Potential dependence, activation energy

Hydrogen under electrochemical conditions was investigated very recently [222, 223]. Santana et al. investigated the electro-oxidation of molecular hydrogen at the Pt(110)-water interface [222]. The Tafel-Volmer mechanism with a homolytic H-H bond cleavage followed by the formation of adsorbed terminal hydrogen atoms and further oxidation of the H atoms was observed by the authors. Furthermore, Santana et al. found the potential dependent activation energies for this process to be in accordance with experimental results. 

The selection of functional directly controls the accuracy of the DFT methods. A study of ORR by hybrid density functional (HDFT) method and MP2 method has been reported by Albu et al. [29, 30]. They evaluated a large number of HDFT methods toward calculation of potential-dependent activation energies for uncatalyzed and Pt-catalyzed oxygen reduction and hydroperoxyl oxidation reactions. In the HDFT methods, the one-parameter hybrid Fock-Kohn-Sham operator is written as... 

Anderson AB (2003) Theory at the electrochemical interface reversible potentials and potential-dependent activation energies. Electrochim Acta 48 3743-3749... 

Anderson and Albu reported potential dependent activation energies for the following four elementary steps ... 

Cai, Y. and Anderson, A.B. The reversible hydrogen electrode Potential-dependent activation energies over platinum from quantum theory. Journal of Physical Chemistry B, 108,9829 9833, 2004. 

However, the situation becomes already more complicated for ternary single crystals like lanthanum-aluminate (LaAlC>3, er = 23.4). The temperature dependence of the loss tangent depicted in Figure 5.3 exhibits a pronounced peak at about 70 K, which cannot be explained by phonon absorption. Typically, such peaks, which have also been observed at lower frequencies for quartz, can be explained by defect dipole relaxation. The most important relaxation processes with relevance for microwave absorption are local motion of ions on interstitial lattice positions giving rise to double well potentials with activation energies in the 50 to 100 meV range and color-center dipole relaxation with activation energies of about 5 meV. 

Because of the functional dependence of electrocatalytic rates on potential and activation energy, Eqs. (10) and (16)—(18), the temperature effect on selectivity will be different from that of conventional processes. In the... 

Thus, the biophysical studies demonstrate that globular proteins have (1) a very large number of conformational states corresponding to many shallow local minima in the potential energy function, (2) very broad continuous distributions of activation energies, and (3) time-dependent activation energy barriers. All these properties are consistent with the physical properties of ion channels derived from the fractal properties observed in the channel data and are inconsistent with the physical properties derived from the Markov model. 

We saw that formal kinetic equations apart from kinetic parameters also contain surface concentrations Cj of electrically active species. It follows from the material presented in previous chapters that differs from the corresponding bulk values because a diffusion layer with certain concentration profiles forms at the electrode surface. Moreover, another reason due to which surface concentrations change is adsorption phenomena, which form a certain structure called a double electrode layer (DEL) at the boundary metal solution. It is clear that in kinetic equations, it is necessary to use local concentrations of reactants and products, that is, concentrations in that region of DEL where electrically active particles are located. The second effect produced by DEL is related to the fact that a potential in the localization of the electrically active complex (EAC) differs from the electrode potential. Therefore, activation energy of the electrochemical process does not depend on the entire jump of the potential at the boundary but on its part only, which characterizes the change in the potential in the reaction zone. In this connection, the so-called Frumkin correction appears in the electrochemical kinetic equations, which is related to the evaluation of the local potential i// [1]. 

Figure 3.5 schematically lists the various approaches taken in the literature to applying DFT methods to electrocatalytic reactions. These methods are first differentiated by cluster and periodic representation of the electrode surface. The reaction center model (Model 1 in Figure 3.5), developed by Anderson and coworkers,is an early attempt to evaluate potential dependent reaction energies and activation barriers. It relies on using a small cluster to represent the reaction center of the electrode and evaluates the electron alfinity of such cluster. We will not detail this method, because its fidelity is questionable given its arbitrarily small representation of the electrode when considering its electronic structure and lack of scalability to a more accurate electrode representation. 

Voltammetry is an important tool for evaluating electrochemical and electro-catalytic processes. In a voltammetric experiment, the potential of a working electrode is varied with time relative to a reference electrode. The current of the working electrode is measured and reported as a function of potential. If the potential is swept linearly with time, peaks or waves are observed, which can be attributed to the various electrochemical processes possible in the system. For comparison with experiment, DFT calculated energetics can be used to predict voltammetry results in much the same way microkinetic models are used to predict catalytic kinetics. In the sections above, we have discussed DFT methods to calculate elementary reaction or adsorption free energies as a function of electrode potential. These free energy differences can be used to calculate potential dependent equilibrium constants. Section 3.2.5 will present a method to calculate potential dependent activation barriers. With these values for all possible elementary reaction steps, we could use microkinetic modeling to simulate voltammetry and compare with experiment. 

Such diagrams make clear the difference between an intermediate and a transition state. An intermediate lies in a depression on the potential energy curve. Thus, it will have a finite lifetime. The actual lifetime will depend on the depth of the depression. A shallow depression implies a low activation energy for the subsequent step, and therefore a short lifetime. The deeper the depression, the longer is the lifetime of the intermediate. The situation at a transition state is quite different. It has only fleeting existence and represents an energy maximum on the reaction path. 

In molecular doped polymers the variance of the disorder potential that follows from a plot of In p versus T 2 is typically 0.1 eV, comprising contributions from the interaction of a charge carrier with induced as well as with permanent dipoles [64-66]. In molecules that suffer a major structural relaxation after removal or addition of an electron, the polaron contribution to the activation energy has to be taken into account in addition to the (temperature-dependent) disorder effect. In the weak-field limit it gives rise to an extra Boltzmann factor in the expression for p(T). More generally, Marcus-type rates may have to be invoked for the elementary jump process [67]. 

Figure 18 shows the dependence of the activation barrier for film nucleation on the electrode potential. The activation barrier, which at the equilibrium film-formation potential E, depends only on the surface tension and electric field, is seen to decrease with increasing anodic potential, and an overpotential of a few tenths of a volt is required for the activation energy to decrease to the order of kBT. However, for some metals such as iron,30,31 in the passivation process metal dissolution takes place simultaneously with film formation, and kinetic factors such as the rate of metal dissolution and the accumulation of ions in the diffusion layer of the electrolyte on the metal surface have to be taken into account, requiring a more refined treatment. 

Dependence of Catalytic Rates and Activation Energies on Catalyst Potential UWRand Work Function [Pg.152]

If one assumes the potential energy curves to have a similar parabolic dependence on the displacement of the atoms, a simple relation can be deduced between activation energy, the crossing point energy of the two curves, and the reaction energy. One then finds for a ... 

This is an example of a reversible reaction the standard electrode potential of the 2PS/PSSP + 2c couple is zero at pH 7. The oxidation kinetics are simple second-order and the presence of a radical intermediate (presumably PS-) was detected. Reaction occurs in the pH range 5 to 13 with a maximum rate at pH 6.2, and the activation energy above 22 °C is zero. The ionic strength dependence of 2 afforded a value for z Zg of 9 from the Bronsted relation... 

The interaction of hydrogen (deuterium) molecules with a transition metal surface c an be conveniently described in terms of a Lennard--Jones potential energy diagram (Pig. 1). It cxxislsts of a shallcw molecular precursor well followed by a deep atomic chemisorption potential. Depending on their relative depths and positions the wells m or may not be separated by an activation energy barrier E as schematically Indicated by the dotted cur e in Fig. 1. 

Figure 5 summarises results for the CO2, N2 and N2O formation rates for the dependence of apparent activation energies on catalyst potential. Although there is a notable increase in activation energy with increased Na coverage in each case, the variation is not as abrupt at that characteristic of EP CO oxidation [24] and NO+CO reactions. 

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