Many equilibrium constants can be determined directly by measuring the concentrations of reactants and products at equilibrium. However, this approach becomes difficult when the equilibrium constant is extremely large.
Consider a complexation reaction
\[ \mathrm{M^{n+}+L} \rightleftharpoons \mathrm{ML^{n+}} \]with a formation constant
\[ K_f = \frac{[\mathrm{ML^{n+}}]} {[\mathrm{M^{n+}}][L]} \]If
\[ K_f \gg 1, \]nearly all of the metal ions become bound to the ligand. The equilibrium concentration of the free metal ion may be far below the detection limit of ordinary analytical techniques.
Electrochemistry provides a way to measure these extremely small concentrations.
The Nernst equation relates the cell potential to the activity of the electroactive species. For the reduction
\[ \mathrm{M^{n+}+ne^- \rightarrow M(s)} \]the Nernst equation is
\[ E = E^\circ + \frac{RT}{nF} \ln[\mathrm{M^{n+}}] \]because the activity of the pure metal is unity.
Measuring the cell potential therefore allows the concentration of the free metal ion to be determined, even when that concentration is extremely small.
Once the free metal-ion concentration has been determined electrochemically, the remaining equilibrium concentrations can be obtained from mass balance.
The formation constant is then calculated using
\[ K_f = \frac{[\mathrm{ML^{n+}}]} {[\mathrm{M^{n+}}][L]} \]In this way, electrochemical measurements make it possible to determine equilibrium constants that may be many orders of magnitude larger than can be measured directly by chemical analysis.
Many important chemical systems involve very large equilibrium constants. Examples include metal-ion complexation, precipitation reactions, and certain biological binding processes.
In these systems, the free concentration of one reactant may be so small that direct measurement is impractical. Electrochemistry circumvents this difficulty because the cell potential depends logarithmically on concentration, allowing extremely small concentrations to produce measurable voltage changes.
Big picture: Electrochemical measurements provide much more than cell potentials. Through the Nernst equation, they allow extremely small equilibrium concentrations to be measured, making it possible to determine very large equilibrium constants such as formation constants and solubility products that would otherwise be difficult to obtain experimentally.
The complex ion \(\mathrm{Cu(CN)_4^{2-}}\) forms according to
\[ \mathrm{Cu^{2+}(aq)+4CN^-(aq)\rightleftharpoons Cu(CN)_4^{2-}(aq)} \]with
\[ K_f=1.0\times10^{25} \]Suppose a solution contains
\[ [\mathrm{Cu(CN)_4^{2-}}]=0.0100\ \text{M} \]and
\[ [\mathrm{CN^-}]=0.100\ \text{M} \]Estimate the equilibrium concentration of free \(\mathrm{Cu^{2+}}\). Then calculate the potential of a copper concentration cell comparing this solution with a standard \(1.00\ \text{M}\ \mathrm{Cu^{2+}}\) half-cell at 25 °C.
The formation constant is
\[ K_f= \frac{[\mathrm{Cu(CN)_4^{2-}}]} {[\mathrm{Cu^{2+}}][\mathrm{CN^-}]^4} \]Rearranging to solve for the free copper(II) ion concentration:
\[ [\mathrm{Cu^{2+}}] = \frac{[\mathrm{Cu(CN)_4^{2-}}]} {K_f[\mathrm{CN^-}]^4} \] \[ [\mathrm{Cu^{2+}}] = \frac{0.0100} {(1.0\times10^{25})(0.100)^4} \] \[ [\mathrm{Cu^{2+}}] = 1.0\times10^{-23}\ \text{M} \]This concentration is far too small to measure easily by ordinary chemical analysis, but it can still affect an electrochemical potential.
For a copper concentration cell,
\[ \mathrm{Cu(s)|Cu^{2+}(aq,\ 1.0\times10^{-23}\ M)||Cu^{2+}(aq,\ 1.00\ M)|Cu(s)} \]the standard cell potential is zero because both half-cells use the same redox couple:
\[ E^\circ=0 \]The cell potential comes entirely from the concentration difference:
\[ E= \frac{0.05916}{2} \log\left( \frac{[\mathrm{Cu^{2+}}]_{\text{high}}} {[\mathrm{Cu^{2+}}]_{\text{low}}} \right) \] \[ E= \frac{0.05916}{2} \log\left( \frac{1.00}{1.0\times10^{-23}} \right) \] \[ E= 0.680\ \text{V} \]Therefore,
\[ \boxed{E=0.680\ \text{V}} \]Physical interpretation: The formation constant is so large that essentially all of the copper(II) ions are tied up as \(\mathrm{Cu(CN)_4^{2-}}\), leaving an extremely small concentration of free \(\mathrm{Cu^{2+}}\). Electrochemistry is useful here because the Nernst equation converts that tiny concentration into a measurable voltage.
Big picture: Electrochemistry is not only a way of generating electrical energy—it is also a powerful analytical tool. By relating cell potentials to ion concentrations through the Nernst equation, electrochemical measurements make it possible to determine equilibrium constants, formation constants, and solubility products for systems in which one or more equilibrium concentrations are far too small to measure directly.