Assuming that no equilibria other than dissolution are involved, calculate the molar solubility of each of the following from its solubility product: (a) \(\mathrm{Ag}_{2} \mathrm{SO}_{4}\) (b) \(\mathrm{PbBr}_{2}\) (c) AgI (d) \(\mathrm{CaC}_{2} \mathrm{O}_{4} \cdot \mathrm{H}_{2} \mathrm{O}\)

The solubility product constant, or simply Ksp, is a crucial principle in understanding the solubility of sparingly soluble salts. It is defined for a salt that partially dissolves in solution and quantifies the degree to which the salt dissociates into its ions at a given temperature. A higher Ksp value indicates that more of the salt can dissolve, leading to greater ion concentrations in solution.

When a chemical equation of a salt dissolving in water is given, like \(Ag_2SO_4 \rightleftharpoons 2Ag^+ + SO_4^{2-}\), the Ksp expression aligns with the stoichiometry of this dissociation. In this example, the Ksp would be given by \([Ag^+]^2[SO_4^{2-}]\). In calculations, we assume molar solubility (s), which represents the moles of salt that can dissolve in 1 liter of water. The relationship between the dissolved ions and solubility allows us to deduce molar solubility from the known Ksp value. For each equation, the ionic products are raised to the power of their stoichiometric coefficients in the balanced equation and are then multiplied together to obtain the Ksp expression.

At chemical equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time. Equilibrium is a dynamic process — reactions continue to occur, but in such a way that the concentrations remain constant.

In the context of solubility, chemical equilibrium involves the balance between the dissolution of a solid and the reformation of the solid from its ions in solution. If we take \(AgI \rightleftharpoons Ag^+ + I^-\), at equilibrium, the rate at which the silver iodide dissolves to form \(Ag^+\) and \(I^-\) ions is equal to the rate at which these ions come together to form solid AgI. This equilibrium is crucial in the molar solubility calculations since it dictates the levels at which the dissolved ions can exist in the solution without precipitating more of the solid.

Dissolution equations represent the process by which a solute breaks down into its constituent ions in a solvent. These equations form the stepping stone of calculating solubility and understanding how different factors might affect it. Starting with the correct dissolution equation is essential, as the stoichiometry will impact how we link the molar solubility to the Ksp value.

For example, the dissolution equation for \(PbBr_2\) is \(PbBr_2 \rightleftharpoons Pb^{2+} + 2Br^-\). This tells us that for each mole of \(PbBr_2\) that dissolves, 1 mole of \(Pb^{2+}\) ions and 2 moles of \(Br^-\) ions are produced in the solution. The coefficients in this equation will directly influence the solubility product expression, as they dictate the relationship between the concentration of the dissolved ions and the molar solubility. Understanding these relationships is critical for accurately calculating the solubility of a compound.