# The Cell Potential and Gibbs Free Energy

As the cell reaction in an electrochemical cell progresses, electrons move through a wire connecting the two electrodes until the equilibrium point of the cell reaction is reached, at which point the flow of electrons ceases.

As long as the reaction has not reached equilibrium, the electrons being driven through the external circuit may be used to do work.

The work that transfer of a given number of electrons may do depends upon the potential difference between the two electrodes. This potential difference is called the cell potential, and is measured in volts. (When the cell potential is large, transfer of a given number of electrons can do more electrical work than when the cell potential is small.)

Note that when the cell reaction is at equilibrium, the cell potential is zero and no work can be done.

The maximum amount of electrical work a galvanic cell can do at constant temperature and pressure is given by ΔG, the change in the Gibbs energy of the system from the starting situation to the final situation. (In practice this will usually be equal to the Standard Gibbs Energy of Reaction, ΔGºr , for the cell reaction, which is given by the sum of the standard molar Gibbs energies of the products, minus the sum of the standard molar Gibbs energies of the reactants, each standard molar Gibbs energy being multiplied by its stoichiometric coefficient.)

Thus to make thermodynamic measurements upon the cell by measuring the work it can do, it is necessary to ensure that the cell is operating reversibly, as only then does it produce the maximum amount of work, which can be equated with ΔG.

Furthermore, another thermodynamic property, the Reaction Gibbs Energy, ΔGr , (which is entirely different from the standard reaction Gibbs energy, ΔGº, defined above) must be evaluated at a specified composition of the reaction mixture. This makes it necessary to ensure that the cell is operating reversibly at a specific, unchanging composition.

Both requirements may be satisfied by measuring the cell potential when a precisely equal opposing potential is being applied, ensuring that the cell reaction is occurring under conditions of thermodynamic reversibility and that the composition of the cell is constant. (It may make more sense to think of this in terms of measuring the minimum potential that needs to be applied to halt the cell reaction, and equating this value with the cell potential. When the opposing and cell potentials are precisely equal, the driving force for the reaction has been neutralised, and no current flows; the cell reaction is poised to occur, but no change is actually taking place – hence the composition remains constant.) The resulting potential difference is called the zero-current cell potential, designated E.

The relationship between the reaction Gibbs energy, ΔGr , and the zero-current cell potential, E, is given by

where ν is the number of electrons transferred in the reaction, and F, the Faraday constant, has the approximate value 96 485.3 C mol-1.

Note that a negative reaction Gibbs energy (spontaneous cell reaction) corresponds to a positive zero-current cell potential.

The equation also implies that the cell potential is proportional to the slope of the Gibbs energy with respect to the extent of reaction (the definition of ΔG). i.e.  the driving force of the cell (the cell potential) is greater the further from equilibrium  the cell reaction is (as the slope of a graph of Gibbs energy against extent of reaction tends to zero the closer one gets to the point of equilibrium, which corresponds to the Gibbs energy minimum.)