Constant current planar electrodes

Starting at t = 0 a constant current is applied to the electrode in order to cause oxidation or reduction of electroactive species, and the variation of the potential of the electrode with time is measured (chronopoten-tiometry). Fick s second law is solved using the Laplace transform as in the previous section the first two boundary conditions are the same, but the third is different  

The third condition expresses the fact that a concentration gradient is being imposed at the electrode surface. 

As in the last section, and following the same arguments, we reach 

The equations that give the variation of concentration with time and the variation of potential with time are 

When co = 0, all species in the zone of the electrode have been consumed, as shown in Fig. 5.6. The corresponding value of t is called the transition time, t. 

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Eddy diffusion as a transport mechanism dominates turbulent flow at a planar electrode ia a duct. Close to the electrode, however, transport is by diffusion across a laminar sublayer. Because this sublayer is much thinner than the layer under laminar flow, higher mass-transfer rates under turbulent conditions result. Assuming an essentially constant reactant concentration, the limiting current under turbulent flow is expected to be iadependent of distance ia the direction of electrolyte flow. 

In the case of mass transport by pure diffusion, the concentrations of electroactive species at an electrode surface can often be calculated for simple systems by solving Fick s equations with appropriate boundary conditions. A well known example is for the overvoltage at a planar electrode under an imposed constant current and conditions of semi-infinite linear diffusion. The relationships between concentration, distance from the electrode surface, x, and time, f, are determined by solution of Fick s second law, so that expressions can be written for [Ox]Q and [Red]0 as functions of time. Thus, for... 

Fig. 2.3 Experimental current-time curve (a) and logarithmic curves (b) for the application of a constant potential to a graphite disc electrode of radius 0.5 mm (planar electrode) for the reduction of Fe(CN)g. ...

Fig. 2.15 (Solid line) Current-time curves for the application of a constant potential to a spherical electrode calculated from Eq. (2.142). D0 = Dr = 10-5 cm2 s 1, co = cr = 1 mM, rs = 0.001 cm, (E — E ) = -0.2 V, 7=298 K. (Dashed line) Current-time curves for the application of a constant potential to a planar electrode of the same area as the spherical one calculated from Eq. (2.28). (Dotted line) Steady-state limiting current for a spherical electrode calculated from Eq. (2.148). The inner figure corresponds to the plot of the current of the spherical electrode versus j ft...

The time evolution of the cathodic limiting current (Eq. 2.147) has been plotted in Fig. 2.15 together with that obtained for a planar electrode (Eq. 2.28) and the constant steady-state limiting current for a spherical electrode given by... 

From this figure, it can be seen that the current decays with time as in the planar case although this decrease leads to a constant value, / lc ss, different from zero, which will be achieved sooner as the electrode radius diminishes. The current for times close to zero is identical to that obtained in a planar electrode given to the prevalence of the term 1 / over the inverse of the radius. For longer times, the opposite happens and the term l/rs is dominant. 

Even in the simplest situation for which a = a2 = 0.5, the global behavior of the response depends upon three parameters, the difference between the formal potentials AEf, and the rate constants of both steps k(j and k. Thus, the observed current-potential curves are the result of the interaction of thermodynamic and kinetic effects so the appearance of two or one waves would not be due solely to thermodynamic stability or instability of the intermediate species but also to a kinetic stabilization or destabilization of the same [4, 31]. This can be seen in Fig. 3.19 in which the current-potential curves of an EE process with AE = 0 mV taking place at a planar electrode with a reversible first step... 

Eq. (4.61) and Table 2.3 of Sect. 2.6), whereas for the second potential pulse the amount of converted charge is much smaller than that obtained at a planar electrode (macroelectrode). Indeed, when the electrode radius becomes small enough the converted charge for the second potential pulse is constant and coincides with (for example, from Eq. (4.62) in the limit rs current-time curves. 

As previously depicted for planar electrodes in reference [50], the decrease of the heterogeneous rate constant gives rise to the decrease of the peak current, the increase of the peak half width and the shift of the peak potential toward more negative values. For fully irreversible systems (very small k° values), it is observed that the current peak and the peak half width become independent of the rate constant, and only the position of the peak (i.e., the peak potential G,peak) changes with k°. 

From Eqs. (5.92)-(5.94), it is clear that K°phe ss < x°phe < xplane, that is, the maximum value of the dimensionless rate constant is that corresponding to a planar electrode (macroelectrode). For smaller electrodes, /c(sphc decreases until it becomes identical to the value corresponding to a stationary response, xpphe ss. In practice, this means that the decrease of the electrode size will lead to the decrease of the reversibility degree of the observed signal. It can be seen in the CV curves of Fig. 5.14, calculated for k ) = 10 eras 1 and v = 0.1 Vs-1, that the decrease of rs causes an increase and distortion of the dimensionless current similar to that observed for Nemstian processes (see Fig. 5.5), but there is also a shift of the curve toward more negative potentials (which can be clearly seen in Fig. 5.14b). 

Fig. 6.9 Cyclic voltagrams corresponding to an EC mechanism at a planar electrode calculated by following the numerical procedure given in [23, 24] (see also Appendix I). (a) Effect of the dimensionless rate constant = k /a on the current-potential response. K = 0. The values of appear in the figure, (b) Effect of the equilibrium constant /sfeq = 1 /K on the current-potential response.

Fig. 7.15 Influence of the dimensionless rate constant Kplane = k° Jr/D on the SWV forward (idashed lines) and reverse (solid lines) currents i//] la"c and i// 3la"e (a) and on the SWV net current (b), corresponding to a non-reversible electrochemical reaction at a planar electrode. sw = 50mV, AEs = 5mV, a = 0.5. The values of Kplane = k°yJr/D appear on the curves...

The evolution of the peak current (/ dlsc,peak) with frequency (/) is plotted in Fig. 7.37 for the first-order catalytic mechanism with different homogeneous rate constants at microdisc electrodes. For a simple reversible charge transfer process, it is well known that the peak current in SWV scales linearly with the square root of the frequency at a planar electrode [6, 17]. For disc microelectrodes, analogous linear relationships between the peak current and the square root of frequency are found for a reversible electrode reaction (see Fig. 7.37 for the smallest kx value). 

The mercury-pool electrode. Mercury pools of sufficient diameter to approach a planar configuration obey the equations derived for linear diffusion to a planar electrode. This has certain theoretical advantages because of the large number of equations that have been derived for the planar electrode geometry, especially in terms of constant-current chronopotentiometry and linear-potential sweep chronoamperometry. 

Sand equation — Consideration of the concentration c(x=o,t) at a planar working electrode in contact with a stagnant (unstirred) electrolyte solution for a reaction, where the oxidized species in the bulk is present at a concentration c and the reduced species is initially absent with an applied constant current 7, yields... 

Assume that a solution (100 ml) containing is reduced at a constant current density, (/), of 100 mA cm employing planar electrodes of 10-cm area. 

Constant cun ents are not obtained in reasonable periods of time with a planar electrode in an unstirred solution because concentration gradients out from the electrode surface are constantly changing with time. In contrast, the DME exhibits constant reproducible currents nearly instantaneously after an applied voltage adjustment. This behavior represents an advantage of the DME that accounted for its widespread use in the early years of voltammetry. 

We consider here a situation where the mass transport of the electroactive species may become rate determining, but all other processes which control the current-potential characteristics can still adjust rapidly. Thus, the concentration of the electroactive species, c, becomes time dependent. Since we allow only for diffusion, its temporal evolution is given by Pick s second law [i.e., in the case of a planar electrode, by dc/dt = D (d c/dz with the diffusion coefficient D, and z the spatial coordinate perpendicular to the electrode]. At the electrode (z = WE), the concentration obeys Pick s fust law, (dc/dz) z=we = Kuc i F). At a certain distance from the electrode, it is assumed that the concentration is at a constant value, c, its bulk value (constituting the second boundary condition). The concept of the Nemst diffusion layer underlies this idea. 

Thus the diffusion current for the spherical case is just that for the linear situation plus a constant term. For a planar electrode. 

The current at any point in ihe electrolysis wc have just discussed is determined by the rale of transport of A from the outer edge of the diffusion layer to the electrode surface. Because the product of the electrolysis P diffuses from Ihe surface and is ultimately swept away by convection, a continuous current is required to maintain the surface concentrations demanded by the Nernst equation. Convection, however, maintains a constant supply of A at the outer edge of the diffu.sion layer, fhus, a steady-siaie current results that is determined by the applied potential. This current is a quantitative measure of how fast A is being brought to the surface of the electrode, and this rate isgivcnhyOc /dj where X is the distance in centimeters from the electrode surface. I or a planar electrode, the current is given by Equation 25-4. 

Imagine a parallelepipedic cell (with unidirectional geometry), comprising two planar, parallel, silver electrodes with a surface area of S= 1 cm separated by a distance of L=2 mm. They are both successively immersed in one of the two aqueous solutions S, or 2 being studied, and connected to a constant current supply ... 

This is the quiet-solution experiment discussed in Section 1.2 with the current response shown in Fig. 2A. App is instantaneously switched from an initial value (open circuit or a value at which no electrolysis is occurring) to slightly past p, held constant for a fixed time, then normally switched off or back to E. If material diffuses to a planar electrode surface in only one direction (linear diffusion), then the exact description of the current-time curve is given by the Cottrell equation ... 

Nowadays, most chemists know the name "Sand" only because it appears attached to a partic ilar equation in texts that describe chronopotentiometry. This technique can be useful in the diagnosis of electrode reactions (l). A current step i is Impressed across an electrochemical cell containing iinstirred solution, the potential of the working electrode is measured with respect to time and the transition time x is noted. According to the Sand equation, the product ix /2 should be constant for an uncomplicated linear diffusion-controlled electrode reaction at a planar electrode. 

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