José Salvador

Reverse pulse polarography of labile metal + macromolecule systems with induced reactant adsorption: theoretical analysis and determination of complexation and adsorption parameters

A model for the interpretation of reverse pulse polarography (RPP) in metal macromolecular ligand systems is developed, including both adsorption of the macromolecule and the induced adsorption of the metal ion. The following basic assumptions are made: reversible charge transfer at a stationary planar electrode (static mercury drop electrode), labile complex, large excess of ligand compared with the total metal concentration, formation of a 1:1 complex and diffusion coefficients for ligand and complex species different from that of the free metal ion. Equations for the limiting current Ilim and for the RPP full wave are deduced for both linear and Langmuirian adsorption. It is found that adsorption does not modify the expressions for Ilim valid for the case without adsorption. Thus the stability constant K can be determined. In contrast, adsorption influences the shape and position of the RPP full wave. In the case of linear adsorption, an analytical expression has been deduced which relates the adsorption parameter KML1 to ΔE12 values, thus allowing the calculation of KML1 from RPP waves. This method has been applied to experimental results obtained for the Cd(II) polymethacrylate system.

Lability and mobility effects on mixtures of ligands under steady-state conditions

Analytical solutions for the steady-state flux arriving at an active surface from a mixture (in which one active species reacts with non-active ligands in the medium) can be helpful in a variety of problems: voltammetric techniques, heterogeneous processes in reactors, toxic or nutrient uptake, techniques of diffusive gradients in thin films (DGT), etc. Under any geometry that sustains steady-state, a convenient combination of the reaction–diffusion equations leads to a simpler formulation of the problem for arbitrary diffusivities of the species and arbitrary rate constants of the first-order conversion between the active species and the non-active species. The resulting problem can be characterised in terms of a list of dimensionless parameters involving the kinetic and mobility properties of each species. A lability degree for each 1∶1 complex in terms of the surface concentrations leads to: (i) a lability criterion specific for each complex in the mixture and (ii) the assessment of the relative contribution of each complex to the resulting flux. Semi-infinite spherical diffusion (as in the Gel Integrated MicroElectrode, GIME, biouptake modelling of micro-organisms, etc.) is specifically considered and some consequences of its full analytical solutions are discussed.

Voltammetry of labile metal-macromolecular systems for any ligand-to-metal ratio, including adsorption phenomena. The role of the stability constant

Abstract 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.

Amalgamation effects in reverse pulse polarography at spherical electrodes. Influence on speciation measurements

Abstract 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.

Voltammetry of labile metal—complex systems with induced reactant adsorption. Theoretical analysis for any ligand-to-metal ratio

Abstract 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.

Use of activity coefficients for bound and free sites to describe metal–macromolecule complexation

A thermodynamic formalism to describe the small molecule–macromolecule complexation equilibrium, based on the concept of free and occupied sites (formal species) is presented. The formalism is particularly useful in systems with a large number of species and allows for the inclusion of either transport phenomena or adsorption on the boundary of the system. The homogeneous and independent complexation behaviour is denoted ideal complexation. All other behaviours of complexation are treated as deviations from such an ideal system by means of their activity coefficients, this allows the definition of a thermodynamical equilibrium constant, K, for any complexation process, written in terms of formal species. Irrespective of the model of complexation considered in the system, the concentration equilibrium relationship for formal species tends to K when the concentration of the small molecule tends to vanish (limit of ligand excess). The experimental recording of activity coefficients is straightforward and there is no need for numerical derivatives of experimental data. The analysis of the activity coefficients vs. the free metal concentration plots allows an easy and general characterization of the complexation process. Two particular cases of non-ideal complexation (interactions between bound sites and the presence of chelates) are selected to illustrate the general characteristics of the activity coefficients, and to relate them to the affinity spectrum. Expressions for the first two moments of the affinity distribution in terms of the characteristics of the activity coefficients are given.

Interpretation of speciation, measurements on labile metal-macromolecular systems by voltammetric techniques

Phenomena such as non-ligand excess, adsorption and heterogeneous or non-independent complexation models are discussed in order to improve the interpretation of the plots of the normalized limiting currents (Φ)versus the total ligand concentration in speciation studies. It is shown that under ligand excess conditions, all complexation models behave like the homogeneous and independent model. This result enhances the reliability of stability constants obtained under excess conditions using an equilibrium relationship between formal species. The convenience of Φversus total metal concentration plots in order to detect the mentioned phenomena and to obtain quantitative information about the complexation and adsorption processes from the limiting Φ value at low metal concentrations is emphasized. Finally, it is shown that non-ligand excess conditions with their corresponding interpretative framework are required for heterogeneity studies.

Influence of the adsorption phenomena on the NPP and RPP limiting currents for labile metal-macromolecule systems

Abstract 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.

Complexation isotherms in metal speciation studies at trace concentration levels. Voltammetric techniques in environmental samples

Is any complexation isotherm valid to analyse macromolecular binding of trace metal ions in environmental samples? To answer this question, a detailed study of the behaviour of some usual complexation isotherms with different underlying affinity spectra at intermediate and low coverages is performed. Metal-complexation is characterised through the behaviour of the average equilibrium function, defined as Kc ≡ [occupied sites]/([free metal][free sites]), which can be interpreted, at any metal concentration, as the average of the microscopic stability constants of the remaining free sites of the system. It is shown that, at intermediate coverages, the value of Kc depends basically on the average, μ, and standard deviation, σ, of the distribution function p(log k), the binding curve showing low sensitivity with respect to the concrete shape of the affinity spectrum. As a result, isotherms with the same values of μ and σ describe binding data of H+ to humic acid (HA) with similar accuracy. In contrast, at low coverages, Kc tends to the average of the microscopic stability constants of all the sites in the naked macromolecule, 〈k〉, which, for a given μ and σ, strongly depends on the shape of the isotherm. This result imposes an important constraint on the isotherms used and warns against extrapolation since isotherms with very similar behaviours at intermediate coverages, can predict very different ones for low coverages. This different behaviour is clearly evidenced by simulated voltammetric experiments, suggesting their suitability for speciation studies of trace metals in environmental samples. Normalised limiting currents of Cd/HA are used to validate isotherms suitable to describe both low and intermediate coverage.

Voltammetric currents for any ligand-to-metal concentration ratio in fully labile metal-macromolecular complexation. Easy computations, analytical properties of the currents and a graphical method to estimate the stability constant

In order to enable a wider use of voltammetric methods in speciation analysis, it is convenient not to be restricted by ligand excess conditions. This work assumes labile ideal complexation of a metal ion by a ligand, planar electrode, no electrodic adsorption and equal diffusion coefficients for the complex and the ligand, but very different from the metal ion. It is shown that the system of non-linear equations describing the diffusion of the species in a potentiostatic experiment for any ligand to metal ratio can be reduced to only one ordinary differential equation by means of a change of variable. Standard numerical methods can then be used in the computation of the solution with a great saving of computational time and resources in comparison with other existing methods. Some properties of the currents are also proved: (i) Cottrellian behaviour for any current in normal pulse polarography (NPP) and for limiting currents in reverse pulse polarography (RPP), (ii) the dependence of the normalised limiting current (φ) on just three parameters, and (iii) the equality of limiting NPP and RPP currents. The normalised current for high stability constant values depends on just two parameters, one of which is the ratio of total metal/total ligand concentrations, and can be found from an implicit algebraic equation. A new representation for the normalised limiting currents is suggested: the iso-φ diagram, which for each ratio of diffusion coefficients, e, describes the currents for any stability constant in a unique drawing. A new graphical procedure arising from this diagram is suggested and then applied to data corresponding to Zn/poly(methacrylic) acid at pH 6 and fixed ionic strength.