Themes > Science > Chemistry > Electrochemistry > Electrochemical impedance spectroscopy > Electrode Kinetics

Introduction
The rate of the electron transfer reaction at the electrode surface can be varied by changing the applied voltage. In this section we note the main relationships between the heterogeneous rate constants for electron transfer and the voltage, and introduce the terms reversible and irreversible in the context of electrolysis reactions.

Electron Transfer and Energy levels
The key to driving an electrode reaction is the application of a voltage (V). If we consider the units of volts

V = Joule/Coulomb

we can see that a volt is simply the energy (J) required to move charge (c). Application of a voltage to an electrode therefore supplies electrical energy. Since electrons possess charge an applied voltage can alter the 'energy' of the electrons within a metal electrode.

The behaviour of electrons in a metal can be partly understood by considering the Fermi-level (EF). Metals are comprised of closely packed atoms which have strong overlap between one another. A piece of metal therefore does not possess individual well defined electron energy levels that would be found in a single atom of the same material. Instead a continum of levels exist with the available electrons filling the states from the bottom upwards. The Fermi-level corresponds to the energy of the highest occupied orbitals.

This level is not fixed and can be moved by supplying electrical energy (see above figure). Electrochemist's are therefore able to alter the energy of the Fermi-level by applying a voltage to an electrode.
Depending upon the position of the Fermi level it may be thermodynamically feasible to reduce/oxidise a species in solution. The figure below shows the Fermi-level within a metal along with the orbital energies (HOMO and LUMO) of a molecule (O) in solution.
 

On the left hand side the Fermi-level has a lower value than the LUMO of (O). It is therefore thermodynamically unfavourable for an electron to jump from the electrode to the molecule. However on the right hand side, the Fermi-level is above the LUMO of (O), now it is thermodynamically favourable for the electron transfer tooccur, ie the reduction of (O. Whether the process occurs depends upon the rate (kinetics) of the electron transfer reaction.


The Rate of Electron Transfer
We begin by considering the one electron transfer reaction

where kred and kox are the rate constants for the reductive and oxidative steps respectively. Assuming there are arbitary amounts of (O) and (R) in the solution the total current flowing i is the sum of the reductive ic and oxidative ia currents

where A is the electrode area, F the Faraday constant, n the number of electrons transferred and [ ]o the surface concentration of either (O) or (R). Using transition state theory from chemical kineticsit is possible to relate the free energies of activation to the rate constants kox or kred. These are predicted by

The free energies of activation for the electrode reaction are related to both the chemical properties of the reactants/transition state and the response of both to potential. With a small amount of rearrangement it is possible to show that the rate constants (reductive constant shown) show the following potential dependent behaviour

the first exponential can be seen to be independent of the voltage. The second contains the term E - Ee, which is the difference between the applied voltage E and the voltage established by the mixture of (O) and (R) at equilibrium. The term alpha reflects the sensitivity of the transition state to the applied voltage. If alpha = 0 then the transition state shows no potential dependence. Typically alpha = 0.5 this means that the transition state responds to potential in a manner half way between the reactants and the products response.

By using the above approach it is possible to derive the Butler-Volmer equation, which is the fundamental relationship between the current flowing and the applied voltage.



This expression shows how the current will respond to changes in potential, the value of alpha and the quantity io which is called the exchange current (density) .

Here we wish to see how the voltage influences the current in the absence of concentration effects. To do this we will assume that the a solution is well mixed ie that the surface and bulk concentrations are identical which will be reasonable under condition of small current flow. Now the Butler-Volmer equation simplifies to




Without concentration and therefore mass transport effects to complicate the electrolysis it is possible to establish the effects of voltage on the current flowing. In this situation the quantity E - Ee reflects the activation energy required to force current i to flow. Plotted below are three curves for differing values of io with alpha =0.5.

For each curve when E - Ee = 0 then no current flows since the system is in total equilibrium. However as a voltage different to that of Ee is applied then different responses are observed depending upon the value of io. When io is 'large' (curve a) then a small change in E -Ee results in a large current change. Essentially there is little or no activation barrier to either of the electrolysis reactions. For this case the electrode reaction is said to be reversible since both kred and kox are large. At the other extreme whenio is very 'small' (curve c) then a large value of E -Ee is needed to alter the current. This reflects the fact that there is now a high barrier to activation and so the rates of the reduction and oxidation processes become slow. Electrode reactions of this type are termed irreversible. Intermediate behaviour is generally referred to as quasi-reversible (curve b). Not surprisingly the different rates of electrode kinetics give rise to substantially different behaviour in voltammetric and impedance analysis.


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