Particles quantum-mechanical tunneling

We have seen that 10" M s is about the fastest second-order rate constant that we might expect to measure this corresponds to a lifetime of about 10 " s at unit reactant concentration. Yet there is evidence, discussed by Grunwald, that certain proton transfers have lifetimes of the order 10 s. These ultrafast reactions are believed to take place via quantum mechanical tunneling through the energy barrier. This phenomenon will only be significant for very small particles, such as protons and electrons. 

Figure 3.5 Graphical representation of the quantum mechanical tunnelling effect between tip and sample. The probability P of a particle with kinetic energy E tunnelling through a potential barrier cf> is shown as a function of sample-tip separation z.

In Chapter 7 general kinetics of electrode reactions is presented with kinetic parameters such as stoichiometric number, reaction order, and activation energy. In most cases the affinity of reactions is distributed in multiple steps rather than in a single particular rate step. Chapter 8 discusses the kinetics of electron transfer reactions across the electrode interfaces. Electron transfer proceeds through a quantum mechanical tunneling from an occupied electron level to a vacant electron level. Complexation and adsorption of redox particles influence the rate of electron transfer by shifting the electron level of redox particles. Chapter 9 discusses the kinetics of ion transfer reactions which are based upon activation processes of Boltzmann particles. 

Quantum mechanical tunneling. Tunneling is the phenomenon by which a particle transfers through a reaction barrier due to its wave-like property.Figure 1 graphically illustrates this for a carbon-hydrogen-carbon double-well system Hydrogen... 

The reason that a compound ion can be field dissociated can be easily understood from a potential energy diagram as shown in Fig. 2.23. When r is in the same direction as F, the potential energy curve with respect to the center of mass, V(rn) is reduced by the field. Thus the potential barrier width is now finite, and the vibrating particles can dissociate from one another by quantum mechanical tunneling effect. Rigorously speaking, it... 

Quantum mechanical tunneling is a fertile idea with many applications in chemistry. It will be seen that in practice calculations using its concepts make a significant difference with very light particles, e.g., electrons and (to a much lesser degree) protons. But these are exactly the particles that hold center stage in interfacial electrochemistiy. Indeed, without quantum mechanical tunneling, no electric currents across interfaces could occur.11... 

A key point that must be made is diat quantum mechanical tunneling through the Marcus-theory barrier when it is non-zero can increase the rate for electron transfer just as is true for any other activated process. Because the electron is so light a particle, tunneling can be a major contributor to die overall rate. Models for electron tunneling will not, however, be presented here. 

Returning to the one-dimensional box of constant width, if the potential does not increase suddenly to infinity at one of the walls then the wave-function does not vanish there. Due to the continuity requirement of the wavefunction, it decreases exponentially to zero inside the wall of finite height. Therefore, there is a non-zero probability that the particle will penetrate the wall, although its kinetic energy is lower than the potential barrier (Fig. 2.6). This effect is called quantum-mechanical tunnelling. 

In electrochemical proton transfer, such as may occur as a primary step in the hydrogen evolution reaction (h.e.r.) or as a secondary, followup step in organic electrode reactions or O2 reduction, the possibility exists that nonclassical transfer of the H particle may occur by quantum-mechanical tunneling. In homogeneous proton transfer reactions, the consequences of this possibility were investigated quantitatively by Bernal and Fowler and Bell, while Bawn and Ogden examined the H/D kinetic isotope effect that would arise, albeit on the basis of a primitive model, in electrochemical proton discharge and transfer in the h.e.r. 

Unfortunately, the quantum-mechanical tunneling approach does not lead to any better understanding of why a should be temperature dependent in many reactions, if only for the reason that in most of the cases where such behavior is observed, the reactions involve heavy particles for which quantum effects are negligible. O2 reduction, however, could be quantally controlled if... 

Quantum mechanical tunneling is a result of the wavelike nature of particles which allows transmission through a reaction barrier. The quantum mechanical transmission probability for energies below is governed by tunneling and reflection at the barrier. The transmission is larger than zero even well below the barrier and will depend crucially on the barrier width. In... 

The ability of a particle to penetrate into a classically forbidden region is the basis of the quantum mechanical tunnel effect. Consider a particle for which the potential function looks like that in Fig. 21.6. Two regions of low potential energy are separated by a... 

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