Consider a general reaction for a galvanic cell such as: 

                    lL + mM qQ + rR   ………  (1)

 Where l, m, q and r are the number of moles of L, M, Q and R respectively.

 The corresponding change in Gibb's free energy ∆G for the reaction above is given by the difference in molal free energy of products and reactants. 

          ∆G = ∆G° (products) - ∆G° (reactants) ………………  (2)

 Where ∆G° is standard free energy,  ∆G is free energy change.

 So:   ∆G = (qG_Q + rG_R ) – (lG_L + mG_M ) ……………………  (3)

If each substance is in standard state;

        ∆G° = (qG°_Q + rG°_R ) – (lG°_L + mG°_M ) ………………  (4)

But from Vanhoff Isotherm, we have

        G_A = G°_A + RT ln a_A ……………………………………. (5)

Where      a_A = Activity of the substance,

                T = Absolute temperature,

And          R = Gas constant.

Activity of a substance is defined as pressure in atmospheres for gases, and is adequately approximated by concentration C; in gram equivalents per liter for many relatively dilute corrosive solutions.

Activity Coefficient = C_A r_A

 Substituting (4) from (5):

         ∆G - ∆G° = (qG_Q - qG°_Q ) + (rG_R - rG°_R ) – (lG_L - lG°_L ) – (mG_M - mG°_M ) .....(5')

 Using this equation (5'):

         ∆G - ∆G° = RT ln a_R + RT ln a_Q – RT ln a_L – RT ln a_M

         ∆G - ∆G° = RT ln [ (aqQ arR ) / ( amM alL )] ………  (6)

   ∆G - ∆G° = RT ln [ (a_products) / (a_reactants)]  ………  (7)  

 But we have     ∆G = - nFE

         ∆G - ∆G° = - nF (E - E°)

                         = RT ln [(a_products) / (a_reactants)]

  E - E° = - {RT/nF} ln [ (a_products) / (a_reactants) ]

 So, we get the NERNST EQUATION as:

              E = E° - {RT/nF} ln [ (a_products) / (a_reactants)]