D22.3 Relationships among ΔG°, K°, and E°

The thermodynamics of a redox reaction can be described in terms of Δ_rG° and K°, and these can be further related to E°_cell because they all describe whether the redox reaction is reactant-favored or product-favored at equilibrium.

In a voltaic cell, the difference in Gibbs free energy between products and reactants allows the cell to do electrical work. We represent electrical work done by the cell as w_elec.

The charge of 1 mole of electrons is known as the Faraday’s constant (F):

Hence, the total quantity of charge transferred per mole of a redox reaction is:

In this equation, n is the number of electrons transferred in the balanced redox reaction, which can be obtained from the balanced half-reactions that are added to produce the overall redox reaction. For example, in the reaction:

6 e⁻ are transferred because each half-reaction, after multiplying by a factor to balance electrons, involves 6 e⁻:

When this redox reaction happens once, 6 e⁻ are transferred. When a mole of this reaction takes place, 6 moles of e⁻ are transferred; that is, 6 mol × 96485 C/mol = 578910 C are transferred. Hence, we have:

If we operate a voltaic cell such that the maximum possible electrical work, w_elec, is done and the only work done is electrical work, then:

The negative sign makes sense because a positive E_cell indicates a spontaneous redox reaction while for Δ_rG a negative value indicates a spontaneous reaction.

If all the reactants and products are in their standard states, then the equation becomes:

This equation also links standard cell potentials to equilibrium constants, since:

Therefore:

Thus, if any one of Δ_rG°, K°, or E°_cell is known or can be calculated for a redox reaction, the other two quantities can be determined using the relationships shown below. Moreover, any of the three quantities can be used to determine whether a reaction is product-favored at equilibrium.