The Symmetry Factor

2) have been redrawn in the inset box and aligned to illustrate the fraction P of the applied overpotential that ends up modifying the height of the cathodic energy barrier. On the basis of the necessary simplifying assumptions that their slopes do not change with ti and that 

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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. ... 

The final expression is the classical limit, valid above a certain critical temperature, which, however, in practical cases is low (i.e. 85 K for H2, 3 K for CO). For a homonuclear or a symmetric linear molecule, the factor a equals 2, while for a het-eronuclear molecule cr=l (Tab. 3.1). This symmetry factor stems from the indistinguishable permutations the molecule may undergo due to the rotation and actually also involves the nuclear partition function. The symmetry factor can be estimated directly from the symmetry of the molecule. 

Table 3.1. The symmetry factor for different symmet groups and examples of molecules belonging to them.

As follows from the Bronsted relationship, the symmetry factors for the cathode and anode processes are related to each other [see Eq. (6.17a)] ... 

A simple quadratic form of Eq. (34.10) is due to an identical parabolic form of the free-energy surfaces f/, and U. Since the dependence of the activation free energy on AF is nonhnear, the symmetry factor a may be introduced by a differential relationship,... 

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]

Further development of the basic model and the detailed analysis of the dependence of the symmetry factor on the potential and the temperature54 have shown that there are additional factors which can affect the elementary act of this reaction. These investigations led to the formulation of the charge variation model (CVM)55 which will be discussed in the next section. 

Figure 8. Dependence of the symmetry factor a on the free energy of the transition for the reaction of hydrogen ion discharge on a metal electrode.

Variation of the symmetry factor with the driving force 126... 

VARIATION OF THE SYMMETRY FACTOR WITH THE DRIVING FORCE... 

The transfer coefficient determines the symmetry - or lack thereof - of the current-potential curves they are symmetric for a = 1/2. For this reason the transfer coefficient is also known as the symmetry factor. 

Values of the Symmetry Factor and Variation with the Driving Force... 

The variations of the symmetry factor, a, with the driving force are much more difficult to detect in log k vs. driving force plots derived from homogeneous experiments than in electrochemical experiments. The reason is less precision on the rate and driving force data, mostly because the self-exchange rate constant of the donor couple may vary from one donor to the other. It nevertheless proved possible with the reaction shown in Scheme 3.3.11... 

FIGURE 3.12. Potential energy profiles for the concerted and stepwise mechanism in the case of a thermal reductive process (E is the electrode potential for an electrochemical reaction and the standard potential of the electron donor for a homogeneous reaction) and variation of the rate constant and the symmetry factor when passing from the concerted to the stepwise mechanism. 

Here kf and kb are the adsorption and desorption constants when 9 —> 0. The derivation of the equation above is similar to establishment of the Butler-Volmer kinetic law for electrochemical electron transfer reactions, where the symmetry factor, a, is regarded as independent from the electrode potential. Similarly, in the present case, the symmetry factor, a, is assumed to be independent of the coverage, 9. 

The rate constants may be expressed as functions of the self-exchange rate constant, k0, and the potential difference, linearizing the activation-driving force law and taking a value of 0.5 for the symmetry factor. Thus,... 

Figure 8-7 shows the anodic and cathodic polarization curves observed for a redox couple of hydrated titanium ions Ti /Ti on an electrode of mercury in a sulfuric add solution the Tafel relationship is evident in both anodic and cathodic reactions. FYom the slope of the Tafel plot, we obtain the symmetry factor P nearly equal to 0.5 (p 0.5). 

Another, and even more striking aspect of the variation of the symmetry factor with the driving force is the prediction, from (9) and (10), that an inverted region should exist at large driving forces, i.e. when the inequality (40) applies. The activation free energy is then predicted to increase with the... 

F is the partition function of the gas. (The symmetry factors that appear in the partition functions and the other parts of the calculation usually cancel and in any case do not contribute a large factor, and so they are not included in this article.) fu and both approximate unity. For the gas... 

Thus, whenever the ligand is very small compared with the adsorbent molecule, one may neglect its effect on and except for the symmetry factor... 

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). 

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