Cell Potential at Equilibrium

The existence of an open circuit potential indicates that although each phase within the cell is in a state of equilibrium with adjacent phases, the cell as a whole is not at a state of equilibrium. If the components of the two half-cells were brought together there would be a spontaneous chemical reaction. If the reaction was allowed to go to completion there would be a decrease in free energy, i.e., a negative DG. If the two terminals are connected with an electronic conductor a current is produced, electrons are transferred, species are oxidized and reduced. The potential difference between the two terminals is proportional to the free energy change. For a spontaneous reaction (a galvanic cell or energy producer) the cell potential is positive. Thus we write

The van’t Hoff isotherm expresses the energy in terms of the standard free energy change and activities. Activities of the initial states (reactants) are taken as negative and the final states (products) are negative.

Substitution gives

The state of equilibrium between adjacent phases means that any species that exists in both phases can move between the two phases without a change in energy. When two phases are brought together, before equilibrium, a given species has a different chemical potential in each phase, being in a different chemical environment. The state of equilibrium is reached by the transfer of charged particles across the interface until the difference in chemical potential is balanced by a potential difference between the two phases.

We then say that the electrochemical potential of species i, is the same in the two phases P and Q.

or

Thus the electrochemical potential of species in a given phase is given by

Given the relation

,

the electrochemical potential of a species in a given phase can also be written as

.

If two phases are in equilibrium we can set the RHS of the above equation for one phase equal to the same expression for the other phase, and written as the sum of the expression for each species that occupies both phases. If we then solve for the difference in potential in each phase we have

,

which gives us the potential change at the interface. Since the cell potential is the sum of the interface potentials, the cell potential is given by

.