Nernst Equation and the Equilibrium Constant of a Cell Reaction

Standard electrode potentials and the standard cell EMF, $E^{\circ}_{cell}$, are only strictly valid under the defined standard conditions — every solute at exactly $1\ \text{M}$, any gas at exactly $1\ \text{atm}$, and the temperature at $298\ \text{K}$. Real cells are very often used away from these conditions, and the Nernst equation is what extends the idea of cell EMF to any actual set of concentrations.

The full form of the Nernst equation, for a general cell reaction transferring $n$ moles of electrons, is: $E_{cell} = E^{\circ}_{cell} - \frac{2.303RT}{nF}\log Q$ where $Q$ is the reaction quotient of the overall cell reaction, written using the same rules as for any chemical equilibrium expression — products over reactants, with pure solids and liquid solvents omitted (since their 'concentration' does not meaningfully vary). At $298\ \text{K}$, evaluating the constant $2.303RT/F$ gives approximately $0.059\ \text{V}$, giving the working form used throughout this chapter: $E_{cell} = E^{\circ}_{cell} - (0.059/n)\log Q$.

Applied to the Daniell cell reaction $\text{Zn}(s) + \text{Cu}^{2+}(aq) \to \text{Zn}^{2+}(aq) + \text{Cu}(s)$ ($n=2$, $Q=[\text{Zn}^{2+}]/[\text{Cu}^{2+}]$), a cell run with $[\text{Zn}^{2+}] = 0.001\ \text{M}$ and $[\text{Cu}^{2+}] = 0.100\ \text{M}$ gives $Q=0.01$, $\log Q = -2$, and $E_{cell} = 1.10 - (0.059/2)(-2) = 1.10 + 0.059 = 1.159\ \text{V}$ — noticeably higher than the standard $1.10\ \text{V}$, because the product ion ($\text{Zn}^{2+}$) is present well below its standard concentration, making the forward reaction even more favourable than at standard state. The same equation, run in reverse, lets an unknown ion concentration be calculated from a measured EMF — this is exactly the working principle behind ion-selective electrodes and pH meters. …