B.1. Reversible Electron Transfer

The term steady state voltammetry is used in this book to denote any voltammetric method in which a steady state is achieved in some range of applied potential between the rate of electrolysis and some other feature of the experiment, such as the rate of diffusion or the rate of a limiting chemical reaction. This condition is signaled by the appearance of a plateau or limiting current in the voltammogram (cf. Figures 11.2 and 11.4). Steady state techniques include rotating disk voltammetry, dc polarography, normal pulse voltammetry, and slow scan ultramicroelectrode voltammetry.

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Figure 11.4 Digital current vs. potential output for a reversible oxidation in normal pulse voltammetry.

Figure 11.4 Digital current vs. potential output for a reversible oxidation in normal pulse voltammetry.

A somewhat general approach to analyzing steady state voltammograms can be used. This is because the regression equations for a reversible electrode reaction (cf. eq. (11.1)) have the same form for each steady state technique. They differ only in the expressions for the limiting current.

Conventional disk-shaped electrodes of radii larger than 0.1 mm develop faradaic currents in analytical electrochemical cells that are described to a good approximation by using a semi-infinite linear diffusion model. Expressions for their limiting currents in steady state voltammetry are available in standard electrochemistry textbooks [1,2]. For reversible electrode reactions, the limiting currents are relatively independent of scan rate. Voltammograms obtained with microelectrodes, on the other hand, may change shape depending on the scan rate used. These electrodes are discussed in more detail in the next section.

We previously encountered the general model for reversible electron transfer in steady state voltammetry in Section 3.B.I. It is summarized in Table 11.2. This table introduces a second background option in addition to the linear one in Section 3.B.I. The linear (b4E + b5) term works well if the background is relatively flat, such as in Figures 11.2 and 11.4. However, a number of background processes on electrodes limit the available potential window, such as electrolysis of hydrogen ions, solvent, electrolyte ions, or the electrode itself, and can contribute to a large exponential background current.

The exponential term (Table 11.2) describes the rising signal caused by the irreversible reaction responsible for the background current at the potential limit of the working electrode. In such cases, the model is i = ;/(l + 6) + 64[exp(65£)]. (11.2)

A voltammogram overlapped with an exponential background is illustrated for an oxidation wave in Figure 11.5. The resolved voltammogram was computed from the regression parameters after nonlinear regression analysis of the overlapped voltammetric data onto eq. (11.2).

Table 11.2 Model for Reversible Electron Transfer in Steady State Voltammetry

Assumptions: Reversible electron transfer, linear diffusion Regression equation (reductions)'.

i = /[/( 1 + 6) + background; 6 = exp[(£ - EJ')(F/RT)], R = gas constant, F — Faraday's constant, and T = temperature in Kelvins Regression parameters: Data:

Special instructions: Linear background = b$E + b5 Exponential background current = ¿)4[exp(65£')] Use e = exp[-(£ - Ef)(F/RT),] for oxidations

Figure 11.5 Steady state voltammogram for an oxidation wave (-) severely overlapped with a large increasing anodic current showing the resolved voltammogram computed from nonlinear regression parameters (--------) using model in Table ll.2 with exponential background and the separate exponentially increasing background ( ).

Figure 11.5 Steady state voltammogram for an oxidation wave (-) severely overlapped with a large increasing anodic current showing the resolved voltammogram computed from nonlinear regression parameters (--------) using model in Table ll.2 with exponential background and the separate exponentially increasing background ( ).

A similar model can be used when two or more voltammograms are overlapped with one and other (Figure 11.6). They can be separated by using nonlinear regression onto the model in Table 11.3, which is simply the sum of two reversible waves. The fraction (/) of component 1 is referred to the total amount of analyte in the mixture. Regression onto this model was used successfully to resolve overlapped dc polarograms for Pb(II) and T1(I), which differ in half-wave potential by only about 50 mV [3].

Figure 11.6 Severely overlapped steady state voltammograms representing two reversible oxidations (-) showing the two resolved voltammograms computed from the regression parameters using model in Table 11.3: (a) first component and (b) second component.

Figure 11.6 Severely overlapped steady state voltammograms representing two reversible oxidations (-) showing the two resolved voltammograms computed from the regression parameters using model in Table 11.3: (a) first component and (b) second component.

Table 11.3 Model for Two Overlapped Reversible Waves in Steady State Voltammetry

Assumptions: Two reversible electron transfers, linear diffusion Regression equation (reductions):

i = ;',(f/(l + <?,) + (1 - /)/(1 + 02); 6j = exp[£ - Ef)(F/RT),], j = 1 or 2; R = gas constant, F = Faraday's constant, T = temperature in Kelvins, /= fraction of component 1 in the solution Regression parameters: Data:

f (', ET E°2' (F/RT)i (F/RT)2 i vs. E Special instructions: Linear background = b-,E + bs Exponential background current = b-,[exp(bsE)] Use 6j = exp[-(£" - Ef)(F/RT)j] for oxidations

B.2. Reversible Electron Transfer at Ultramicroelectrodes in Highly Resistive Media

Electrodes with one dimension in the micrometer range have become popular tools in electrochemistry [4], Platinum, gold, or carbon disks of radii 1-25 ¡xm imbedded in glass or polymer insulating materials are typical of this class. Because very small currents are generated at electrodes of such very small size, they can be used in highly resistive media and at very high rates of scanning. Studies in organic solvents without electrolyte become possible. Efficient diffusion and small capacitance at these tiny electrodes leads to signal to noise that is often better than for larger solid electrodes of the same material. The possibility for scan rates on the order of 1 million V 1 provides the ability to study very fast electrochemical and chemical reactions.

Microdisk electrodes of radii 1-25 ¿¿m give a steady state voltammetric response at low scan rates (e.g., <25 mV s"1) when the electrode reaction is controlled by diffusion. The model for diffusion under such experimental conditions is essentially spherical, resulting in a steady state sigmoid shape of the current vs. potential curve. As the scan rate in such a system increases, the shape of the curve begins to show a peak, until at very large scan rates the response is peak shaped as in cyclic voltammetry. The curve shape changes because, as the scan rate increases, the time scale of the experiment becomes smaller and the spherical diffusion field collapses to the semiinfinite planar model. A steady state is no longer achieved, and depletion of electroactive material very near the electrode leads to a decrease in current after the peak [4],

The regression model for the sigmoid-shaped reversible response at low scan rates is the same as for any reversible steady state voltammogram (Table 11.2). Excellent fits of this model to microelectrode voltammograms representing standard reversible electrochemical reactions have been reported [5, 6].

Even though microelectrodes can be used in resistive fluids without adding electrolyte, the resulting voltammograms are somewhat broadened by the ohmic or iRu drop of the cell, where i = current and Ru = uncompensated cell resistance. Resistance is large in solvents and oil-based media not containing sufficient electrolyte and is difficult to compensate for electronically. Therefore, the resulting ohmic broadening can be significant.

The ohmic drop of a cell is difficult to distinguish from slow electrode kinetics. However, in a microelectrode cell, a known reversible redox couple may be used to obtain Ru. Although the dependence of ohmic drop on potential in a cell containing a microdisk electrode may be complex, a simple approximate expression [5, 6] is effective when the ohmic drop is not too large:

where E is the actual potential at the microelectrode, and V is the applied voltage.

Equation (11.3) is combined with the model in Table 11.2 to yield the model in Table 11.4 for microelectrode voltammetry in a resistive medium. Note the special pair of logical statements that must be included in the subroutine that computes the current to avoid computer overflows or underflows. An overflow will occur when the result of a calculation exceeds the largest number the computer can store, thus triggering an error message and program termination. Analogously, some computers do not automatically assign a value of 0 to results that are too small for them to represent. This results in an underflow error message.

The model in Table 11.4 was tested with data obtained at scan rates of <10 mV s"1 for the reversible oxidation of ferrocene in acetonitrile without added electrolyte at a carbon disk electrode of 6 ¡xm radius [6], The model in Table 11.4 fixes RT/Fat its theoretical value and involves five parameters,

Table 11.4 Model for Reversible Ultramicroelectrode Voltammetry in a Resistive Medium

Assumptions: Reversible electron transfer, resistive media Regression equation (reductions):

i = /,/(1 + 8) + b4E + b5; 6 = exp[(K + iRu - E°')(F/RT)], R = gas constant, F = Faraday's constant, and T = temperature in Kelvins Regression parameters: Data:

Special instructions: To avoid possible overflow or underflow, include the following in subroutine for computing i: IF (£ - E°') s -10 THEN i = i, + b4E +b5 IF (E - E°') > -10 THEN i = b4E + b, use 6 = exp[-V + iRu - E°')(F/RT)] for oxidations including a linear background. In testing the model, a modified steepest descent algorithm gave better convergence than the Marquardt-Levenberg algorithm because matrix inversion errors were generated with the latter. Analysis of 10 data sets with a program employing the steepest descent algorithm gave an average Ru of 3.47 ± 0.20 X 106 ohms. Graphical representation of a typical data set shows that the points fit the regression line quite well (Figure 11.7). A resistance-free voltammogram computed from the results of the regression analysis is also presented.

The model in Table 11.4 has also been used to find the resistance of a microemulsion consisting of tiny surfactant-coated water droplets suspended in oil [7], This medium had an Ru approximately 40-fold larger than acetonitrile in the preceding example. Knowledge of Ru allowed computation of resistance-free voltammograms for catalysts dissolved in the water droplets of the microemulsion and estimation of their electrochemical parameters.

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