Introduction. Basic Equations. Approximations to Derivatives. Ordinary Differential Equations. The Explicit Method. Boundary Conditions. Unequal Intervals. The Commonly Used Implicit Methods. Other Methods. Adsorption. Effects Due to Uncompensated Resistance and Capacitance. Two-Dimensional Systems. Convection. Performance. Programming. Simulation Packages. Some Mathematical Proofs. Useful Procedures. Example Programs.

Digital Simulation in Electrochemistry

Rotating disk electrodes

The Physical Environment in Soil Microbiology: An Attempt to Extend Principles of Microbiology to Soil Microorganisms

A variety of generator-collector systems are reviewed, from the original rotating ring-disc electrodes developed in the 1950s, to very recent developments using new geometries and microelectrodes. An overview of both theoretical and experimental aspects are given, and the power of these double electrode systems in analytical electrochemistry is illustrated with a range of applications.

Generator-collector double electrode systems: A review

Despite decades of active attention, important problems remain pending in the catalysis of dioxygen reduction by iron porphyrins in water in terms of selectivity and mechanisms. This is what happens, for example, for the distinction between heterogeneous and homogeneous catalysis for soluble porphyrins, for the estimation of H2O2/H2O product selectivity, and for the determination of the reaction mechanism in the two situations. With water-soluble iron tetrakis(N-methyl-4-pyridyl)porphyrin as an example, procedures are described that allow one to operate this distinction and determine the H2O2/H2O product ratio in each case separately. It is noteworthy that, despite the weak adsorption of the iron(II) porphyrin on the glassy carbon electrode, the contribution of the adsorbed complex to catalysis rivals that of its solution counterpart. Depending on the electrode potential, two successive catalytic pathways have been identified and characterized in terms of current-potential responses and H2O2/H2O selectivity. These observations are interpreted in the framework of the commonly accepted mechanism for catalytic reduction of dioxygen by iron porphyrins, after checking its compatibility with a change of oxygen concentration and pH. The difference in intrinsic catalytic reactivity between the catalyst in the adsorbed state and in solution is also discussed. The role of heterogeneous catalysis with iron tetrakis(N-methyl-4-pyridyl)porphyrin has been overlooked in previous studies because of its water solubility. The main objective of the present contribution is therefore to call attention, by means of this emblematic example, to such possibilities to reach a correct identification of the catalyst, its performances, and reaction mechanism. This is a question of general interest, so that reduction of dioxygen remains a topic of high importance in the context of contemporary energy challenges.

Molecular Catalysis of O2 Reduction by Iron Porphyrins in Water: Heterogeneous versus Homogeneous Pathways.

If an intermediate generated on the disc electrode decomposes homogeneously in the solution, then the collection efficiency at the ring electrode will be reduced. This paper examines the effects of such a decomposition which takes place by first-order kinetics. For thin-ring, thin-gap electrodes, two limiting solutions are obtained; one when the collection efficiency is very small corresponding to most of the generated material being lost by reaction, and the other when the collection efficiency is close to N, the kinetic loss being small. It is also demonstrated that for any particular electrode the collection efficiency is a function only of (kν⅓D–⅓ω–1).

Ring-disc electrodes. Part 5.—First-order kinetic collection effciencies at the ring electrode

The ring-disc electrode has been used as the basis of a sensitive analytical technique.1 Reagent is generated at the disc electrode and the ring electrode is used to determine any excess which has not been consumed. The ring-current against disc-current curve is calculated by the complete solution of the convective diffusion equation using the proper boundary conditions. Experimental results for the titration of As(III) with electrogenerated Br2 are found to be in complete agreement with theory.

Ring-disc electrodes. Part 4.—Diffusion layer titration curves

The size of the ring current observed when a reagent generated on the disc electrode reacts rapidly with a reagent of comparable concentration in the bulk of the solution, has been calculated and shown to be in good agreement with experiment.2 Depending on the second-order rate constant for the reaction, more or less of the disc reagent will penetrate through the bulk reagent to the ring electrode. An approximate theory is developed, using the method of moments and a reaction layer treatment, which relates the size of the ring current to the second-order rate constant. The method seems capable of measuring a wide range of homogeneous second-order rate constants with an upper limit of 108 l. M–1 sec–1. It is also demonstrated for any particular electrode and for any particular placing of the reaction zone that the kinetic collection efficiency is only a function of (ka∞ω–1D⅓ν–⅓).

Ring disc electrodes. Part 6.—Second-order reactions

In this paper we calculate the value of N for the system where the intermediate formed on the disc electrode is reversibly but almost wholly converted into a second intermediate in a homogeneous reaction; this second intermediate returning through the first intermediate is destroyed at the ring electrode We also calculate the effect of homogeneous kinetics and of slow electrochemical kinetics at a simple ring electrode and the effect of slow electrochemical kinetics for the ring-disc electrode The results are discussed and a table summarizing the chemical and electrochemical systems dealt with in this series of papers is given

Ring-disc electrodes. Part 7.—Homogeneous and heterogeneous kinetics

In this paper we calculate the value of N for the system where the intermediate formed on the disc electrode is reversibly but almost wholly converted into a second intermediate in a homogeneous reaction; this second intermediate returning through the first intermediate is destroyed at the ring electrode. We also calculate the effect of homogeneous kinetics and of slow electrochemical kinetics at a simple ring electrode and the effect of slow electrochemical kinetics for the ring-disc electrode. The results are discussed and a table summarizing the chemical and electrochemical systems dealt with in this series of papers is given.

W. J. Albery

Papers

Ring-disc electrodes. Part 5.—First-order kinetic collection effciencies at the ring electrode

Ring-disc electrodes. Part 4.—Diffusion layer titration curves

Ring disc electrodes. Part 6.—Second-order reactions

Ring-disc electrodes. Part 7.—Homogeneous and heterogeneous kinetics

Ring-disc electrodes: Part 7. - Homogeneous and heterogeneous kinetics