Determine the \(K_{\text {sp }}\) for the following sparingly soluble compounds, given their molar solubilities: (a) AgI, \(9.1 \times 10^{-9} \mathrm{~mol} \cdot \mathrm{L}^{-1}\); (b) \(\mathrm{Ca}(\mathrm{OH})_{2}\), \(0.011 \mathrm{~mol}-\mathrm{L}^{-1}\); (c) \(\mathrm{Ag}_{3} \mathrm{PO}_{4}, 2.7 \times 10^{-6} \mathrm{~mol}^{-\mathrm{L}}^{-1}\) (d) \(\mathrm{Hg}_{2} \mathrm{Cl}_{2}, 5.2 \times 10^{-7} \mathrm{~mol} \cdot \mathrm{L}^{-1}\).

Molar solubility refers to the number of moles of a substance that can dissolve in one liter of solution until an equilibrium is reached. At equilibrium, no more solute can dissolve because the process of dissolution and crystallization occurs at an equal rate. For example, if the molar solubility of AgI is given as \(9.1 \times 10^{-9} \mathrm{mol/L}\), this means that at equilibrium, \(9.1 \times 10^{-9}\) moles of AgI can be dissolved in one liter of water before the solution becomes saturated.

Understanding molar solubility is important because it helps in predicting the extent to which a solute can dissolve in a solvent, especially for sparingly soluble or insoluble compounds. This concept is widely applied in areas such as pharmacology, environmental science, and material science, where the solubility of compounds affects their usability and impact.

The solubility product constant (Ksp) quantifies the solubility of a compound under equilibrium conditions. It is specific to each compound and temperature dependent. The Ksp is calculated by multiplying the concentrations (in molarity) of the ions in a saturated solution raised to the power of their stoichiometric coefficients from the dissolution equation.

Take, for instance, the compound AgI, with its dissolution represented by the reaction \(\text{AgI}(s) \rightleftharpoons \text{Ag}^+(aq) + \text{I}^-(aq)\). The Ksp calculation would involve squaring the molar solubility of AgI since the stoichiometric coefficients of Ag+ and I- are both 1. The process for calculating Ksp varies depending on the compound's dissolution reaction and must consider the stoichiometry for products correctly to yield accurate results.

Dissolution of compounds is the process by which a solid, liquid, or gaseous solute forms a homogeneous mixture with a solvent. The result of this process, in the context of a solid solute and liquid solvent, is a solution. During dissolution, molecules of the solute separate from each other and become surrounded by solvent molecules.

The dissolution rate and extent can be influenced by factors such as temperature, pressure, molecular interactions, and the nature of the solute and solvent. Solubility is a key measurement in this process and is governed by thermodynamics, specifically the free energy changes related to the process. Not all compounds dissolve equally - while some dissolve readily, others are classified as 'sparingly soluble,' an attribute captured by their solubility product constant (Ksp).

Equilibrium concentration is defined as the concentration of reactants and products in a chemical reaction that remain constant over time when the system is at equilibrium. In the context of solubility, it refers to the concentration of dissolved ions in a saturated solution at a given temperature where the rate of dissolution equals the rate of precipitation.

For instance, in the dissolution of Ca(OH)₂, where the solute dissociates into Ca²⁺ and OH^- ions, the equilibrium concentrations would be the molar solubility for Ca²⁺ and twice the molar solubility for OH^-, because of the stoichiometry 1:2. Those values are then used in the calculation of the Ksp. Understanding equilibrium concentration is crucial for predicting the behavior of a chemical system and for calculating key properties such as the solubility product constant.