Free energy, Cell EMF, and Equilibrium Constants

We have observed that voltaic cells use redox reactions that occur spontaneously.  Any reaction that can occur in a voltaic cell to produce a positive emf must be spontaneous.  Consequently, it is possible to decide whether a redox reaction will be spontaneous by using half-cell potentials to calculate the emf associated with it.  

We have seen that the change in Gibbs free energy, ΔG is a measure of the spontaneity of a process that occurs at constant temperature and pressure.  Since the emf of a redox reaction indicates whether the reaction is spontaneous, we would expect some relationship to exist between emf and free energy change.  This relationship is given by the following equation.

ΔG  = -nFE_cell

In this equation n is the number of moles of electrons transferred in the reaction and F is Faraday's constant, named after Michael Faraday.  Faraday's constant is the quantity of electrical charge on one mole of electrons.  This quantity of charge is called a faraday, F:

1 F = 96,500 C/mol e^- or 96,500 J/V-mol e^-

Both n and F are positive quantities.  Thus, a positive value of E, the cell potential, leads to a negative value of  ΔG.  So keep in mind that a positive value of E and a negative value of ΔG both indicate that a reaction is spontaneous.  

When both the reactants and the products are in their standard states, the equation above can be modified to give:

ΔG^o  = -nFE^o_cell

The measurement of cell potentials gives us another way to obtain equilibrium constants.  We can take the equation above and an equation relating free energy and the equilibrium constant, that we discussed in thermodynamics and combine them as shown below.

ΔG^o  = -nFE^o_cell

ΔG^o  = -RT ln K

nFEocell = RT ln K

  
         RT        2.303 RT
Eocell = --- ln K = -------- log K
         nF           nF