# The Cottrell Experiment and Diffusion Limitation 2/3 – The Cottrell Experiment

This chapter is part of the series â€˜The Cottrell experiment and diffusion limitation â€™. In this chapter, we discuss the Cottrell experiment.

## Experiments

Simple experiments often lead to intensive theoretical observations, when performed under real conditions. Suddenly temperature, pressure, etc. influence the outcome of the experiment. The Cottrell experiment is one of the most basic potential control experiments and fortunately well understood. The basic idea is to keep an electrode at a potential that wonâ€™t lead to any reaction at the electrode and when a stable state is reached a potential step is made that will lead to a chemical reaction.

In detail, this means the potential E of the electrode is before the potential step below the formal potential E^{0}â€™of an electrochemically active species. This species is present in the solution surrounding the electrode. The potential before the potential step is named E_{1} in Figure 4.6. In the following pictures and explanations it is assumed that the species is in its reduced form (Red). The concentration of Red is c*_{Red} everywhere before the potential step, meanwhile the concentration of the oxidized form c*_{Ox} is 0.

## The Cottrell experiment

The potential step is raising the potential so that it results in

According to the Nernst equation (see Nernst equation), the ratio Ox/Red will become bigger. Red is consumed by the electrode by accepting an electron, that is Red is oxidized, and the oxidized form (Ox) is produced (see Figure 4.6). If E_{2} is significantly higher than E^{0}â€™, c_{Red} is 0 at the electrode surface and all Red close to the electrode is consumed.

The measured current after the potential step is a result of Redâ€™s oxidation to Ox. The current will flow as long as Red is consumed. Before the potential step, the lack of potential as a driving force was limiting the reaction and by increasing the potential, it is no longer the limiting factor for this reaction. Since all molecules of Red reaching the electrode are consumed immediately, the mass transfer of Red towards the electrode becomes the limiting factor. The species Red is completely depleted in front of the electrode but at a distance x_{1} from the electrode surface the concentration is still c*_{Red} (see Figure 1.2a).

This concentration gradient leads to diffusion of Red towards the electrode surface. The diffusion will also deplete all the Red in the distance x_{1}, so Red from a distance x_{2} will diffuse to x_{1}. This means that the layer of liquid in which diffusion occurs, the diffusion layer, is growing from the electrode into the bulk (see Figure 1.2a). There are limits to the growth of the diffusion layer. For usual macro-electrodes (i.e. electrodes with a diameter above 25 Âµm) the limit is reached when the diffusion layer is disturbed by convection, that is movement of the liquid. This can be self-induced or due to stirring. In a stirred solution a diffusion layer will have a smaller final thickness than in a non-stirred one.

The current measured is always depending on the flux J of molecules towards the electrodes. Actually the current I is proportional to the flux J.Â

If the flux J is known the current I canÂ alsoÂ be calculated. The flux is described by Fickâ€™s lawÂ

which shows that the flux J is proportional to the gradient or in a one dimensional case slope of the concentration curve âˆ‚c/âˆ‚x with the diffusion coefficient D as proportional factor. D is a property and thus a fixed value for a molecule, ion, or complex in a medium. In the literature one can find tables for D, but one should take care that this value is also referring to the right medium.Â

Directly after the potential jump in the Cottrell experiment, the concentration of Red is 0 at the electrode surface only. With increasing time the diffusion layer grows into the solution and the resulting gradient is less steep. As a result the flux J is getting smaller and smaller with time (seeÂ Figure 1.2b). To describe the change of I with time, the change of J or c with time t is needed. The time differential of the concentration is described by Fickâ€™s second law:Â

Fickâ€™s laws are treating general diffusion;Â solving it for the particular case of the Cottrell experiment requires some conditions for the initial state and the boundaries. These conditions are:Â

- Before the potential step the concentration is c*RedÂ everywhere

*cRed(x, 0) = c*RedÂ* - After the potential step the concentration at the electrode surface is 0

*cRed(0, t) = 0Â* - In an infinite distance from the electrode the concentration is always c*Red

*cRed(âˆž, t) = c*RedÂ*

Using these conditions to solve Fickâ€™s law leads to the Cottrell equation:Â

The z is the number of electrons transferred per molecule, F is the Faraday constant, and A the area of the electrode.

## The diffusion layer

This equation can be applied to many processes in electrochemistry when depletion of observed species startsÂ and the reaction gets diffusion controlled.Â As we now knowÂ equation 1.5, we expect a currentÂ respondingÂ to the potential step as seen inÂ Figure 1.3a.Â There will be a current step with the potential step and afterwards the currents decays with t-Â½.Â

In most real life experiments, at a certain time the change of current becomes negligible and the current can be considered constant. A reason for this is that the diffusion layer cannot grow infinitely. At some point the disturbances and movement of the liquid will destroy the outer layers of the diffusion layer.Â

AlthoughÂ thisÂ seemsÂ to be a simpleÂ experimentÂ withÂ little practical value, it is actually still used to determine diffusion coefficients or the electrodeâ€™s surface area. This is done by turning the I-t-curve into a linear form. From the Cottrell equation it is clear that I vs t-Â½Â should lead to a linear plot with a slopeÂ

that can be used to calculate A, c*, D,Â or z, if the other ones are known (seeÂ Figure 1.3b).Â

If the experiment is performed and the linear plot is made, it often doesÂ not show a perfect line, especially in the beginning of the curve andÂ atÂ the very end. The end of the curve is not perfectly linear since the diffusion layer doesÂ not expand perfectly into infinityÂ asÂ discussed before.Â

The beginning of the curve is influenced by two factors. First, it is impossible to perform an ideal potential step. Potentiostats have rise-times in ns or even Âµs, that is they can reach a set potential in that time period, but this is still not perfectly instantaneous. So the beginning of the curve deviates from the ideal potential step. The second factor, the capacitive current or capacitive charging current, will be explained in the next chapter.Â