Problem 21 - Introductory Chemistry Atoms First

Question

Sparingly soluble magnesium hydroxide dissolves in water to yield an equilibrium magnesium ion concentration of \(1.44 \times 10^{-4} \mathrm{M}\). (a) Write the balanced equilibrium equation for magnesium hydroxide dissolving in water. (b) Write the \(K_{\mathrm{sp}}\) expression for magnesium hydroxide. (c) What is the equilibrium concentration of hydroxide ion? (d) Calculate the value of \(K_{\mathrm{sp}}\) for magnesium hydroxide (show your calculation).

Step-by-Step Solution

Answer

(a) The balanced equilibrium equation for magnesium hydroxide dissolving in water is: \(Mg(OH)_2(s) \rightleftharpoons Mg^{2+}(aq) + 2OH^-(aq)\) (b) The Ksp expression for magnesium hydroxide is: \(K_{sp} = [Mg^{2+}][OH^-]^2\) (c) The equilibrium concentration of hydroxide ion is \(2.88 \times 10^{-4} \mathrm{M}\). (d) The value of Ksp for magnesium hydroxide is approximately \(1.194 \times 10^{-11}\).

Step 1: Write the balanced equilibrium equation for magnesium hydroxide dissolving in water.

Magnesium hydroxide is a sparingly soluble salt which partially dissolves in water. When it dissolves, it forms magnesium ions (Mg²⁺) and hydroxide ions (OH⁻). The balanced equilibrium equation for magnesium hydroxide dissolving in water is given by: \[Mg(OH)_2(s) \rightleftharpoons Mg^{2+}(aq) + 2OH^-(aq)\]

Step 2: Write the Ksp expression for magnesium hydroxide.

The solubility product constant, Ksp, represents the extent to which a compound dissolves in water. For the reaction in Step 1: \[ Mg(OH)_2(s) \rightleftharpoons Mg^{2+}(aq) + 2OH^-(aq)\] The expression for Ksp would be: \[ K_{sp} = [Mg^{2+}][OH^-]^2\]

Step 3: Calculate the equilibrium concentration of hydroxide ion.

Given that the equilibrium magnesium ion concentration is \(1.44 \times 10^{-4} \mathrm{M}\), let's denote this value as [Mg²⁺]. Since for every magnesium ion, there are two hydroxide ions produced, the concentration of hydroxide ion [OH⁻] can be calculated as follows: \[ [OH^-] = 2 \times [Mg^{2+}] = 2 \times 1.44 \times 10^{-4} \mathrm{M} = 2.88 \times 10^{-4} \mathrm{M}\]

Step 4: Calculate the value of Ksp for magnesium hydroxide (show your calculation).

We can now use the Ksp expression and the equilibrium concentrations of Mg²⁺ and OH⁻ to calculate the Ksp value for magnesium hydroxide: \[ K_{sp} = [Mg^{2+}][OH^-]^2 = (1.44 \times 10^{-4})(2.88 \times 10^{-4})^2\] \[ K_{sp} = (1.44 \times 10^{-4})(8.2944 \times 10^{-8})\] \[ K_{sp} = 1.194 \times 10^{-11}\] Thus, the solubility product constant, Ksp, for magnesium hydroxide is approximately \(1.194 \times 10^{-11}\).

Key Concepts

Equilibrium Concentration

Equilibrium concentration is crucial in understanding the solubility of substances in a solution. It characterizes the amount of solute present in a dynamic equilibrium where the rate of dissolving solute is equal to the rate of precipitating solute. When dealing with sparingly soluble compounds like magnesium hydroxide, equilibrium is established between the undissolved solute and the ions in solution.

Magnesium hydroxide, for instance, dissolves in water to a small extent, and the equilibrium concentrations of its ions can be determined experimentally or through subsequent calculations. It's crucial to grasp that even sparing solubility indicates there's a certain, quantifiable amount of the substance in dissolution. The concentrations of the ions at equilibrium become part of a precise mathematical relationship described by the Ksp expression.

Magnesium Hydroxide Solubility

Magnesium hydroxide is often referred to as sparingly soluble. Its solubility is low, meaning it doesn't dissolve well in water, but there is still a definite equilibrium between solid and dissolved species. Solubility is influenced by temperature and the presence of other ions in the solution that might form complexes. The low solubility is important in many applications, including medicinal use where magnesium hydroxide acts as an antacid.

Understanding solubility is key to many fields, including environmental science, medicine, and engineering. When it comes to magnesium hydroxide, knowing its solubility behavior allows predictions about its interactions in different conditions, which can be essential for water treatment processes or the formulation of pharmaceuticals.

Balanced Chemical Equation

A balanced chemical equation is an expression that represents the reactants and products in a chemical reaction with their respective coefficients that balance the number of atoms for each element. This balance is a stipulation of the Law of Conservation of Mass, indicating that atoms are neither created nor destroyed in a chemical reaction.

For magnesium hydroxide's dissolution in water, the balanced equation is: \[Mg(OH)_2(s) \rightleftharpoons Mg^{2+}(aq) + 2OH^-(aq)\]. This equation shows that one molecule of solid magnesium hydroxide dissociates into one magnesium ion and two hydroxide ions in aqueous solution. Balancing equations is an essential skill in chemistry, as it lays the foundation for further calculations such as those related to concentration and stoichiometry.

Concentration Calculations

Concentration calculations involve determining the amount of solute in a given volume of solution, often expressed in molarity (M), which is moles of solute per liter of solution. These calculations are pivotal for predicting the behavior of solutes in solution and for preparing solutions with precise concentrations for reactions or products.

For instance, the solubility product expression is derived from the equilibrium concentrations of the ions, which can be calculated once the equilibrium concentration of one ion is known. In the case of magnesium hydroxide, if the magnesium ion concentration at equilibrium is known, it allows us to work out the hydroxide ion concentration since the stoichiometry is known from the balanced equation. Once the individual concentrations are known, substituting them into the Ksp expression yields the solubility product constant, which is a valuable indicator of the solute's solubility under the given conditions.