Lecture 14 Simple Electrode Kinetics

Introduction

Most of the systems so far considered have assumed facile electron transfer kinetics, with the reaction rate determining step being the mass transport of material to the electrode surface. This lecture considers the opposite scenario: fast mass transport but rate limiting electron transfer kinetics. This is important as it gets to the heart of the power of electrochemistry: one can effectively control the height of the kinetic activation barriers (DG^‡).

Electrode transfer kinetics

When one considers the kinetics of electron transfer reactions it is important to note that the situation is almost identical to ‘normal’ homogenous bulk phase kinetics, but with the difference being that the free energy of the reactant can be controlled by applying a potential.

· Why can an applied voltage affect the kinetics of a reaction?

This question can be answered at a number of different levels. The simplest explanation is to recall that:

and

In words this is to say that the free energy of a reaction is related to both the equilibrium constant K_a of a system and the free energy of a reaction is also related to the electrochemical oxidation potential, E^o. But these are thermodynamic effects – the question is about the kinetics of the reaction.

Transition State Theory (TST)

This theory will be considered in detail in lectures on chemical (bulk) kinetics, but it is a useful model here.

Consider a reduction reaction:

O + ne- g R

By TST, the species, O with gain an electron and go through a transition state. The energy barrier to forming this is called DG^‡_c. The c subscript denotes this to be a cathodic reaction – that is a reduction reaction.

In thermodynamics we have seen that the standard potential and the standard free energy of a reaction are related. Similarly, the free energy barrier of an electrochemical reaction is linked to the applied potential.

This can be seen more clearly in the diagram below:

The whole (dark lines) show the energy barrier when the system is at equilibrium as no potential has been applied. When a potential is applied, the free energy of reactants (O + ne^-) is raised by an amount nFE where E is the applied potential. The reaction coordinate changes to the dotted line and it can be seen that the energy required to reach the transition state is lowered:

At zero applied potential: Energy to reach transition state = DG^‡_c.

When potential E applied: New energy to transition state = DG^‡_c(2)

It can be clearly seen that DG^‡_c(2) is lower than DG^‡_c. As a result the reaction proceeds faster when a negative potential is applied. The converse is true for the back reaction:

R g O + ne^-.

To quantify this further we can see that on applying a potential E:

For the reduction reaction:

DG^‡_c(2) = DG^‡_c + anFE Remember E will have a negative sign (relative to the equilibrium E) so the energy barrier is lower

For the back reaction (oxidation of R) we can write:

DG^‡_a (2) = DG^‡_a – (1-a)nFE So the energy barrier is higher

The quantity a relates to the symmetry of the energy barrier. It has a value between 0 and 1, but is often taken to be 0.5.

• It can be seen how the value DG^‡ relates to the applied potential, but how does this affect the rate of electron transfer?

The Arrhenius equation can be written to take into account the value DG^‡ if we recall that DG^‡ is effectively the activation energy of the reaction by:

E_a = DG^‡ + 2k_bT

We can write the Arrhenius equation for the forward and back reactions:

Forward reaction:

Back reaction:

So now we can substitute into these equations the new values of DG^‡ after a potential has been applied and rearrange to get:

DG^‡_c = DG^‡_c(2) - anFE

and

DG^‡_a = DG^‡_a(2) + (1-a)nFE

So we can write:

But since the first part of both equations are potential independent we do not need to consider them further and can write:

That is:

Now we have two expressions that relate how the forward and back rate constants for an electron transfer reaction at an electrode are affected by the applied potential.

Recall now that:

(cathodic)

(anodic)

and the total current in a reaction is given by:

i = i_c - i_a

So we can write:

Where C_O and C_R are the surface concentrations of the oxidised and reduced species. However, since mass transport is fast we can say that: C_bulk = C_surface.

This means we can substitute into this generic equation our rate constant equations. Also we can fiddle with the quantity E to define this relative to the equilibrium potential E^o and write:

This is the Butler-Volmer equation and very important in understanding electrode kinetics. It is particularly important in situations such as corrosion, where knowing the value of the equilibrium rate constant k° allows one to determine the rate of corrosion (see lecture 15).

This is a slightly simplified treatment of electrode kinetics. Further details and other models are described in:

Bard and Faulkner: Electrochemical Methods

Other ways to change the rate of electrochemical reactions

The situation discussed has only examined the effect of the applied potential. However, it should be intuitively obvious that the electrode structure and material may also have a role in the kinetics of electron transfer. This is because the electron transfer is in fact a tunnelling of an electron from the conduction band of the electrode into the LUMO of the molecule being reduced.

Since different electrodes will have different energy conduction bands, it is clear that changing an electrode material will affect electron transfer rates. This is especially true when one considers semi-conductor electrodes.

A good example are photoconductors. These are semiconductors which conduct electricity when light of the correct wavelength is shone on them. Examples are CdSe and CdS nanoparticles on TiO₂ films as well as p doped silicon.

Such systems can be studied by measuring the oxidation or reduction current of a species on the semiconductor electrode under different levels of illumination.