D40.1 Relationships among Δ_rG°, K°, and E°_cell

A redox reaction can be described in terms of Δ_rG° and K°, and these thermodynamic variables can be 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 mol 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 in Figure 1. Moreover, any of the three quantities can be used to determine whether a reaction is product-favored at equilibrium.

Exercise 1: Standard Gibbs Free Energy Change for a Redox Reaction

D40.2 Nernst Equation

Now that the connection has been made between standard Gibbs free energy change and standard cell potentials, we can consider the situation under nonstandard conditions. Recall that Δ_rG is related to Δ_rG° by the reaction quotient, Q, and a similar relationship can be applied to cell potentials:

Rearranging the variables gives the Nernst equation:

The Nernst equation can be used to calculate the cell potential at concentrations or partial pressures that differ from standard conditions, which is often the case for voltaic cells.

D40.3 Concentration Cells

A concentration cell is a special type of voltaic cell where the electrodes are the same material but the half-cells have different concentrations of soluble species. Because one or both half-cells are not under standard-state conditions, the half-cell potentials are unequal, and there is a potential difference between the half-cells. That potential difference can be calculated using the Nernst equation.

For example, consider this concentration cell at 25°C:

The standard cell potential is 0 V because the anode and cathode involve the same reaction; however, the process is spontaneous because if equal volumes of the two half-cell solutions were mixed, the concentration of Zn²⁺ would change to the average of the initial concentrations, namely, to 0.30 M. The cell can do work because the concentrations of Zn²⁺ change.

The Nernst equation verifies that the process is spontaneous at the given conditions, because it shows that E_cell > 0 V:

In a concentration cell, the standard cell potential (E°_cell) is always zero. In order to have a spontaneous forward reaction, and hence have a positive cell potential (E_cell), the reaction quotient Q must be less than 1 (when Q < 1, ln(Q) < 0). As the reaction proceeds, the concentrations change, Q approaches 1 and E_cell approaches 0 V.