So far we have cited the possibility of doing useful electrical work as the major reason for constructing galvanic cells. Now, we will define exactly what "useful electrical work" means and derive a relationship between work, free energy (G), and cell potential.
>From physics, we know that: Potential (E) = - Work (w) / Charge (q) therefore, w = -qE
Let us now define a quantity called a faraday (F) and let F equal the charge in coulombs per mole of electrons (96,485 C). Then q = nF and w= -nFE.
From thermodynamics we know that ΔG = ΔU -TΔS + Δ(PV) and U = heat + w. Therefore, at constant T and P: ΔG = w. Therefore: ΔG = -nFE and at standard state: ΔGo = -nFEo
Because the sign of ΔG predicts the direction of spontaneous reaction and G and E are directly related by the above equation, we can also use E to predict the direction of spontaneous reaction. As you know, if ΔG is greater than zero, the reaction is non-spontaneous but spontaneous in the reverse direction but if ΔG is less than zero, the reaction is spontaneous as written. ΔG and E have opposite signs from the above equation, therefore, spontaneous reactions will have positive cell potentials.
As alluded to in Galvanic Cells, Heading , E's are intrinsic properties of reactions and therefore do not need to be multiplied by any factors when computing the overall cell potential. However, when adding reduction potentials to calculate the potential of a new reduction reaction there are additional mathematical complications. Those complications arise because potentials are not thermodynamic quantities. According to Hess's Law, only state functions (G, H, S) and not path functions (w, q, E) of a series of reactions may be summed to generate a new value for the overall reaction. Because G is a state function and we have a relationship between G and E (ΔG = -nFE) we can add the ΔG's of the reactions to compute the G of the overall reaction which can then be converted into a value of E. For example, that you can compute the overall cell potential by adding the reduction and oxidation potentials of the half-reactions.