over an activation free-energy barrier, AGf, as illustrated by the Morse curves in Figure 2.14. The rate of reaction will be proportional to exp (-AGf /RT). 

The amount of free energy required to reach the transition state is called the activation free energy, AG. From equation (13) of chapter 2, we can equate AG to —RT In Kx, where A4 is an equilibrium constant for the formation of the transition state from the reactants. The fraction of the reactants that are in the transition state at any given moment is given approximately by A4, or e AGyRT. We therefore can write the overall rate constant for a reaction as... 

A solvent may play a decisive role when a reaction chooses the ion radical pathway. For instance, the solvent effect on the thermodynamic contribution to the activation free energy causes an increase in the cleavage rate constant for the chloroanthracene anion radical with an increase in the solvent acceptor number. There are examples of similar solvent effects in a review by Jaworski (1998). 

Activation entropies are useful because they can give information on the structure of a transition state (as stated above, a more confined transition state is signalled by a negative, unfavorable, activation entropy), but the ab initio calculation of rate constants [148] from activation free energies is not as straightforward as... 

---