Equation Cottrell

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)  

In an unperturbed solution, when a potential step is applied that causes a surface reaction to occur, the current decays according to the Cottrell equation. One application is the determination of the diffusion coefficient for an electrochemically active species O, with the initial concentration of 0.1 M, that undergoes a single electron reduction at a 1 mm diameter electrode. The current decays to a diffusion limited current of 68 pA after 2 sec. When applying the Cottrell equation it is important to note that the concentration has to be converted to mol cm (in our case 0.1 M corresponds to 10 mol cm ). The electrode area is 7.85 X 10 cm, and the value for the diffusion coefficient can be calculated as 

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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. 

Cottrell equation Cottrell unit Couchman equation Couette flow Couette viscometers Cough drops Coughlozenges... 

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... 

In the case of a fractal electrode, the relationship between the current and the time follows generalized Cottrell equation derived... 

From the Laplace transforms of Eqs. (19) and (21) to (23), a generalized Cottrell equation which describes the current response on the applied potential step is obtained as following equation... 

Figures 8a and 8b present the simulated current transients obtained from the self-affine fractal interfaces of r/ = 0.1 0.3 0.5 and r] = 1.0 2.0 4.0, respectively, embedded by the Euclidean two-dimensional space. It is well known that the current-time relation during the current transient experiment is expressed as the generalized Cottrell equation of Eqs. (16) and (24).154 So, the power exponent -a should have the value of - 0.75 for all the above self-affine fractal interfaces.

For the contiguous fractal with dy < 2.0, Pajkossy and Nyikos gave the first experimental evidence of the validity of the generalized Cottrell equation.121 They prepared two kinds of partially active electrodes a regular fractal pattern with cly = (log 8)/(log 3) = 1.893 and a Sierpinski gasket1 with dF =... 

Concentration Profiles. Cottrell Equation. As previously mentioned, the region close to the electrode surface where the concentrations... 

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 ... 

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