Barrier symmetry factor

In certain cases encountered experimentally, for example, for the HER at Ni or Ni-Mo alloys (75), the electrochemical barrier symmetry factor for the initial proton-discharge step [Eq. (4)] may be close to that for the electrochemical desorption step [Eq. (5)] then a limiting coverage ([Pg.42]

FIGURE 4.2.1. Effect of change of electrode potential, as VF, on energy of activation, A of an electron transfer reaction in relation to the barrier symmetry factor, fS [11]. 

In Section 1.4 it was assumed that the rate equation for the h.e.r. involved a parameter, namely the transfer coefficient a, which was taken as approximately 0-5. However, in the previous consideration of the rate of a simple one-step electron-transfer process the concept of the symmetry factor /3 was introduced, and was used in place of a, and it was assumed that the energy barrier was almost symmetrical and that /3 0-5. Since this may lead to some confusion, an attempt will be made to clarify the situation, although an adequate treatment of this complex aspect of electrode kinetics is clearly impossible in a book of this nature and the reader is recommended to study the comprehensive work by Bockris and Reddy. ... 

Thus, for a transition between any two vibrational levels of the proton, the fluctuation of the molecular surrounding provides the activation. For each such transition, the motion along the proton coordinate is of quantum (sub-barrier) character. Possible intramolecular activation of the H—O chemical bond is taken into account in the theory by means of the summation of the probabilities of transitions between all the excited vibrational states of the proton with a weighting function corresponding to the thermal distribution.3,36 Incorporation in the theory of the contribution of the excited states enabled us in particular to improve the agreement between the theory and experiment with respect to the independence of the symmetry factor of the potential in a wide region of 8[Pg.135]

The following expressions of the transfer coefficient (symmetry factor) result from the quadratic activation-driving force relationship in equation (3.3) and from the definition of the intrinsic barrier in equation (3.4) ... 

A Simple Picture of the Symmetry Factor. In order to employ simple geometry, one now ignores the curvature of the Morse curves and considers that the potential energy barrier near the intersection point is made up of straight lines (Fig. 9.11). This simplifying analogue of the barrier is useful for a first-base discussion of the symmetry factor p. 

The first approach (Section 8.2.4) at this computation ran along the following lines The electrical work of activation arises because in the activation process charges have to be moved through the difference of potential between the initial and activated states, i.e., from xl + x2 to xl in Fig. 8.17. It was necessary, therefore, to know what fraction of the total j ump distance is the distance between the initial state and the barrier peak. This distance ratio was defined as the symmetry factor P, i.e.,... 

The essential point that emerges from this first discussion of P is that only a fraction of the potential difference across the double layer, not the whole potential difference, is operative on the reaction. That there is a fraction P becomes clear what the fraction is remains a problem as long as the barrier shape is not known. This point of view must only be considered as the first murmuring of a theory of p, the symmetry factor. 

Fig. 9.32. A linear analogue to the potential energy barrier for electron tunneling shows that the ratio of tan y to tan y+tan 6 is equal to the ratio of energies AH to AH0, the ratio of the tangents being equal to the symmetry factor p. This gives the symmetry factor new physical meaning.

The statement a chemical reaction is symmetry allowed or symmetry forbidden, should not be taken literally. When a reaction is symmetry allowed, it means that it has a low activation energy. This makes it possible for the given reaction to occur, though it does not mean that it always will. There are other factors which can impose a substantial activation barrier. Such factors may be steric repulsions, difficulties in approach, and unfavorable relative energies of orbitals. Similarly, symmetry forbidden means that the reaction, as a concerted one, would have a high activation barrier. However, various factors may make the reaction still possible for example, it may happen as a stepwise reaction through intermediates. In this case, of course, it is no longer a concerted reaction. 

Note that we have derived here Eq. 6E, which was written intuitively in Section 11.1. It is also interesting that the symmetry factor p disappeared from Eq. 35E in the process of linearization. Thus, the rate of reaction near equilibrium does not depend on the detailed shape of the energy barrier for activation (which determines the value of p, as we shall see). It does, however, depend on the magnitude of the energy of activation, which manifests itself in the value of i. ... 

But the symmetry factor p has been defined strictly for a single step and is related to the shape of the free-energy barrier and to the position of the activated complex along the reaction coordinate. To describe a multi step process, p must be replaced by an experimental parameter, which we call the cathodic transfer coefficient a. Instead of Eq. 41E we then write ... 

First, the variation in the intrinsic barriers, AG, for related electrochemical reactions can be expected to be closely similar to those for the same series of homogeneous reactions using a fixed coreactant. If the comparison is made at a fixed electrode potential, E, the (often unknown) driving-force terms cancel provided that the free-energy profiles are symmetrical (the symmetry factor a. = 0.5) so that ... 

The "symmetry factor fi expresses the fraction of the contribution of electrical energy to the activation energy of the electrodic reaction. Its magnitude depends on the position of the energy barrier and varies between 0 and 1. Most often, a symmetrical energy barrier is assumed, for which p - 0.5. 

In this equation, and represent the surface concentrations of the oxidized and reduced forms of the electroactive species, respectively k° is the standard rate constant for the heterogeneous electron transfer process at the standard potential (cm/sec) and oc is the symmetry factor, a parameter characterizing the symmetry of the energy barrier that has to be surpassed during charge transfer. In Equation (1.2), E represents the applied potential and E° is the formal electrode potential, usually close to the standard electrode potential. The difference E-E° represents the overvoltage, a measure of the extra energy imparted to the electrode beyond the equilibrium potential for the reaction. Note that the Butler-Volmer equation reduces to the Nernst equation when the current is equal to zero (i.e., under equilibrium conditions) and when the reaction is very fast (i.e., when k° tends to approach oo). The latter is the condition of reversibility (Oldham and Myland, 1994 Rolison, 1995). 

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