Current Equations - Part 1
The current in an electrolytic cell is a measure of the rate of the electrochemical reaction and is proportional to the concentration of the electroactive species. Taking the reversible reduction of an oxidized species to a reduced species, both in solution and not accompanied by purely chemical reactions
with k1 and k2 standing for the forward and backward rate constants, we have the rates of each reaction expressed as
where cO and cR represent the concentrations of the oxidized species and reduced species. These rates can be expressed in terms of current density i by multiplying by the Faraday constant F and the number of electrons transferred n. The Faraday is 96,485 coul mol-1. So the rate equations can be written
and
where k1 and k2 are a function of potential. If the forward and backward current densities are equal there is no net current but there are varying rates of electron transfer in both directions referred to as the exchange current. If the exchange current density is high the net and measurable current density represents a slight bias in the ongoing exchange of electrons in both directions, especially if the potential is near the equilibrium potential. So the net current density is the difference between the cathodic and anodic currents
.
We have written the reaction as a reduction and in this case a cathodic current density is positive. The current density is an exponential function of the potential. Written in terms of a potential E versus a reference electrode the current density expression becomes
where the rate constants kf and kb are independent of potential.
and
The term a is called the symmetry coefficient and usually is 0.5±0.2. R is the gas constant and T is the absolute temperature.
At a certain potential the forward and backward rates are equal and there is no net current. The system is then at equilibrium. The potential E in this case is the equilibrium potential Eeq. Of course we could substitute a single (heterogeneous) rate constant k0 for the forward and backward rate constants. So now we can write a general current density equation as
where E0 is the standard potential for the system. If we separate the cathodic and anodic current densities and set them equal, as they would be at equilibrium, we have
.
This can be rearranged to give
.
When this is solved for Eeq we have
which is our old friend the Nernst equation, derived from a current density equation. This applies to the special case where there is no net current and the system is at equilibrium.
For a system not at equilibrium we can write
.
The term (E-E0) is called the overpotential represented by h. A plot of this equation is given in Figure 1.
Figure 1.
The curve intersects the zero current density line at Eeq. At this point we still have the exchange current density. If we plot the cathodic and anodic current densities separately we can see the exchange currents (Figure 2.).
Figure 2.