Chronoamperometry Cottrell equation

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

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

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,... 

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. 

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. 

We describe here that the redox oligomer wires fabricated with the stepwise coordination method show characteristic electron transport behavior distinct from conventional redox polymers. Redox polymers are representative electron-conducting substances in which redox species are connected to form a polymer wire.21-25 The electron transport was treated according to the concept of redox conduction, based on the dilfusional motion of collective electron transfer pathways, composed of electron hopping terms and/or physical diffusion.17,18,26-30 In the characterization of redox conduction, the Cottrell equation can be applied to the initial current—time curve after the potential step in potential step chronoamperometry (PSCA), which causes the redox reaction of the redox polymer film ... 

In practice two methods are used for stationary planar electrodes in quiescent solution chronoamperometry and chronopotentiometry. By use of an electroactive species whose concentration, diffusion coefficient, and n value are known, the electrode area can be calculated from the experimental data. In chronoamperometry, the potential is stepped from a value where no reaction takes place to a value that ensures that the concentration of reactant species will be maintained at essentially zero concentration at the electrode surface. Under conditions of linear diffusion to a planar electrode the current is given by the Cottrell equation [Chapter 3, Eq. (3.6)] ... 

Cottrell equation — Consider a large planar - electrode, of surface area A, initially at rest, in contact with a semiinfinite layer of unstirred solution containing excess electrolyte and some small amount of electroactive species R with bulk concentration Cr. At the instant t = 0, the potential of the electrode is suddenly changed (see -> chronoamperometry) to a value at which the reaction... 

Diffusion current is usually measured in the case of several transient electrochemical methods, e.g., in chronoamperometric and chronocoulometric experiments (see chronoamperometry, -> chronocoulometry, -> Cottrell equation) as well as it determines the shape of... 

Finite diffusion — Finite (sometimes also called -> limited) diffusion situation arises when the -> diffusion layer, which otherwise might be expanded infinitely at long-term electrolysis, is restricted to a given distance, e.g., in the case of extensive stirring (- rotating disc electrode). It is the case at a thin film, in a thin layer cell, and a thin cell sandwiched with an anode and a cathode. Finite diffusion causes a decrease of the current to zero at long times in the - Cottrell plot (-> Cottrell equation, and - chronoamperometry) or for voltammetric waves (see also - electrochemical impedance spectroscopy). Finite diffusion generally occurs at -> hydrodynamic electrodes. 

In an electrochemical system with a stationary working electrode and a quiescent solution, when the potential of the working electrode is increased from a value at which no electrochemical reaction is occurring to a value at which the electrochemical reaction is diffusion controlled, the current will decay with time, as shown in Figure 1.13. This technique is called chronoamperometry. At this potential the surface concentration of the electroactive species decreases according to the potential, and the diffusion layer expands with time and according to the depletion of the reactant, which causes the current decay. The Cottrell equation can be used to express current decay with time ... 

When conventional electrodes with diameters between 0.1 and 2 mm are used, the latter quantity has usually decayed to zero after 0.5 ms or less and may be neglected in experiments lasting 1 ms or more. This decay time is reduced to the microsecond time regime when ultramicroelectrodes are used [94,125,202]. According to Eq. (64), which for i = 0 is known as the Cottrell equation, the current approaches zero when the time approaches infinity. However, undisturbed linear diffusion can be maintained only over rather short time intervals unless special precautions are taken (see Sec. II.D.l), and the measurements of current-time curves, called chronoamperometry (CA), are often complicated by additional modes of transport. Therefore, the use of properly shielded electrodes [140] should be considered in chronoamperometric experiments exceeding approximately 1 s. The mathematical formalism for chronoamperometry has been developed also for the application of ultramicroelectrodes [203]. 

If charge diffusion is significantly slower so that the distance of charge transport, L, (=2(Dt) ) is clearly smaller than the thickness of the lamina, 5, the electrochemical response will be equivalent to that recorded when reactants freely diffuse from an infinite volume of solution to the electrode. This situation, often termed as thick-layer behavior, corresponds to semi-infinite boundary conditions, and concentration profiles such as that shown in Figure 2.5c are then predicted. Accordingly, Cottrell-type behavior is observed, for instance, in cyclic voltammetry (CV) and chronoamperometry (CA). In this last technique, a constant potential sufficiently cathodic for ensuring diffusion control in the reduction of Ox to Red is applied. The resulting current-time (i-t) curves should verify the Cottrell equation presented in the previous chapter (Equation (1.3)). 

In most chronoamperometry, with measurement times of 1 ms to 10 s, the diffusion layer is several micrometers to even hundreds of micrometers thick. These distances are much larger than the scale of roughness on a reasonably polished electrode, which will have features no larger than a small fraction of a micrometer. Therefore, on the scale of the diffusion layer, the electrode appears flat the surfaces connecting equal concentrations in the diffusion layer are planes parallel to the electrode surface and the area of the diffusion field is the geometric area of the electrode. When these conditions apply, as in Figure 5.23a, the geometric area should be used in the Cottrell equation. 

In a potential-step experiment, the potential of the working electrode is instantaneously stepped from a value where no reaction occurs to a value where the electrode reaction under investigation takes place and the current versus time (chronoamperometry) or the charge versus time (chronocoulometry) response is recorded. The transient obtained depends upon the potential applied and whether it is stepped into a diffusion control, in an electron transfer control or in a mixed control region. Under diffusion control the transient may be described by the Cottrell equation obtained by solving Tick s second law with the appropriate initial and boimdary conditions [1, 2, 3, 4, 5 and 6] ... 

The analysis of chronoamperometry data is based on the Cottrell equation, which defines the current-time dependence for linear diffusion control ... 

Figure 5.6 Chronoamperometry, semi-infinite condition, and current-time dependence which follows the Cottrell equation.

In chronoamperometry, after switching on an overpotential, the time dependence of the current is monitored. For purely diffusion-controlled processes, the current depends on time according to the Cottrell equation in Chapter 5, Eq. (5.20). The current decreases proportional to 1/Vt. The combination with charge transfer control leads to the following... 

Potential-Step chronoamperometry (CA), chronocoulometry (CC), and sampled current voltammetry were used to probe the nature of the charge transport process close to the electrode surface. By manipulating the time scale of the current decay response at sufficiently short times, semi-infinite diffusion is observed. The resulting current response conforms to the Cottrell equation ... 

Double-potential step chronoamperometry This method was proposed in 1965 by Schwarz and Shain [18] for the investigation of follow-up reactions especially for the mechanism. During the first potential pulse the product B is produced at a stationary electrode under diffusion-controlled conditions for a timed interval tp. During this interval substance B diffuses into the solution and simultaneously undergoes a chemical reaction. Then, the potential is suddenly switched to a value where B is converted back into A. The backward current indicates the amount of B which has not reacted and can be related to the rate constant kf. The forward current-time dependence is given by the Cottrell equation... 

The Cottrell equation is derived from Pick s second law of diffusion (Section 1.5) and predicts the variation of the current in time, when a potential step is applied under conditions of large overpotential. For this equation to be valid the current must be limited by diffusion of the analyte to the electrode surface, and thus the solution has to be unstirred. The overpotential at which the reaction is driven must be large enough to ensure the rapid depletion of the electroactive species (O) at the electrode surface, such that the process would be controlled by the diffusion to the electrode. This equation is most often applied to potential step methods (e.g., chronoamperometry see Chapter 11) ... 

The Cottrell equation was developed in 1903 by solving a simple model system for chronoamperometry at a large electrode and at high overpotential [F.G. Cottrell, Z. Physik. Chem. 42 (1903) 385]. 

Limited diffusion — applies to a thin film or a thin layer cell. Limited diffusion causes a decrease of the current to zero at long times (i.e., the -> Cottrell equation will then not any more be followed -> chronoamperometry) or at the voltammetric waves -> cyclic voltammetry) because there is not an infinite reservoir of electroactive species. (See also electrochemical impedance spectroscopy.)... 

The most useful equation in chronoamperometry is the Cottrell equation, which describes the observed current (planar electrode of infinite size) at any time following a large forward potential step in a reversible redox reaction (or to large overpotential) as a function of t. ... 

Fig. 10.9 Chronoamperometry at a potential corresponding to Co(0,t) = 0 (a) perturbation signal (b) concentration profiles at increasing times (c) signal recorded i =f(t) according to Cottrell s equation normalization in (b) is performed by dividing Co(x,t) by Cq with respect to the bulk concentration, which is quite common practice...

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