Operator transfer, potential exchange-electron

Finally, the potential exchange-electron transfer operator can be expressed as follows... 

The equations to be fulfilled by momentum space orbitals contain convolution integrals which give rise to momentum orbitals ( )(p-q) shifted in momentum space. The so-called form factor F and the interaction terms Wij defined in terms of current momentum coordinates are the momentum space counterparts of the core potentials and Coulomb and/or exchange operators in position space. The nuclear field potential transfers a momentum to electron i, while the interelectronic interaction produces a momentum transfer between each pair of electrons in turn. Nevertheless, the total momentum of the whole molecule remains invariant thanks to the contribution of the nuclear momenta [7]. 

The electrochemical potential of a redox mediator (EJJ,) must be such that it provides a thermodynamic driving force to facilitate electron transfer with the enzyme to be used. Because of this requirement, the redox potential of the mediator determines the operational potential of the biocatalytic electrode. Thus for an enzymatic oxidation reaction, must be higher than the redox potential of the enzyme ( en) whereas the reverse is true for an enzymatic reduction reaction. The difference between and is defined as the mediator-induced overpotential (A et) and is the potential required for electron transfer to occur between the enzyme and mediator. In the context of a biosensor, a large overpotential can lead to an artificially inflated signal due the oxidation of biological interferants such as ascorbate. Additionally a large overpotential limits the open circuit potential in the context of a biofuel cell. It is therefore desirable to minimize the electrochemical overpotential however, there exists a limit to the minimum overpotential required to facilitate rapid electron exchange between the enzyme and mediator. 

In many MFCs, the proton transfer efficiency from the anode to the cathode is the rate-limiting step and a major cause of internal resistance. Although equivalent amounts of protons and electrons are produced at the anode in MFCs, the migration rate of protons to the cathode is much slower than that of the electrons. It arises from the fact that the migration of electrons is forced by the potential difference between the two electrodes, while the migration of protons is caused by diffusion. A proton exchange membrane (PEM), if present, functions as a proton transfer barrier, and further decreases the proton diffusion rate. Since proton transport inside the fuel cell is slower than its production rate in the anode and its consumption rate in the cathode, a pH difference between the two electrodes occurs without buffer. For example, in the absence of any buffer solution, Gil et al. [76] detected a pH difference of 4.1 (9.5 at cathode and 5.4 at anode) after a 5-h operation with an initial pH of 7 in both chambers. Accumulation of protons at the anode will suppress the microbial activity, thus the electricity production, whereas a limited proton concentration at the cathode may reduce the cathodic reduction rate. 

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