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Page 49 of the course reader may explain this better, but basically the maximum work (wmax) is related to the cell potential E by E = -w/charge. Rearranging gives you -charge x E = w. Faraday's constant gives the charge of 1 mol of electrons and multiplying by the number of moles of electrons transferred gives you the charge (charge = nF). This gives the equation -nFE = w = deltaG.
In order for an electrochemical reaction to occur/be useful, it has to occur spontaneously, right? Powerful batteries have a super negative deltaG, which is what allows them to do so much useful work. Electricity naturally flows from high to low electric potential (E) in a similar manner to heat flowing from a region of high to low temperature. For example, a battery using two substances with a relatively small difference in E (low voltage) will not generate much current (electrical flow), and therefore can do less work than a battery with a larger deltaE. In this way, G and E are related: G tells you whether the electricity will flow in the direction you want it to on its own, and E tells you how much the electricity "wants" to flow from one substance to another.
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