Pure water is saturated with slightly soluble \(\mathrm{PbI}_{2}\) Which of the following is a correct statement concerning the lead ion concentration in the solution, and what is wrong with the others? (a) \(\left[\mathrm{Pb}^{2+}\right]=\left[\mathrm{I}^{-}\right]\); (b) \(\left[\mathrm{Pb}^{2+}\right]=K_{\mathrm{sp}}\) of \(\mathrm{PbI}_{2} ;(\mathrm{c})\left[\mathrm{Pb}^{2+}\right]=\sqrt{K_{\mathrm{sp}}}\) of \(\mathrm{PbI}_{2}\); (d) \(\left[\mathrm{Pb}^{2+}\right]=0.5\left[\mathrm{I}^{-}\right]\)

The ionic compound \(\mathrm{PbI}_{2}\) dissociates into its ions in the solution. First thing in understanding the process is to write down the general dissociation equation for the \(\mathrm{PbI}_{2}\) compound. It's shown as:\(\mathrm{PbI}_{2} \leftrightarrow \mathrm{Pb}^{2+} + 2\mathrm{I}^{-}\)Where, on the LHS we have \(\mathrm{PbI}_{2}\) solid and on the RHS, it's broken down into its individual ions.

Evaluate all given relations between the concentration of \(\mathrm{Pb}^{2+}\)and \(\mathrm{I}^{-}\), respectively the solubility product (Ksp) of \(\mathrm{PbI}_{2}\). (a) If the concentrations of \(\mathrm{Pb}^{2+}\) and \(\mathrm{I}^{-}\) are equal, it contradicts with the stoichiometry of the dissociation. As per the equation, one molecule of \(\mathrm{PbI}_{2}\) gives two \(\mathrm{I}^{-}\) ions. So, this is incorrect.(b) The concentration of \(\mathrm{Pb}^{2+}\) equals to the \(K_{sp}\) of \(\mathrm{PbI}_{2}\). But the \(K_{sp}\) equals the multiplication of the concentrations of \(\mathrm{Pb}^{2+}\) and \(\mathrm{I}^{-}\) , not just the concentration of \(\mathrm{Pb}^{2+}\). So, this is incorrect.(c) The concentration of \(\mathrm{Pb}^{2+}\) equals the square root of \(K_{sp}\) of \(\mathrm{PbI}_{2}\). This is also incorrect based on the same reasons as for (b).(d) The concentration of \(\mathrm{Pb}^{2+}\) equals half the concentration of \(\mathrm{I}^{-}\). Given one \(\mathrm{PbI}_{2}\) molecule yields one \(\mathrm{Pb}^{2+}\) and two \(\mathrm{I}^{-}\) ions when it dissociates, this relation is correct.

Given the relations examined in step 2, only (d) is correct based on the stoichiometry of the dissociation.