Write equations to show whether the solubility of either of the following is affected by pH: (a) \(\mathrm{PbI}_{2} ;\) (b) \(\mathrm{Hg}_{2}(\mathrm{CN})_{2}\).

The Solubility Product, or Ksp, is a special kind of equilibrium constant used for equilibrium between a solid and its respective ions in a solution. For a sparingly soluble compound like \(\text{PbI}_2\), Ksp helps us determine how much of it can dissolve in water. The general formula for Ksp is \[K_{sp} = [product\text{ }ions]_{coefficients}\text{ }multiplied\text{ }together\text{ }at\text{ }equilibrium \]. For example, lead(II) iodide (PbI2) dissociates into one lead ion (Pb^2^+) and two iodide ions (I^-). Its Ksp expression is \[K_{sp} \text{ }\text{ }\text{ }\text{ } = [Pb^{2+}][I^-]^2 \]. In this case, if given a value for Ksp, calculating the exact concentration of ions in solution becomes straightforward.

An Equilibrium Constant (K) expresses the relationship between the concentrations of reactants and products at equilibrium in a reversible chemical reaction. For solubility equilibria, Ksp is the particular equilibrium constant used. The larger the Ksp, the more soluble the substance is in water. Conversely, a small Ksp indicates limited solubility. For \(\text{Hg}_2(\text{CN})_2\), at equilibrium, we have \(\text{Hg}_2(\text{CN})_2 (\text{s}) \rightleftharpoons 2Hg^+(\text{aq}) + 2CN^-(\text{aq})\) and its Ksp expression would be \[K_{sp} = [Hg^+][CN^-]^2 \]. This tells us the ratio of the ion concentrations at equilibrium.

The Common Ion Effect describes the decrease in solubility of an ionic compound when a solution already contains one of the ions present in the compound. Imagine adding more I^- to a solution where \(\text{PbI}_2\) is trying to dissolve. According to Le Chatelier's Principle, the system shifts to re-establish equilibrium by reducing the dissolution of \(\text{PbI}_2\). This doesn't affect every solubility case, only when the ion added directly influences the equilibrium equation of the dissolving compound. However, in analyzing pH, we look at how the ions like \(I^-\text{ }or\text{ }CN^-\)\t respond to H+ or OH- concentrations in the solution.

Acid-Base Reactions can significantly affect the solubility of compounds, especially if the ions formed during dissociation are either weak bases or acids. In the case of \(\text{Hg}_2(\text{CN})_2\), the cyanide ion (CN^-) is a weak base that will readily react with hydrogen ions (H^+) in an acidic environment to form hydrocyanic acid (HCN): \( (\text{CN}^-) + (\text{H}^+) \rightarrow (\text{HCN}) \). This reaction reduces the concentration of CN^- ions in the solution and prompts more \(\text{Hg}_2(\text{CN})_2\) to dissolve to maintain equilibrium, hence, increasing its solubility. Conversely, since iodide (I^-) from \(\text{PbI}_2\)\t doesn't react significantly with H+ or OH-, pH changes won’t notably alter its solubility.