Question

For which of these reactions is the equilibrium constant called a solubility product? (a) \(\mathrm{Zn}(\mathrm{OH})_{2}(s)+2 \mathrm{OH}^{-}(a q) \rightleftharpoons \mathrm{Zn}(\mathrm{OH})_{4}^{2-}(a q)\) (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

The equilibrium constant can be called a solubility product for reaction (d) \(\mathrm{PbI}_{2}(s) \rightleftharpoons \mathrm{Pb}^{2+}(a q)+2 \mathrm{I}^{-}(a q)\)

Step-by-step solution

Identify the reactions involving solid substances dissolving into ionic constituents

Analyzing each reaction, it can be noticed that not all of them involve a solid substance dissolving into ions. Some involve ions in the solution reacting with each other. So the reactions involving solid substances initially are (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) \(\mathrm{CaCO}_{3}(s)+2 \mathrm{H}^{+}(a q) \rightleftharpoons \mathrm{Ca}^{2+}(a q)+\mathrm{H}_{2} \mathrm{O}(l)+\mathrm{CO}_{2}(g)\), and (d) \(\mathrm{PbI}_{2}(s) \rightleftharpoons \mathrm{Pb}^{2+}(a q)+2 \mathrm{I}^{-}(a q)\)

Identify the reactions in which a solid is dissolving into its ions in water

The reactions identified in the previous step are checked for whether a solid substance is dissolving into its ions in water or not. Here, (a) is not considered since it already excluded in the previous step. Among these reactions, (b) is not a case of a solid dissolving rather it's being formed through reaction of ions. Hence, the reactions (c) and (d) meet the condition of a substance dissolving into ions.

Verification

The remaining reactions (c) and (d) involve a solid in the reaction. However, they need further verification to confirm whether they can be classified under the definition of equilibrium constant known as solubility product. Notice that reaction (c) is an acid-base reaction and the solid CaCO3 is reacting with 2H+ and hence this equilibrium does not signify dissolution into ions in water only. Therefore, among the given reactions, reaction (d) is a true depiction of a solid dissolving into its ions in water and is a classic example where the Ksp, equilibrium constant would be called a solubility product.

Key concepts

Equilibrium Constant

The equilibrium constant, often denoted as \( K \), is a fundamental concept in chemical reactions. It represents the ratio of the concentrations of products to reactants at equilibrium. Each concentration is raised to the power that matches its coefficient in the balanced chemical equation.
For a general reaction \( aA + bB \rightleftharpoons cC + dD \), the equilibrium constant \( K \) is expressed as: \[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]This formula helps determine the direction in which a reaction will proceed under different conditions.
It's important to note that equilibrium constants are specific to a particular temperature, meaning any changes in temperature will affect the value of \( K \). Understanding how to use the equilibrium constant is crucial for solving problems related to chemical equilibrium.

Ionic Dissolution

Ionic dissolution is the process where an ionic compound separates into its individual ions upon dissolving in a solvent, such as water. This process is especially significant in reactions involving salts and minerals.
When a solid ionic compound is placed in water, the solid lattice breaks apart. The ions then become surrounded by water molecules, leading to their separation into individual ions.
For example, the dissolution of lead(II) iodide, \( \mathrm{PbI}_2 \), in water can be represented by the reaction: \[ \mathrm{PbI}_2(s) \rightleftharpoons \mathrm{Pb}^{2+}(aq) + 2\mathrm{I}^-(aq) \] Understanding ionic dissolution is crucial for predicting the solubility of compounds and for applications in various chemical processes.

Ksp

The solubility product constant, represented as \( K_{sp} \), is a specific type of equilibrium constant applied to the dissolution of sparingly soluble salts. It expresses the product of the ionic concentrations at equilibrium, where each concentration is raised to the power of its stoichiometric coefficient.
For a salt \( AB \) that dissociates into ions \( A^+ \) and \( B^- \), the solubility product is given by: \[ K_{sp} = [A^+][B^-] \]Only the concentrations of the ions appear in the \( K_{sp} \) expression because the concentration of the solid does not change.
Therefore, \( K_{sp} \) allows us to calculate the solubility of the compound in a particular solvent and understand the equilibrium states for dissolution processes.

Solid Solubility

Solid solubility refers to the extent to which a solid substance can dissolve in a solvent before reaching saturation. This concept is integral in solutions involving ionic compounds. A solid reaches its solubility limit when the solution holds the maximum amount of dissolved ions, forming a saturated solution.
At this point, the rate of dissolution and the rate of precipitation reach equilibrium. Understanding solid solubility requires acknowledging both the physical and chemical interactions between the solute and solvent.
For example, the solubility of lead(II) iodide in water is limited, which is characterized by the equilibrium description: \[ \mathrm{PbI}_2(s) \rightleftharpoons \mathrm{Pb}^{2+}(aq) + 2\mathrm{I}^-(aq) \] Calculating solid solubility requires knowledge of the \( K_{sp} \), as it directly influences the amount of solid that will dissolve before reaching saturation.