Ivica Ruić

The heterogeneous equivalent: A method for digital simulation of electrochemical systems with compact reaction layers⋆

Summary Digital simulation of an electrochemical system with homogeneous chemical kinetics and a compact reaction layer ( i.e. the reaction layer thickness is small compared to the diffusion layer thickness) can require impractically long computation times. By invoking steady-state mathematics it is possible to deduce an equivalent system involving only heterogeneous phenomena. This “heterogeneous equivalent” circumvents the need to simulate the details of the reaction layer thereby avoiding excessive computation times.

On the theory of stepwise electrode processes

Summary The equations for d.c. polarographic waves of two-step electrode reaction are derived with the stationary plane, expanding plane model of diffusion, as well as with a steady-state approach. The equations for total current and currents due to individual steps are derived and the corresponding logarithmic diagrams are discussed. The procedures for characterization of individual steps are proposed and illustrated by the analysis of data for Cu 2+ ion reduction in perchlorate media. The results are compared with those of Jacq et al. who used the same data and the same model. The meaning of overall parameters of the stepwise electrode reaction is discussed and, using the Marcus theory of charge transfer, the relation between the overall standard heterogeneous rate constant and the corresponding rate constants of homogeneous exchange reactions is presented. The meaning of Vetters “exchange currents of individual steps” is also discussed.

On the influence of coupled homogeneous redox reactions on electrode processes in d.c. and a.c. polarography: I. Theory for two independent electrode reactions coupled with a homogeneous redox reaction

Summary A theoretical study of the d.c. and a.c. polarographic responses with the mechanism A + e ⇌ k s , 1 , α 1 B C + e ⇌ k s , 2 , α 2 D A + D ⇌ k 2 k 1 C + B is presented. It is found that the coupling of the two heterogeneous electrode processes by the homogeneous redox reaction under many conditions will lead to significant effects on the a.c. polarographic response which can be exploited to enable measurement of the rate parameters k 1 and/or k 2 of the homogeneous redox reaction. Other unique measurement possibilities also arise, such as deduction of the E 0 and k s -values for totally irreversible electrode processes. In the extreme where k 1 ≫ k 2 , the theoretical predictions confirm the conclusions of Yamaoka regarding the enhancement of an irreversible a.c. polarographic wave by a second, more easily reduced, redox couple. In the opposite extreme ( k 1 ≪ k 2 ), which corresponds to the classical catalytic mechanism, the theory provides for the first time a.c. polarographic rate laws for the case where the homogeneous process is second-order. The theoretical equations are applicable to all combinations of homogeneous and heterogeneous kinetic-thermodynamic parameters, encompassing situations involving both resolved and unresolved polarographic waves. the usual assumption of equal diffusion coefficients is relaxed and the equations developed should account for moderate differences in diffusion coefficient under most circumstances, and large differences if the homogeneous redox reaction is reasonably rapid. Computer programs developed invoke the expanding sphere electrode model. Guidelines for experimental evaluation of various rate and thermodynamic parameters are presented. It is found that some rather simple measurement procedures based on appropriate theoretical working curves will permit evaluation of the homogeneous kinetic parameters. Implications of the results for a.c. polarographic analysis of multi-component systems are considered.

Logarithmic analysis of two overlapping D.C. polarographic waves: I. Reversible and totally irreversible processes

Summary Logarithmic analysis of a composite d.c. polarographic wave, which includes two overlapping waves, gives a composite curve with one inflection point and two linear parts. The ratio of diffusion currents, real half-wave potentials and the slopes of each separate wave can be calculated using the general equations evaluated and the experimental data of inflection point, and intersection points of the extrapolated linear parts with the abscissa. The analysis should be performed first by graphical estimation of the inflection point, and then by applying several iterations. Depending on the condition fulfilled, the appropriate simpler relation is used. The possibilities of application of logarithmic analysis of two overlapping d.c. polarographic waves are discussed with respect to the ratio of diffusion currents and to the difference between the half-wave potentials.

On the influence of coupled homogeneous redox reactions on electrode processes in d.c. and a.c. polarography: II. The e.e. mechanism with a coupled homogeneous redox reaction

Summary A theoretical study of the d.c. and a.c. polarographic response with the mechanism A + e ⇌ k s , 1 , α 1 B B + e ⇌ k s , 2 , α 2 C A + C ⇌ k 2 k 1 2 B is presented. The rate law derivation encompasses the expanding sphere electrode model, systems where diffusion coefficients of the reacting species are unequal, all possible combinations of the heterogeneous charge transfer rate parameters, and most values of the homogeneous rate parameters (excluding very large k1 values). Calculations performed with a FORTRAN representation of the derived rate laws reveal that the homogeneous reaction can substantially influence the d.c. and a.c. polarographic behavior, not only in the well-recognized situation where the disproportionation step is significant, but also under certain conditions when the direction of the homogeneous step corresponds to reproportionation. A number of experimentally-significant situations are examined with emphasis on obtaining guidelines regarding when the homogeneous process is kinetically-significant and how its rate parameters might be characterized using d.c. and a.c. polarographic observables.

Fundamental harmonic a.c. polarography with disproportionation following the charge transfer step: Theory and experimental results with the U(VI)/U(V) couple

Summary A theoretical and experimental study is presented for the small amplitude a.c. polarographic response with systems involving second-order disproportionation following the charge transfer step. The theoretical derivation combines wellestablished analytical procedures for solving the a.c. boundary value problem with various nummerical methods to account for effects of d.c. polarization. The resulting rate law predicts significant influences of the disproportionation step on a.c. polarographic observables, including several novel effects not predicted or observed with first-order coupled chemical reactions. The basis for kinetic assessment of rate parameters associated with the heterogeneous charge transfer step and the homogenous disproportionation mechanism is clearly indicated by the theoretical rate law. The computer programs developed on the basis of the derivation provide the means for calculating predicted responses on the basis of most commonly-used mathematical models for the DME, including the expanding sphere model. The electrode reduction of UO 2 2+ at the mercury-aqueous 6 M HClO 4 interface is used as a model for the mechanism in question in an experimental assay of the theorys fidelity. Various predicted trends are verified experimentally. Analysis of the experimental data using the theoretical rate law generated rate parameters for the heterogeneous electrode reaction, the homogeneous disproportionation step and the diffusion process. Agreement between theory and experiment obtained with this set of rate parameters is quite satisfactory for a variety of observables. Finally, the disproportionation rate constant obtained is consistent with previous measurements by a variety of authors using both electrochemical and non-electrochemical methods.

Fundamental harmonic a.c. polarography with irreversible dimerization following the charge transfer step: Theory and experimental results with the benzaldehyd system

Summary A theoretical and experimental study is presented for the small amplitude fundamental harmonic a.c. polarographic response with systems involving dimerization following the charge transfer step. The theoretical derivation invokes the procedure recently presented for the disproportionation case, which combines numerical methods with conventional functional analysis procedures. Computer programs were developed which provide the option of calculating the a.c. polarographic response on the basis of various commonly-used models for the DME, including the expanding sphere model. The electrode reduction of benzaldehyde in alkaline aqueous ethanol is employed as a model system in an experimental evaluation of the theoretical equations. Excellent agreement is obtained between predictions of the theoretical a.c. polarographic rate law and experimental results.

A note on the form of current-potential curves

Summary The forms of the current-potential curves obtained for different models of diffusion by rigorous and approximate methods are compared. It is found that for instantaneous current the irreversible part of the logarithmic diagram (log( i/i d −i) vs. potential) obtained by rigorous methods at sufficiently positive potentials agree very well with approximate results (with corresponding diffusion layer thickness (π Dt ) 1/2 for stationary and (3π Dt/7 ) 1/2 for expanding plane model). However, at more negative potentials the slope of the logarithmic diagram is no longer equal to α nF/RT in the case of a stationary plane model of diffusion. In the case of an expanding plane at these potentials the diffusion layer thickness is no longer (3π Dt/7 ) 1/2 but equal to that of the stationary plane (π Dt ) 1/2 . Therefore the analysis of quasi-reversible polarographic waves and related curves should be performed without the factor (3/7) 1/2 (erroneously) used earlier, because the irreversible part of the logarithmic diagram at potentials near the reversible potential has no effect on the observed current (which is controlled by diffusion and not by heterogeneous charge transfer) and it is only at these low potentials that the expanding plane diffusion layer solution should be used. The analysis of mean current is also discussed.

Logarithmic analysis of two overlapping D.C. Polarographic waves: IV. Quasireversible electrode processes

Summary The application of logarithmic analysis of two quasireversible overlapping d.c. polarographic waves is discussed. In this case, only the reversible part of the first quasireversible wave and the irreversible part of the second quasireversible wave can usually be separated from the original experimental logarithmic curve. One of the quasireversible waves can be completely constructed only in the case when the other is relatively small. The logarithmic analysis can also be applied to the anodic-cathodic quasireversible wave and a method for the determination of both transfer coefficients, reversible half-wave potential, and the corresponding rate constant of the electrode reaction is proposed.

On the influence of electrode curvature and growth in d.c. and a.c. polarography: The e.e. mechanism with amalgam formation

Summary Predictions of d.c. and a.c. polarographic responses with the e.e.-mechanism obtained from various electrode models, ranging from the stationary plane to the expanding sphere, are compared. Electrode curvature effects are found to be substantial, not only in d.c. polarography where this effect is long-acknowledged, but also in a.c. polarography where the problem is usually ignored. Large curvature effects attend electrode reactions which lead to amalgam formation, the a.c. polarographic current amplitude being most susceptible. The results presented suggest that proper quantitative analysis of a.c. data obtained with a DME often will require use of rate laws based on rigorous treatment of the expanding sphere boundary value problem.