Relation Between Free Energy and Potential of a Galvanic Cell
Given a cell with emf E (in volts), if it is operated reversibly at constant temperature and pressure the electrical work done on passage of an infinitesimal quantity of electricity dq (in coulombs) is the product dqE (in volt-coulombs or joules). If -DG (joules /mole) is the decrease in free energy due to the passage of n faradays of electricity nF (in coulombs), the change in free energy due to the passage of dq coulombs is -DG dq/nF which is equivalent to the electrical work or -DG dq/nF=Edq. Thus
For the general process aA+bB+...=lL+mM+...the change in free energy with the passage of n faradays of charge is DG=lmL+mmM+...-amA-bmB-..., where mi is the partial molar free energy or chemical potential, and since
then
or
which is called the Van't Hoff isotherm.
At equilibrium DG=0 and
where K is called the thermodynamic equilibrium constant.
Combining the above two equations we have
Using the expression 
, the above equation becomes
This can separated into two parts representing the two electrode reactions which comprise the cell
or
