The Nernst Equation

The use of standard reduction potentials to predict the course of a reaction is somewhat limited, in that they are only valid under standard conditions. However, they can be used to predict the course of redox reactions under nonstandard conditions.

This process of extrapolation of the properties of a reaction under one set of conditions to those under another set of conditions uses the Nernst equation.

The Nernst Equation

If we know the Gibbs free energy of reaction at under one set of conditions, it can be used to predict the Gibbs free energy of reaction, and from that the reduction potential, under a new set of conditions.

Derivation of the Nernst equation
Consider the reaction:
The Gibbs free reaction energy can be related to the standard Gibbs free reaction energy:
Q is the reaction quotient:
The reduction potentials may be related to the free energies of reaction:  
Therefore, the Nernst equation is the result:

The Nernst equation relates the non-equilibrium properties of a reaction to those at equilibrium.  At equilibrium, the reaction free energy, and hence the potential for the reaction, is zero, and the reaction quotient, Q, is the equilibrium constant, K. We can therefore calculate the equilibrium constant from the standard reduction potential.

The exponential dependence of the equilibrium constant on the reaction potential means that a change in reaction potential of one volt results in a change in the equilibrium constant of seventeen orders of magnitude: hence, E* = +2 V means K = 1034, E* = 0 V means K = 1,  and E* = -2 V means K = 10-34.

When we introduced half-reactions, we saw that we could split up the overall potential for a cell into contributions from the oxidant and the reductant. We can also do this for the Nernst equation, and therefore write a similar expression for each of the reduction couples.

pH dependence of reduction potentials

The Nernst equation can be used to determine the pH dependence of the potential for a given reaction.

Consider the reaction:
The Nernst equation gives the potential:
But, E* is the reduction potential for the Ox/Red couple, E*(Ox/Red), as the standard reduction potential for the H+/H2 couple is zero, and [H2] is given by the partial pressure of H2, which is 1 under standard conditions, and the number of electrons transferred is 2.
The Nernst equation becomes:
The definition of pH: pH = -log10[H+]
So the reduction potential is related to the pH:

In general this equation should be adapted for the number of electrons transferred.

Stability Fields: Water

The values of the reduction potentials for the H+/H2 and O2,H+/H2O redox couples, and their pH dependences, can be used to predict where a given species may be stable in aqueous solution. This stability can be expressed in terms of the stability field (aka Pourbaix diagram).

The Stability Field of Water

When a reducing agent that can reduce water to H2, or an oxidizing agent which can oxidize water to O2, is placed in water, the reduction or oxidation reaction will take place, and the species will decompose, and hence it is unstable. A species is therefore stable if its reduction potential lies in the range 0 < E < +1.23 V, 0 V being E*(H+/H2) and +1.23 V being E*(O2,H+/O2). However, the reduction potentials change with pH, and so the range of stable reduction potentials also changes.

The variation of stable reduction potentials with pH is shown in the stability field plot, and in water, the stable species are those which have reduction potentials lying between the solid black lines.

If the effects of kinetic control are taken into account, the need for the presence of the overpotential predicts stability for those species with reduction potentials in the range between the dashed black lines.

The vertical lines on the diagram have been added to show the range of pH values commonly found in lakes and streams, pH values between 4 and 9. Hence the area in the middle of the diagram represents the area of stability in natural waters.