# Theory

The waveform of the voltage applied to a working electrode in CV is triangular shaped (i.e., the forward and reverse scan). Since this voltage varies linearly with time, the scan rate is the slope (V/s). An example of a CV for the reduction of ferricyanide to ferrocyanide is shown in Figure 1. The experiment conditions are listed in the figure caption.

Figure 1. CV for the reduction of 4.8 mM ferricyanide in 0.1 M KCl at a 1.0 mm glassy carbon electrode. Scan rate is 100 mV/s.

The peak shape of the reductive and reverse oxidative current vs. electrode potential curve (I-E) in Figure 1 is typical of an electrode reaction in which the rate is governed by diffusion of the electroactive species to a planar electrode surface. That is, the rate of the electron transfer step is fast compared to the rate at which ferricyanide is transported (diffuses) from the bulk solution to the electrode surface due to a concentration gradient, as ferricyanide is reduced to ferrocyanide. In such a case the peak current, Ip, is governed by the Randle-Sevcik relationship

$\mathrm{I_p = k\, n^{3/2} A\, D^{1/2}\, C^b\, \upsilon^{1/2}} \tag{1}$

where the constant k = 2.72 x 105; n is the number of moles of electrons transferred per mole of electroactive species (e.g., ferricyanide); A is the area of the electrode in cm2; D is the diffusion coefficient in cm2/s; Cb is the solution concentration in mole/L; and υ is the scan rate of the potential in volt/s.

The Ip is linearly proportional to the bulk concentration, Cb, of the electroactive species, and the square root of the scan rate, υ1/2. Thus, an important diagnostic is a plot of the Ip vs. υ1/2. If the plot is linear, it is reasonably safe to say that the electrode reaction is controlled by diffusion, which is the mass transport rate of the electroactive species to the surface of the electrode across a concentration gradient. The thickness, δ, of the "diffusion" layer can be approximated by δ ~ [D t]1/2, where D is the diffusion coefficient and t is time in seconds. A quiet (i.e. unstirred solution) is required. The presence of supporting electrolyte, such as KNO3 in this example, is required to prevent charged electroactive species from migrating in the electric field gradient.

Another important diagnostic for characterizing the electrode reaction is the value of the peak potential, Ep. When the rate of electron transfer is fast, the Ep value will be independent of the scan rate; indicating a reversible electrode reaction. Then the difference between the anodic peak potential, Epa, and the cathodic, Epc value will be equal to 57 mV/n. The thermodynamically reversible potential, Eo, is found at E0.85. That is, 85% of the way-up the I-E wave to Ep. This value should also be independent of the scan rate. There are a few issues to consider -

To compare your E0.85 value with values listed in books, it must be corrected since you used an Ag/AgCl reference electrode rather than the Normal Hydrogen Electrode (NHE) that was used when the Standard Values of Electrode Potentials, Eo, were defined from the Nernst expression.

Also the measured electrode potential must be corrected if there is a significant solution resistance (ohmic) between the working and reference electrode. The measured potential then contains an additional component, E (ohmic) = iR. It is best to minimize the possibility of such iR by use of a high background electrolyte concentration (ions that carry the current in the solution) and/or by placing the tip of the reference electrode close to (but not touching) the surface of the working electrode.

The electrode reaction during the scan from +600 mV to 0.0 mV is

$\mathrm{Fe^{III}(CN)_6^{3-} + e^- \rightarrow Fe^{II}(CN)_6^{4-} \hspace{40px} E^\circ = 0.361\: V\: vs.\: NHE\: at\: 25^\circ C} \tag{2}$

The electrode potential, E, is thermodynamically determined by the Nernst relationship

$\mathrm{E = E^\circ + \dfrac{0.0591}{n} \log \left( \dfrac{a_{ox}}{a_R} \right )\tag{3}}$

where aox is the activity of the oxidized species, ferricyanide in the present case, and aR is the activity of the reduced species, ferrocyanide. Experimentally, the activity is affected by the presence of other ions. For practicality, we therefore define a “formal potential” that uses concentration in mole/L rather than activity, which we do not know. The Nernst becomes

$\mathrm{E = E°' + \dfrac{0.0591}{n} \log \left[\dfrac{C_{ox}}{C_R}\right] \tag{4}}$

The “formal” potential, Eo’, depends on the nature of electrolytes in the solution as seen in this list of Formal Potentials for ferri/ferrocyanide in aqueous solutions at 25ºC vs. NHE:

\begin{alignat}{3} & && &&\mathrm{\underline{E°'\: values}}\\ &\mathrm{Fe(CN)_6^{3-} + e^- = Fe(CN)_6^{4-}} &&\hspace{100px}\textrm{0.1 M HCl} &&\hspace{10px}\textrm{0.56 V}\\ & &&\hspace{100px}\textrm{1.0 M HCl} &&\hspace{10px}\textrm{0.71 V}\\ & &&\hspace{100px}\mathrm{1.0\: M\: HClO_4} &&\hspace{10px}\textrm{0.72 V} \end{alignat}

The potential, E, at any point along the I-E wave should reflect the concentration of the ferricyanide and ferrocyanide at the electrode surface in the presence of whatever background electrolyte you are using.

Irreversibility is when the rate of electron transfer is sufficiently slow so that the potential no longer reflects the equilibrium activity of the redox couple at the electrode surface. In such a case, the Ep values will change as a function of the scan rate. The computer-controlled potentiostat has algorithms to evaluate the Ip and EP values, calculate the area under the I-E curves (the integrated charge), and compute an estimated electron transfer rate constant. A unique feature of an electrochemical reaction is that a "reversible" electrode reaction at low scan rates can become "irreversible" at high scan rates. Why is this? [Click on “Concepts” on the home page for additional discussion of fundamentals.]