Electrochemical measurements can also be used to determine the solubility products of sparingly soluble salts. The key idea is that the dissolution of an ionic solid can be constructed by combining appropriate electrochemical half-reactions.
Consider the dissolution equilibrium
\[ \mathrm{PbCl_2(s)} \rightleftharpoons \mathrm{Pb^{2+}(aq)} + 2\mathrm{Cl^-(aq)} \]whose equilibrium constant is the solubility product
\[ K_{sp} = [\mathrm{Pb^{2+}}] [\mathrm{Cl^-}]^2 \]Rather than measuring these extremely small equilibrium concentrations directly, we can determine the standard Gibbs energy change for the dissolution reaction from electrochemical measurements.
Suppose the following standard reduction half-reactions are known:
\[ \mathrm{Pb^{2+}(aq)+2e^- \rightarrow Pb(s)} \]and
\[ \mathrm{PbCl_2(s)+2e^- \rightarrow Pb(s)+2Cl^-(aq)} \]The second half-reaction already contains the sparingly soluble salt. Reversing the first half-reaction gives
\[ \mathrm{Pb(s)\rightarrow Pb^{2+}(aq)+2e^-} \]Adding the two reactions cancels both the electrons and the metallic lead, leaving
\[ \boxed{ \mathrm{PbCl_2(s)} \rightleftharpoons \mathrm{Pb^{2+}(aq)} + 2\mathrm{Cl^-(aq)} } \]which is exactly the dissolution reaction whose equilibrium constant is \(K_{sp}\).
Once the standard cell potential for the dissolution reaction has been determined, the standard Gibbs energy change follows from
\[ \Delta G^\circ = -nFE^\circ \]The solubility product is then obtained from
\[ \Delta G^\circ = -RT\ln K_{sp} \]Combining these relationships gives
\[ \ln K_{sp} = \frac{nFE^\circ}{RT} \]Thus, a single electrochemical measurement provides the information needed to determine the solubility product.
Many sparingly soluble salts have extremely small solubility products. As a result, the equilibrium concentrations of their dissolved ions may be difficult to measure accurately by direct analytical methods.
Electrochemistry avoids this difficulty because the cell potential depends on the Gibbs energy change rather than on directly measuring the tiny equilibrium concentrations. Once the cell potential is known, the corresponding value of \(K_{sp}\) follows immediately from thermodynamics.
Big picture: Electrochemical measurements provide an indirect but powerful method for determining solubility products. By constructing the dissolution reaction from appropriate half-reactions, the standard cell potential can be converted into a standard Gibbs energy change and ultimately into the solubility product, linking electrochemistry directly to solution equilibrium.
Use the following standard reduction potentials to calculate \(K_{sp}\) for \(\mathrm{PbCl_2(s)}\) at 25 °C.
| Half-reaction | \(E^\circ\) |
|---|---|
| \(\mathrm{Pb^{2+}+2e^- \rightarrow Pb(s)}\) | \(-0.126\ \text{V}\) |
| \(\mathrm{PbCl_2(s)+2e^- \rightarrow Pb(s)+2Cl^-}\) | \(-0.268\ \text{V}\) |
The target reaction is the dissolution of lead(II) chloride:
\[ \mathrm{PbCl_2(s)\rightleftharpoons Pb^{2+}(aq)+2Cl^-(aq)} \]To obtain this reaction, reverse the lead reduction half-reaction:
\[ \mathrm{Pb(s)\rightarrow Pb^{2+}+2e^-} \]and add it to
\[ \mathrm{PbCl_2(s)+2e^- \rightarrow Pb(s)+2Cl^-} \]After canceling \(\mathrm{Pb(s)}\) and the electrons, the net reaction is
\[ \mathrm{PbCl_2(s)\rightarrow Pb^{2+}(aq)+2Cl^-(aq)} \]The standard cell potential for this reaction is
\[ E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} \] \[ E^\circ_{\text{cell}} = (-0.268)-(-0.126) = -0.142\ \text{V} \]Now use the relationship between \(E^\circ\) and the equilibrium constant:
\[ \ln K = \frac{nFE^\circ}{RT} \]For this reaction, \(n=2\), and \(K=K_{sp}\):
\[ \ln K_{sp} = \frac{(2)(96485)(-0.142)} {(8.314)(298)} \] \[ \ln K_{sp} = -11.1 \] \[ K_{sp} = e^{-11.1} \] \[ \boxed{K_{sp}\approx1.6\times10^{-5}} \]Physical interpretation: The negative value of \(E^\circ_{\text{cell}}\) corresponds to a positive \(\Delta G^\circ\), so the dissolution of \(\mathrm{PbCl_2}\) is not strongly favored under standard conditions. This is consistent with the small value of \(K_{sp}\), which tells us that \(\mathrm{PbCl_2}\) is only sparingly soluble in water.
Big picture: Electrochemistry provides a powerful indirect method for determining equilibrium constants. By combining appropriate half-reactions and relating the resulting cell potential to Gibbs energy, electrochemical measurements can be used to determine solubility products, formation constants, and other equilibrium constants that are difficult to obtain by direct chemical analysis.