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A self consistent procedure which uses the limiting currents of a titration experiment for the rigorous computation of the binding curve of the labile complexes formed with a metal and a heterogeneous ligand is presented. The homogeneous complexation model corresponding to the mean stability constant of all the affinity spectra is used as a first trial complexation model to compute the fractions of bound or free metal in bulk conditions along the titration. Afterwards, the binding curve so obtained is used in an iterative procedure to improve the computation of the fraction of free metal in bulk conditions at each metal-to-ligand ratio up to convergence. The resulting numerical process constitutes a rigorous way to obtain the binding curve. Its application to some cases with discrete or continuous affinity spectra is presented.

This paper discusses the determination of the binding curve from limiting currents. Recently, an analytical approximation for the limiting current in terms of the bulk complex concentration has been proposed. This expression is so simple that it can be easily inverted allowing the computation of the binding curve from normalised limiting currents corresponding to the titration of a fixed amount of ligand with metal. Different complexation models with discrete or continuous affinity spectra are analysed. An affinity window associated with a given titration is recognised and it is shown that an accurate binding curve can be obtained by combining the meaningful affinity windows of several titration experiments with different amounts of ligand. The global aim of the paper is to enhance the possibilities of binding studies from voltammetric limiting currents providing some criteria to succeed in this analysis.

We present approximate expressions and rigorous numerical simulations corresponding to the voltammetric limiting currents of a titration experiment of a heterogeneous macromolecular ligand with a metal ion under fully labile conditions. We highlight the accuracy of an approximate expression based on a polynomial expansion of the limiting current in terms of the fraction of the bound metal concentration in the bulk solution. Values of the coefficients of the polynomial expansion for macromolecular complexation are provided. The accuracy of this polynomial expression for systems exhibiting different degrees of heterogeneity together with its easy computation, suggests its application in the finding of the bulk value of the bound metal concentration from the limiting current for a given pair of total ligand and metal concentrations. Thus, straightforward calculations lead to the binding curve which can then be interpreted using the standard methods.

The impact of adsorption effects on limiting currents obtained in reverse pulse (RPP) and normal pulse polarography (NPP) is analysed critically for induced reactant adsorption systems with any ligand to metal ratio. Qualitative explanations in terms of the concentration profiles are provided. We demonstrate that, for labile systems, no influence on the RPP limiting current can be found from the complex adsorption. However, ligand adsorption increases the RPP limiting current, this influence decreasing as the total metal concentration decreases. So, RPP normalised limiting current values ( φ ) are suggested to improve the fitting of the stability constant at very low total metal concentrations. If the total ligand concentration is chosen for maximum sensitivity, the remaining effect of the ligand adsorption leads to a bias in the stability constant of less than 14%, provided that the diffusion coefficient of the complex and ligand is 20-fold lower than the diffusion coefficient of the metal ion. Because the limiting NPP and RPP currents are equal for any ligand to metal ratio and without adsorption, approximate expressions derived for NPP facilitate the determination of the stability constant from the φ versus total metal concentration plot. This procedure is applied to some experimental results of Zn and Cd complexed with poly(methacrylic) acid.

An alternative rigorous expression for the limiting current in reverse pulse polarography (RPP) under semi-infinite diffusion of the amalgamated product inside the spherical electrode is presented, together with comparison of previously reported expressions. For typical parameters (radius 0.02 cm, drop time 1 s, pulse time 50 ms) the neglecting of the sphericity can introduce errors of around 27% in the limiting current value and the error increases with decreasing drop radius and increasing drop time. Some approximations (from literature and newly found) lead to very simple and accurate expressions which are then compared. RPP is shown to be particularly suitable for determining the stability constant in metal–macromolecule complexation within the excess ligand framework, as complex adsorption does not affect the limiting current. Inclusion of sphericity on the normalised limiting current is straightforward and an error of around 8% in the planar normalised current is found for typical parameters in macromolecular systems. Two simple approximate expressions to account for sphericity yield sufficiently accurate estimates (under 1%) of the normalised current for these typical cases. A spherically corrected normalised current, which allows the fitting of the stability constant with the same procedure as usually carried out assuming planar diffusion, is presented.

Systems containing metal ions which bind to macromolecular ligands usually contain a large number of chemical species with different numbers of metal ions bound to each macromolecule. Nevertheless, the formulation of macromolecular complexation in the voltammetric studies of such systems usually starts by considering the presence of only three formal species M, ML and L. The conditions under which the mathematical formulation of complexation, transport and adsorption processes for the real species reduces to the formulation of the same processes in a system with only the formal species are analyzed.

The interpretation of the normal-pulse polarographic (NPP) and the reverse-pulse polarographic (RPP) reduction of a metal ion with induced reactant adsorption in a labile metal-macromolecular system for any ligand-to-metal ratio is discussed. The results obtained from the theory here developed are compared with those obtained assuming ligand excess. The Galerkin Finite Element Method has been used in the numerical resolution. The RPP results show that the limiting current is independent of the adsorption phenomena, which allows the determination of the stability constant from a set of current data obtained at different ligand concentrations. For the case of non-excess ligand, a suitable procedure is reported as well as the comparison with the results obtained assuming ligand excess. For any ligand-to-metal ratio there is an influence of the total metal concentration on the plots of the normalized limiting current in relation to the total ligand concentration; this influence increases as the stability constant increases.

The validity of the hypothesis of ligand excess is discussed for the voltammetric reduction of a metal ion (M) in the presence of a ligand (L). The following basic assumptions are made: (i) formation of the electroinactive ML complex, (ii) equal mobility of species M, ML and L, (iii) reversible charge transfer, (iv) labile complex and (v) Langmuirian adsorption of the ligand and the complex. The model proposed is solved rigorously for normal pulse polarography (NPP) in the four possible cases assuming either ligand excess or ligand deficiency or either adsorption or non-adsorption. For the case without adsorption, the assumption of ligand excess affects only the increasing part of the NPP wave independent of the total ligand-to-metal ratio. Then ( I NNP ) lim has the same value for both ligand excess and ligand deficiency while E 1 2 differs, thus preventing the use of the DeFord—Hume method to obtain the stability constant. An analytical expression for E 1 2 is performed, which allows the evaluation of the stability constant at any total ligand-to-metal ratio and provides a quantitative estimation of the error made by applying the classical procedure assuming ligand excess. In the presence of adsorption, the assumption of ligand excess modifies the whole wave. The discrepancy between the currents obtained from the hypothesis of ligand excess with respect to that with ligand deficiency is even higher for the case with adsorption than for the case without adsorption.

The theoretical models proposed for the study of electrode processes under diffusion control on rough surfaces are tested using quasi-two-dimensional experimental electrodes obtained by zinc electrodeposition. Exponents of a generalized scaling law similar to the classical Cottrell equation, which depend on the fractal dimension of the mass distribution measure of the electrode, are calculated using Monte Carlo simulation. The validity of the algorithms of the simulation has been tested previously on flat electrodes.

A model for the interpretation of the normal pulse polarographic reduction of a metal ion in a metal polyelectrolyte system, including both the polyelectrolyte adsorption and the induced adsorption of the metal ion, is developed. The influence of the physical parameters included in the model on the maximum and limiting currents has received special attention. This knowledge is of great importance for the development of a physical picture of the behaviour of the system. The following restrictions have been assumed: (i) large excess of the macromolecular ligand; (ii) labile complex; (iii) reversible charge transfer; (iv) Langmuirian adsorption of both the complex and the ligand species; and (v) the adsorption coefficients of the ligand and the complex are potential-independent.