10.2: The connection to ΔG

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Recall that in addition to being used as a criterion for spontaneity, \(\Delta G\) also indicated the maximum amount of non p-V work a system could produce at constant temperature and pressure. And since w_e is non p-V work, it seems like a natural fit that

\[\Delta G = -nFE \nonumber \]

If all of the reactants and products in the electrochemical cell are in their standard states, it follows that

\[\Delta G^o = -nFE^o \nonumber \]

where \(E^o\) is the standard cell potential. Noting that the molar Gibbs function change can be expressed in terms of the reaction quotient \(Q\) by

\[\Delta G = \Delta G^o + RT \ln Q \nonumber \]

it follows that

\[-nFE = -nFE^o + RT \ln Q \nonumber \]

Dividing by \(–nF\) yields

\[E = E^o - \dfrac{RT}{nF} \ln Q \nonumber \]

which is the Nernst equation. This relationship allows one to calculate the cell potential of a electrochemical cell as a function of the specific activities of the reactants and products. In the Nernst equation, n is the number of electrons transferred per reaction equivalent. For the specific reaction harnessed by Volta in his original battery, E^o = 0.763 V (at 25 ^oC) and \(n = 2\). So if the Zn²⁺ and H⁺ ions are at a concentration that gives them unit activity, and the H₂ gas is at a partial pressure that gives it unit fugacity:

\[ E = 0.763\,V - \dfrac{RT}{nF} \ln (1) = 0/763 \nonumber \]