Problem 52 - Chemical Principles: The Quest for Insight

Question

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) \(\mathrm{Hg}_{2} \mathrm{Cl}_{2}, 5.2 \times 10^{-7} \mathrm{~mol} \cdot \mathrm{L}^{-1}\).

Step-by-Step Solution

Answer

Ksp for A) AgI: 8.28 x 10^-17, B) Ca(OH)2: 5.88 x 10^-6, C) Ag3PO4: 1.85 x 10^-18, and D) Hg2Cl2: 1.41 x 10^-17.

Step 1: Calculate AgI Ksp

The dissolution of AgI in water can be represented by the equation \[ \text{AgI}(s) \rightleftharpoons \text{Ag}^+(aq) + \text{I}^-(aq) \] Each mole of AgI that dissolves produces 1 mole of Ag+ and 1 mole of I-. If the molar solubility is 9.1 x 10^-9 mol/L, then the concentrations of Ag+ and I- at equilibrium are each 9.1 x 10^-9 M. The Ksp is the product of these concentrations: \[ Ksp = [\text{Ag}^+][\text{I}^-] = (9.1 \times 10^{-9}) \times (9.1 \times 10^{-9}) \] \[ Ksp = 8.28 \times 10^{-17} \]

Step 2: Calculate Ca(OH)2 Ksp

The dissolution of Ca(OH)2 can be represented by the equation \[ \text{Ca(OH)}_2(s) \rightleftharpoons \text{Ca}^{2+}(aq) + 2\text{OH}^-(aq) \] For each mole of Ca(OH)2 that dissolves, there is 1 mole of Ca2+ and 2 moles of OH- produced. The molar solubility of Ca(OH)2 is 0.011 mol/L, thus the concentrations of Ca2+ and OH- at equilibrium are 0.011 M and 2(0.011) M respectively. The Ksp can be calculated as: \[ Ksp = [\text{Ca}^{2+}][\text{OH}^-]^2 = (0.011) \times (2 \times 0.011)^2 \] \[ Ksp = 5.88 \times 10^{-6} \]

Step 3: Calculate Ag3PO4 Ksp

The dissolution of Ag3PO4 can be written as: \[ \text{Ag}_{3}\text{PO}_{4}(s) \rightleftharpoons 3\text{Ag}^+(aq) + \text{PO}_4^{3-}(aq) \] The molar solubility of Ag3PO4 is 2.7 x 10^-6 mol/L, producing 3 times as many Ag+ ions and one PO43- ion per formula unit dissolved. The concentrations of Ag+ and PO43- are therefore 3(2.7 x 10^-6) M and 2.7 x 10^-6 M, respectively. The Ksp is: \[ Ksp = [\text{Ag}^+]^3[\text{PO}_4^{3-}] = (3 \times 2.7 \times 10^{-6})^3 \times (2.7 \times 10^{-6}) \] \[ Ksp = 1.85 \times 10^{-18} \]

Step 4: Calculate Hg2Cl2 Ksp

The dissolution of Hg2Cl2 can be represented by: \[ \text{Hg}_2\text{Cl}_2(s) \rightleftharpoons \text{Hg}_2^{2+}(aq) + 2\text{Cl}^-(aq) \] The molar solubility of Hg2Cl2 is 5.2 x 10^-7 mol/L, and the dissolution produces 5.2 x 10^-7 M Hg22+ and twice that amount in Cl- ions. The Ksp is: \[ Ksp = [\text{Hg}_2^{2+}][\text{Cl}^-]^2 = (5.2 \times 10^{-7}) \times (2 \times 5.2 \times 10^{-7})^2 \] \[ Ksp = 1.41 \times 10^{-17} \]

Key Concepts

Molar Solubility

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.

Ksp Calculation

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

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

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.