Electrochemical Cell and Gibbs Energy of the Reaction

Electrical work and Gibbs energy

The electrical work obtainable from a cell in one second equals the potential multiplied by the charge that flows. To extract the maximum possible work from a galvanic cell, the charge must be passed reversibly — under that ideal condition, the reversible work done by the cell equals the decrease in its Gibbs energy.

If the cell's potential is $E_{(\text{cell})}$ and the charge passed corresponds to $n$ moles of electrons (charge $= nF$), then the Gibbs energy change of the reaction, $\Delta_r G$, is

The minus sign reflects that a spontaneous cell reaction (positive $E_{(\text{cell})}$) corresponds to a negative $\Delta_r G$, as thermodynamics requires.

Why $n$ matters even though $E_{(\text{cell})}$ does not

$E_{(\text{cell})}$ is an intensive quantity — it does not depend on how much material reacts — but $\Delta_r G$ is extensive, and its numerical value depends on how the balanced equation is written. For instance, writing the Daniell reaction as

gives a different $\Delta_r G$ than doubling every coefficient,

even though $E_{(\text{cell})}$ itself is identical in both cases, because $n$ (the electrons transferred per mole of the reaction as written) has doubled.

Standard Gibbs energy of the reaction

When every reacting species is at unit concentration, $E_{(\text{cell})} = E^\ominus_{(\text{cell})}$, and the relation becomes

where:

• $\Delta_r G^\ominus$ — the standard Gibbs energy change of the cell reaction
• $n$ — moles of electrons transferred per mole of reaction as balanced
• $F$ — the Faraday constant
• $E^\ominus_{(\text{cell})}$ — the standard cell potential

This is significant beyond electrochemistry itself: it turns a simple voltage measurement on a cell into a direct route to $\Delta_r G^\ominus$, a core thermodynamic quantity for the reaction — no calorimetry required.

Linking back to the equilibrium constant …