Question

For which of the following reactions is the equilibrium constant called a solubility product? (a) \(\mathrm{Zn}(\mathrm{OH})_{2}(s)+2 \mathrm{OH}^{-}(a q) \rightleftarrows \mathrm{Zn}(\mathrm{OH})_{4}^{2-}(a q)\) (b) \(3 \mathrm{Ca}^{2+}(a q)+2 \mathrm{PO}_{4}^{3-}(a q) \rightleftharpoons \mathrm{Ca}_{3}\left(\mathrm{PO}_{4}\right)_{2}(s)\) (c) \(\mathrm{CaCO}_{3}(s)+2 \mathrm{H}^{+}(a q) \rightleftharpoons\) \(\mathrm{Ca}^{2+}(a q)+\mathrm{H}_{2} \mathrm{O}(l)+\mathrm{CO}_{2}(g)\) (d) \(\mathrm{PbI}_{2}(s) \rightleftharpoons \mathrm{Pb}^{2+}(a q)+2 \mathrm{I}^{-}(a q)\)

Short answer

Option (d) \(\mathrm{PbI}_{2}(s) \rightleftharpoons \mathrm{Pb}^{2+}(a q)+2 \mathrm{I}^{-}(a q)\), has a solubility product.

Step-by-step solution

Understand Solubility Product

The solubility product constant ( K_{sp} ) applies to the dissolution of ionic compounds into their constituent ions in a saturated solution. This typically refers to the case where a solid salt dissolves into its ions in solution.

Identify Reaction Formats

A reaction appropriate for K_{sp} is one that involves the dissolution of a solid into its ions. Specifically, the equation should start with a solid (often denoted as (s) ) on the left side and produce ions in aqueous form (denoted as (aq) ) on the right side.

Analyze Each Option

(a) involves both solid and ions on one side, which is not typical for K_{sp} . (b) shows the formation of solid from ions, which describes precipitation, not dissolution. (c) is a dissolution involving multiple species but includes substances beyond just water and ions. (d) shows a solid dissociating into its ions in aqueous form, matching the typical solubility product format.

Determine the Correct Reaction

Option (d), \(\mathrm{PbI}_{2}(s) \rightleftharpoons \mathrm{Pb}^{2+}(a q)+2 \mathrm{I}^{-}(a q)\), represents a typical solubility product scenario with a solid dissolving into its constituent ions in a saturated solution.

Key concepts

Equilibrium Constant

The equilibrium constant, often denoted as \( K \), is a vital concept in chemistry that describes the balance point of a chemical reaction. When a reaction reaches equilibrium, the rate of the forward reaction equals the rate of the reverse reaction. This means that the concentrations of the reactants and products remain constant over time. For any given reaction, the equilibrium constant is expressed as a ratio of the concentrations (or pressures) of the products to the reactants, each raised to the power of their stoichiometric coefficients. This constant provides insight into the position of equilibrium, indicating whether products or reactants are favored.In the context of solubility products, a specific type of equilibrium constant, \( K_{sp} \), is used to describe the dissolution of a solid into ions. For example, in the process of dissolving lead iodide \( \mathrm{PbI}_{2} \rightarrow \mathrm{Pb}^{2+}(aq) + 2\mathrm{I}^{-}(aq) \), the equilibrium constant \( K_{sp} \) would be written as:\[K_{sp} = [\mathrm{Pb}^{2+}] [\mathrm{I}^{-}]^2 \]This expression only includes the ions in solution, as the concentration of the pure solid \( \mathrm{PbI}_{2} \) remains constant, which is why it is not included in the \( K_{sp} \) expression.

Ionic Compounds

Ionic compounds are chemical compounds composed of cations and anions bound together by electrostatic forces. These compounds usually form a crystalline lattice structure that exhibits high melting and boiling points. Common examples include salts like sodium chloride (\( \mathrm{NaCl} \)) and calcium carbonate (\( \mathrm{CaCO}_{3} \)). The ions in these compounds dissociate in water, meaning they separate from the solid structure and disperse as individual, charged particles:- Cations: Positively charged ions- Anions: Negatively charged ionsThis dissociation is a key feature in understanding processes like dissolution, where the ionic compound dissolves to form a homogeneous solution. Ionic compounds typically dissolve in water because the polar water molecules stabilize the ions as they break free from the lattice structure, enabling the ions to interact freely in the solution.

Saturated Solution

A saturated solution is a solution that holds the maximum concentration of solute that can be dissolved at a given temperature and pressure. Beyond this point, any additional solute will not dissolve and will rather remain as a precipitate. Saturation is specific to the solute-solvent combination and vastly depends on factors like temperature. Generally, solubility increases with temperature, allowing more ions to enter into solution before reaching saturation. The concept of a saturated solution is crucial when discussing the solubility product (\( K_{sp} \)), as it represents a state where the dissolved ions are in equilibrium with undissolved solid. This means:- The dissolution rate of the solid equals the rate of crystallization.- Calculations of \( K_{sp} \) depend on the concentrations of ions in this equilibrium state.

Dissolution

Dissolution is the process by which a solute incorporates into a solvent, forming a solution. It involves the breaking of intermolecular forces in the solute and solvent, and the formation of new interactions between the solute and solvent molecules. For ionic compounds, dissolution in water is particularly interesting:- Water molecules surround and interact with the ions, reducing the electrostatic forces keeping them in the lattice.- The ions then disperse throughout the water, resulting in a homogeneous mixture.Dissolution is fundamental when considering the solubility product, \( K_{sp} \), which is only applicable when a solid ionic compound dissolves to form a solution in equilibrium with its solid form. If we consider the reaction \( \mathrm{PbI}_{2}(s) \rightarrow \mathrm{Pb}^{2+}(aq) + 2 \mathrm{I}^{-}(aq) \), dissolution is represented by the change from solid \( \mathrm{PbI}_{2} \) to its dissolved ions, indicating the dynamic balance achievable in solutions.