Question

Write the \(K_{\mathrm{sp}}\) expressions for each of the following compounds: (a) \(\mathrm{Fe}_{3}\left(\mathrm{PO}_{4}\right)_{2},\) (b) \(\mathrm{Ag}_{3} \mathrm{PO}_{4}\), (c) \(\mathrm{PbCrO}_{4}\) (d) \(\mathrm{Al}(\mathrm{OH})_{3}\), (e) \(\mathrm{ZnCO}_{3}\) (f) \(\mathrm{Zn}(\mathrm{OH})_{2}\)

Short answer

\(K_{\mathrm{sp}}\) expressions: (a) \([\mathrm{Fe}^{2+}]^3 \cdot [\mathrm{PO}_{4}^{3-}]^2\) (b) \([\mathrm{Ag}^{+}]^3 \cdot [\mathrm{PO}_{4}^{3-}]\) (c) \([\mathrm{Pb}^{2+}] \cdot [\mathrm{CrO}_{4}^{2-}]\) (d) \([\mathrm{Al}^{3+}] \cdot [\mathrm{OH}^{-}]^3\) (e) \([\mathrm{Zn}^{2+}] \cdot [\mathrm{CO}_{3}^{2-}]\) (f) \([\mathrm{Zn}^{2+}] \cdot [\mathrm{OH}^{-}]^2\).

Step-by-step solution

Understanding the Solubility Product Constant

The solubility product constant, or \(K_{\mathrm{sp}}\), is the equilibrium constant for the dissolving process of a sparingly soluble ionic compound. It is expressed as the product of the concentrations of the ions each raised to the power of its coefficient in the balanced equation for its dissolution.

Writing the Balanced Dissolution Equation for \(\mathrm{Fe}_{3}(\mathrm{PO}_{4})_{2}\)

First, write the balanced dissociation equation for \(\mathrm{Fe}_{3}(\mathrm{PO}_{4})_{2}\) dissolving into its constituent ions: \[\mathrm{Fe}_{3}(\mathrm{PO}_{4})_{2}(s) \rightleftharpoons 3\mathrm{Fe}^{2+}(aq) + 2\mathrm{PO}_{4}^{3-}(aq).\]

Determining \(K_{\mathrm{sp}}\) for \(\mathrm{Fe}_{3}(\mathrm{PO}_{4})_{2}\)

For \(\mathrm{Fe}_{3}(\mathrm{PO}_{4})_{2}\), the \(K_{\mathrm{sp}}\) expression would be the product of the concentration of iron ions cubed and that of phosphate ions squared: \[K_{\mathrm{sp}} = [\mathrm{Fe}^{2+}]^3 \cdot [\mathrm{PO}_{4}^{3-}]^2.\]

Determining \(K_{\mathrm{sp}}\) for \(\mathrm{Ag}_{3}\mathrm{PO}_{4}\)

For \(\mathrm{Ag}_{3}\mathrm{PO}_{4}\), the dissolution equation is \[\mathrm{Ag}_{3}\mathrm{PO}_{4}(s) \rightleftharpoons 3\mathrm{Ag}^{+}(aq) + \mathrm{PO}_{4}^{3-}(aq).\] Hence, the solubility product expression is \[K_{\mathrm{sp}} = [\mathrm{Ag}^{+}]^3 \cdot [\mathrm{PO}_{4}^{3-}].\]

Determining \(K_{\mathrm{sp}}\) for \(\mathrm{PbCrO}_{4}\)

The dissolution of lead chromate, \(\mathrm{PbCrO}_{4}\), is described by the equation \[\mathrm{PbCrO}_{4}(s) \rightleftharpoons \mathrm{Pb}^{2+}(aq) + \mathrm{CrO}_{4}^{2-}(aq).\] The \(K_{\mathrm{sp}}\) expression will therefore be \[K_{\mathrm{sp}} = [\mathrm{Pb}^{2+}] \cdot [\mathrm{CrO}_{4}^{2-}].\]

Determining \(K_{\mathrm{sp}}\) for \(\mathrm{Al}(\mathrm{OH})_{3}\)

For aluminum hydroxide, \(\mathrm{Al}(\mathrm{OH})_{3}\), the dissolution is \[\mathrm{Al}(\mathrm{OH})_{3}(s) \rightleftharpoons \mathrm{Al}^{3+}(aq) + 3\mathrm{OH}^{-}(aq).\] Thus, its solubility product is \[K_{\mathrm{sp}} = [\mathrm{Al}^{3+}] \cdot [\mathrm{OH}^{-}]^3.\]

Determining \(K_{\mathrm{sp}}\) for \(\mathrm{ZnCO}_{3}\)

Zinc carbonate, \(\mathrm{ZnCO}_{3}\), dissociates as \[\mathrm{ZnCO}_{3}(s) \rightleftharpoons \mathrm{Zn}^{2+}(aq) + \mathrm{CO}_{3}^{2-}(aq).\] The \(K_{\mathrm{sp}}\) for zinc carbonate will be \[K_{\mathrm{sp}} = [\mathrm{Zn}^{2+}] \cdot [\mathrm{CO}_{3}^{2-}].\]

Determining \(K_{\mathrm{sp}}\) for \(\mathrm{Zn}(\mathrm{OH})_{2}\)

The solubility reaction for zinc hydroxide, \(\mathrm{Zn}(\mathrm{OH})_{2}\), is \[\mathrm{Zn}(\mathrm{OH})_{2}(s) \rightleftharpoons \mathrm{Zn}^{2+}(aq) + 2\mathrm{OH}^{-}(aq).\] Therefore, the solubility product is \[K_{\mathrm{sp}} = [\mathrm{Zn}^{2+}] \cdot [\mathrm{OH}^{-}]^2.\]

Key concepts

Solubility Product Constant

When a sparingly soluble ionic compound is placed in a solvent such as water, it may reach a state of dynamic equilibrium between the solid and dissolved ions. The extent of dissolution at this equilibrium point is represented by the solubility product constant, denoted as K_sp. This value is crucial, as it helps predict the solubility and helps in comparing the solubility of different compounds under similar conditions.

It's important to remember that K_sp does not indicate the exact solubility of the substance in terms of mass. Instead, it provides information about the concentration of ions in the solution at equilibrium. The higher the value of K_sp, the more soluble the compound is. Calculating the K_sp involves writing the equilibrium expression, where the concentrations of the dissolved ions are raised to the power of their stoichiometric coefficient. Take, for example, the compound calcium fluoride, CaF₂. At equilibrium, the expression would be K_sp = [Ca²⁺][F⁻]². This formula represents the dissolved state of each ion in a saturated solution.

Dissolution Equation

Before we can understand K_sp, let's delve into the dissolution equation, which describes how a compound dissociates into its constituent ions in a solvent. A balanced dissolution equation is essential because it allows us to understand the mole ratios in which the ions are produced and thus construct the K_sp expression accurately.

For instance, when writing the equation for the dissolution of silver chloride, AgCl, we state AgCl(s) \(\rightleftharpoons\) Ag⁺(aq) + Cl⁻(aq). The coefficients in this balanced equation tell us the ratio in which the ions will appear in the solution, and, in turn, these coefficients will be the exponents in the K_sp expression. It is a simple yet powerful way to relate the solid phase to its ionic constituents in the solution.

Ionic Concentration

After understanding the dissolution equation and the K_sp, it's imperative to grasp ionic concentration, as it too plays a critical role in the solubility equilibrium. Ionic concentration is essentially the molarity of ions in a solution, usually shown as moles per liter (M).

For a fully dissolved ionic compound, such as sodium chloride (NaCl), we can assume that the concentrations of Na⁺ and Cl⁻ ions will be equal because they dissociate in a 1:1 ratio. However, for compounds that don't fully dissolve, the determination of ionic concentrations is not as straightforward, and one must account for the limited solubility as dictated by the K_sp.

In practice, to calculate the solubility of a compound, you might set up a reaction table with 'initial', 'change', and 'equilibrium' concentrations, commonly referred to as an ICE table. This systematic approach helps you track changes in concentration from the time you first mix your reactants to when equilibrium is established, ensuring a clear understanding of the system at each stage. A solid grasp of these concepts is fundamental for studying solubility equilibria and their applications in fields such as chemistry and environmental science.