The Cottrell Equation

A potential step experiment was carried out in a solution containing 0.05 M ferrocyanide ([FejCNje] ) dissolved in a solution containing a large excess of inert electrolyte. Care was taken to ensure that there was no stirring of the solution during the experiment. The potential was stepped from a value where there was no reaction to a potential at which the [FejCNje] was oxidised to [FejCNje] at a mass transport controlled rate, and the following currents were recorded  

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This expression is the sum of a transient tenu and a steady-state tenu, where r is the radius of the sphere. At short times after the application of the potential step, the transient tenu dominates over the steady-state tenu, and the electrode is analogous to a plane, as the depletion layer is thin compared with the disc radius, and the current varies widi time according to the Cottrell equation. At long times, the transient cunent will decrease to a negligible value, the depletion layer is comparable to the electrode radius, spherical difhision controls the transport of reactant, and the cunent density reaches a steady-state value. At times intenuediate to the limiting conditions of Cottrell behaviour or diffusion control, both transient and steady-state tenus need to be considered and thus the fiill expression must be used. Flowever, many experiments involving microelectrodes are designed such that one of the simpler cunent expressions is valid. 

Derive the Cottrell equation by combining Fick s first law of diffusion with the tune-dependent change of the concentration gradient during a potential-step experiment. 

However, in contrast to the condition of a constant potential and current sampling in pulse polarography, which yields36 the Cottrell equation ... 

Together with the boundary condition (5.4.5) and relationship (5.4.6), this yields the partial differential equation (2.5.3) for linear diffusion and Eq. (2.7.16) for convective diffusion to a growing sphere, where D = D0x and = Cqx/[1 + A(D0x/T>Red)12]- As for linear diffusion, the limiting diffusion current density is given by the Cottrell equation... 

This is known as the Cottrell equation. It shows that the faradaic transient current, it, decays t 1/2. In contrast, the capacitance current decays exponentially and much faster. According to Eq. (18b.16) a plot of it vs. t 1/2 is a straight line, the slope of which can be used to calculate the D of the analyte if the area of the electrode is known. Eq. (18b. 16) is also used to measure the active area of an electrode by using species with known D. At a spherical electrode (such as HMDE) of radius, r, the Cottrell equation has an added spherical term... 

The fundamental law of chronoamperometry is the Cottrell equation (see Chapter 1, Section 4.2.4) ... 

Clearly, for times tpotential step response. In this time interval (/current-time curve follows the Cottrell equation therefore, the cathodic current is expressed by the relationship ... 

If the kinetics of the preceding reaction are slow, the current-time response is governed by the virtually unchanged initial concentration of Ox hence, the Cottrell equation is valid. 

The value of id at various times is determined by measuring the current value at the longest times (at which the conversion Y/Ox is more likely to be complete) and applying the Cottrell equation (in practice, considering its linear decay with t1/2) at the shorter times of interest. 

The kinetics of following chemical reactions cannot be studied by the single potential step technique in that the response would simply obey the Cottrell equation. In contrast, the double potential step technique, that measures the response exhibited by either the reagent Ox or the product Red, is sensitive to the chemical fate of Red. The cathodic response before the inversion of the applied potential (t < x) is expressed by the Cottrell equation ... 

The responses of the forward and reverse steps are both affected by the chemical complication in fact, the reduction current will be greater than that predicted by the Cottrell equation, whereas the reoxidation current will be lower than that predicted by the Cottrell equation. This implies that to gain information on the regeneration of the reagent it is sufficient to use the single potential step technique. 

For the same process, in chronoamperometry, the Cottrell equation holds ... 

For a quantitative description of the way that a chronoamperometric current trails off with time, we employ the Cottrell equation, as follows ... 

The Cottrell equation, as written here, relates to an electrode in the form of a cylindrical wire. One end of the wire will be embedded in a non-conductive sleeve (e.g. glass), so that only one end of the wire will ever be in contact with the analyte solution. If the wire has a length h and a diameter r, then the surface area A of the wire is given by ... 

The Cottrell equation is derived with the assumption that diffusion is the sole form of mass transport. How can we confirm that it actually is ... 

The derivation of the Ilkovic equation assumes that the drops of mercury are spherical the rate of flow of mercury is constant, the way analyte diffuses toward the DME obeys the Cottrell equation, and the diflusion current is truly limiting, i.e. that Csurfacc = 0-... 

Occasionally, we want to know the value of a diffusion coefficient D. The Cottrell equation (equation (6.3)) is one simple means of obtaining D, i.e. by plotting fiimu) against and obtaining a value for the gradient. (/ im( ) here means the limiting current as a function of time.)... 

The potential step provides the theoretical background for any potentiostatic regulation experiment and a basic understanding is necessary for the mathematical solution of any controlled potential, nonsteady-state voltammetric response, such as LSV, pulse or a.c. experiments. At a stationary electrode, the current response to a potential step is described by the Cottrell equation [eqn. (83)] but at hydro-dynamic electrodes, it needs to be modified to take account of forced convection. 

Just as the fundamental equation for a potential step experiment is the Cottrell equation, for the current step it is, in quiescent solution, the Sand equation [245]... 

The product D0 (dCo/dx)x=0 t is the flux or the number of moles of O diffusing per unit time to unit area of the electrode in units of mol/(cm2 s). (The reader should perform a dimensional analysis on the equations to justify the units used.) Since (3Co/3x)x=01 is the slope of the concentration-distance profile for species O at the electrode surface at time t, the expected behavior of the current during the chronoamperometry experiment can be determined from the behavior of the slope of the profiles shown in Figure 3. IB. Examination of the profiles for O at x = 0 reveals a decrease in the slope with time, which means a decrease in current. In fact, the current decays smoothly from an expected value of oo at t = 0 and approaches zero with increasing time as described by the Cottrell equation for a planar electrode,... 

The Cottrell equation states that the product it,/2 should be a constant K for a diffusion-controlled reaction at a planar electrode. Deviation from this constancy can be caused by a number of situations, including nonplanar diffusion, convection in the cell, slow charging of the electrode during the potential step, and coupled chemical reactions. For each of these cases, the variation of it1/2 when plotted against t is somewhat characteristic. 

Chronoamperometry has proven useful for the measurement of diffusion coefficients of electroactive species. An average value of it1/2 over a range of time is determined at an electrode, the area of which is accurately known, and with a solution of known concentration. The diffusion coefficient can then be calculated from it1/2 by the Cottrell equation. Although the electrode area can be physically measured, a common practice is to measure it electrochemically by performing the chronoamperometric experiment on a redox species whose diffusion coefficient is known [6]. The value of A is then calculated from it1/2. Such an electrochemically measured surface area takes into account any unusual surface geometry that may be difficult to measure geometrically. 

Figure 3.4B displays the data in terms of normalized i versus t 1/2. For the le and 2e cases, the two straight lines are simple plots of the Cottrell equation (Eq. 3.6) for which the slope is determined by the respective n values. The shape... 

Response to pulse C Now we have the limiting case of chronoamperometry. The surface concentration has become effectively zero and the response will follow the Cottrell equation. 

Several approaches to solving this expression for various boundary conditions have been reported [25,26]. The solutions are qualitatively similar to the results at a hemisphere at very short times (i.e., when (Dt),y4 rD), the Cottrell equation is followed, but at long times the current becomes steady-state. Simple analytical expressions analogous to the Cottrell equation for macroplanar electrodes or Equation 12.9 for spherical electrodes do not exist for disk electrodes. For the particular case of a disk electrode inlaid in an infinitely large, coplanar insulator, the chronoamperometric limiting current has been found to follow [27] ... 

It is now possible to write a computer program that will simulate the currenttime behavior that occurs following a single step to diffusion-limiting conditions. The solution to this problem (the Cottrell equation) is well known and there is no need to perform this simulation to obtain any new electrochemical information. However, it is instructive to see how the principles developed earlier may be used to write a program and how the results of that simulation may be verified by comparing them to known results. Any additional programming will be left as an exercise to the reader. 

Selected results obtained with this program appear in Table 20.1. The exact value for Z(K) as obtained from the Cottrell equation is shown here also. [The Cottrell equation may be written... 

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