Question

Write the solubility product expression for each of the following slightly soluble ionic compounds in a saturated aqueous solution: (a) \(\mathrm{Cu}_{2} \mathrm{CO}_{3}(s) \rightleftarrows 2 \mathrm{Cu}^{+}(a q)+\mathrm{CO}_{3}^{2-}(a q)\) (b) \(\mathrm{ZnCO}_{3}(s) \rightleftarrows \mathrm{Zn}^{2+}(a q)+\mathrm{CO}_{3}^{2-}(a q)\) (c) \(\mathrm{Al}_{2}\left(\mathrm{CO}_{3}\right)_{3}(s) \rightleftarrows 2 \mathrm{Al}^{3+}(a q)+3 \mathrm{CO}_{3}^{2-}(a q)\)

Short answer

(a) \(K_{sp} = [\mathrm{Cu}^{+}]^2 [\mathrm{CO}_{3}^{2-}]\); (b) \(K_{sp} = [\mathrm{Zn}^{2+}][\mathrm{CO}_{3}^{2-}]\); (c) \(K_{sp} = [\mathrm{Al}^{3+}]^2 [\mathrm{CO}_{3}^{2-}]^3\).

Step-by-step solution

Understanding the Concept of Solubility Product

The solubility product constant, denoted as \(K_{sp}\), is used to describe the equilibrium between a slightly soluble ionic compound and its ions in a saturated aqueous solution. It is the product of the concentrations of the ions, each raised to the power of their coefficient in the balanced equation.

Write the General Expression for \(K_{sp}\)

For a general solubility reaction where \(aA_{m}B_{n}(s) \rightleftarrows mA^{n+}(aq) + nB^{m-}(aq)\), the solubility product expression is \(K_{sp} = [A^{n+}]^m [B^{m-}]^n\). This equation applies to any ionic compound, using its dissociation into its constituent ions.

Part (a) - Solving for \(\mathrm{Cu}_{2} \mathrm{CO}_{3}\)

Given the reaction: \(\mathrm{Cu}_{2} \mathrm{CO}_{3}(s) \rightleftarrows 2 \mathrm{Cu}^{+}(aq) + \mathrm{CO}_{3}^{2-}(aq)\), we identify the ions and their stoichiometry: 2 moles of Cu⁺ and 1 mole of CO₃²⁻. Thus, the solubility product expression is \(K_{sp} = [\mathrm{Cu}^{+}]^2 [\mathrm{CO}_{3}^{2-}]\).

Part (b) - Solving for \(\mathrm{ZnCO}_{3}\)

The equilibrium reaction for this compound is \(\mathrm{ZnCO}_{3}(s) \rightleftarrows \mathrm{Zn}^{2+}(aq) + \mathrm{CO}_{3}^{2-}(aq)\). There is 1 mole of each ion produced, so the solubility product is \(K_{sp} = [\mathrm{Zn}^{2+}] [\mathrm{CO}_{3}^{2-}]\).

Part (c) - Solving for \(\mathrm{Al}_{2}\left(\mathrm{CO}_{3}\right)_{3}\)

For this reaction: \(\mathrm{Al}_{2}\left(\mathrm{CO}_{3}\right)_{3}(s) \rightleftarrows 2 \mathrm{Al}^{3+}(aq) + 3 \mathrm{CO}_{3}^{2-}(aq)\), recognize the ions formed: 2 moles of Al³⁺ and 3 moles of CO₃²⁻. The solubility product is given by \(K_{sp} = [\mathrm{Al}^{3+}]^2 [\mathrm{CO}_{3}^{2-}]^3\).

Key concepts

Ionic Compounds

Ionic compounds consist of positive and negative ions held together by strong ionic bonds. These compounds typically form crystalline structures and are known for their high melting and boiling points. A common property of ionic compounds is their ability to dissolve in water, producing ions in solution.

When discussing solubility, we often refer to slightly soluble ionic compounds. These are compounds that do not dissolve completely in water yet still form a small amount of ions. Let's say you have an ionic compound like \(\mathrm{Cu}_2 \mathrm{CO}_3\), which dissociates into ions in aqueous solution:
\[ \mathrm{Cu}_2 \mathrm{CO}_3(s) \rightleftarrows 2 \mathrm{Cu}^+(aq) + \mathrm{CO}_3^{2-}(aq) \].
This means that in a saturated solution, the ionic compound is in dynamic equilibrium with its ions. The balance between the undissolved solid and the dissolved ions is described by the solubility product expression.

Saturated Solution

A saturated solution is a state where the maximum amount of solute has dissolved in the solvent at a given temperature and pressure. No more solute can dissolve unless the conditions change. This is a critical concept when talking about ionic compounds dissolving in water.

At saturation, the rate of dissolution of the ionic compound equals the rate of precipitation of the ions back into the solid form. This balance defines the point at which a solution becomes saturated.

• In a saturated solution of \(\mathrm{ZnCO}_3\), for example, zinc ions \(\mathrm{Zn}^{2+}\) and carbonate ions \(\mathrm{CO}_3^{2-}\) are in equilibrium with the solid \(\mathrm{ZnCO}_3\) compound.
• Changes in temperature or pressure can alter the solubility, leading to either more dissolution or more precipitation of the solute.

Understanding the properties of saturated solutions helps in explaining phenomena like the extent to which an ionic compound will dissolve in water.

Equilibrium Expression

The equilibrium expression for the dissolution of an ionic compound is represented by the solubility product constant, \(K_{sp}\). This expression involves the concentrations of the ions when equilibrium is reached in a saturated solution.

To construct the \(K_{sp}\) expression, take the balanced equation for the dissolution reaction and write out the product of the ion concentrations, each raised to the power of their stoichiometric coefficient from the balanced equation.

• For example, for \(\mathrm{Al}_{2}(\mathrm{CO}_{3})_{3}\):
  The dissociation equation is \[\mathrm{Al}_2(\mathrm{CO}_3)_3(s) \rightleftarrows 2 \mathrm{Al}^{3+}(aq) + 3 \mathrm{CO}_3^{2-}(aq)\].
  The \(K_{sp}\) expression is \([\mathrm{Al}^{3+}]^2 [\mathrm{CO}_3^{2-}]^3\).

This expression quantifies the equilibrium position between the undissolved solid and the ions in solution.

A higher \(K_{sp}\) value indicates greater solubility, while a lower \(K_{sp}\) suggests limited solubility. Understanding \(K_{sp}\) expressions is crucial for predicting the solubility behavior of various ionic compounds.