Equilibrium Constant from Nernst Equation

What happens as the cell runs down

Consider the Daniell cell with its circuit closed, driving the reaction

As current flows, $[\text{Zn}^{2+}]$ keeps rising and $[\text{Cu}^{2+}]$ keeps falling, and the voltmeter reading keeps dropping along with it. Eventually a point is reached where the ion concentrations stop changing and the voltmeter reads exactly zero. Nothing has stopped physically — the forward and reverse processes at each electrode are now proceeding at matched rates. This is simply chemical equilibrium being reached inside the cell.

Setting the Nernst equation to zero

At that point $E_{(\text{cell})} = 0$, so the Nernst equation for this cell becomes

which rearranges to

But the ratio of concentrations at equilibrium is, by definition, the equilibrium constant of the reaction:

so at $T = 298\ \text{K}$,

For the Daniell cell the standard cell potential is $E^\ominus_{(\text{cell})} = 1.1\ \text{V}$, so this relation gives its equilibrium constant directly:

The general relation

The same argument, run on any electrochemical reaction transferring $n$ electrons, gives the standing result

where:

• $E^\ominus_{(\text{cell})}$ — the standard cell potential
• $n$ — number of electrons transferred in the balanced cell reaction
• $K_C$ — the equilibrium constant of that reaction
• $R, T, F$ — as before, with $\dfrac{2.303RT}{F} = 0.059\ \text{V}$ at 298 K …