Thermodynamics of Cell Reactions

A galvanic cell converts chemical energy into electrical energy, and the amount of electrical energy it produces can be calculated exactly, once two things are known: the total quantity of electric charge moved and the cell's emf driving that charge between the electrodes. If n is the number of moles of electrons exchanged in the overall cell reaction, the electrical energy produced is

The charge carried by exactly one mole of electrons is called one faraday (1 F). Since the charge on a single electron is $1.602\times10^{-19}$ C, one faraday works out to $1\,F = (6.023\times10^{23})\times(1.602\times10^{-19})\ \text{C} \approx 96500$ C. The charge moved by n moles of electrons is therefore $nF$, so the electrical energy produced becomes

This electrical energy is exactly what does electrical work, so the maximum work obtainable from a galvanic cell is

where the negative sign is a bookkeeping convention indicating that the work is done BY the system ON the surroundings. From the second law of thermodynamics, the maximum work obtainable from a process at constant temperature and pressure equals the change in Gibbs free energy of the system, $W_{max} = \Delta G$, so combining the two relations gives one of the single most important equations in electrochemistry:

For a spontaneous cell reaction, $\Delta G$ must be negative — and this equation shows immediately that this requires $E_{cell}$ to be positive. When every species in the cell sits at its standard state, this becomes $\Delta G^{o} = -nFE^{o}_{cell}$. Finally, recalling the standard thermodynamic relationship between free energy change and the equilibrium constant, $\Delta G^{o} = -RT\ln K_{eq}$, and comparing the two expressions for $\Delta G^{o}$ gives $nFE^{o}_{cell} = RT\ln K_{eq}$, which rearranges to …