91Ó°ÊÓ

The Solubility Product Constant, denoted as \( K_{sp} \), is a measure of the solubility of a sparingly soluble ionic compound. It represents the maximum product of the ionic concentrations at equilibrium, for ions coming from a dissolved solid. In the context of magnesium hydroxide, \( \text{Mg(OH)}_2 \), its dissociation in water establishes a relationship between the concentrations of its ions. This is represented by the equation:

• \[ \text{Mg(OH)}_2 \rightleftharpoons \text{Mg}^{2+} + 2\text{OH}^- \]

Where the solubility product expression is:

• \( K_{sp} = [\text{Mg}^{2+}][\text{OH}^-]^2 \)

This equation means that if you know the solubility, \( s \), of \( \text{Mg(OH)}_2 \), you can determine the concentrations of its ions in a saturated solution. This directly provides insight into how much of each ion is present when the compound is at equilibrium, particularly under given temperature conditions such as at \( 25^{\circ}C \).

The concentration of hydroxide ions \((\text{OH}^-)\) is crucial to understanding the basic nature of a solution. In the case of \( \text{Mg(OH)}_2 \), when it dissociates in water, it contributes hydroxide ions according to the reaction:

• \[ \text{Mg(OH)}_2 \rightleftharpoons \text{Mg}^{2+} + 2\text{OH}^- \]

The concentration of \( \text{OH}^- \) ions depends on the solubility of the compound. Given a solubility \( s \) of \( 3.2 \times 10^{-4} \ \text{M} \), the concentration of hydroxide ions is calculated as:

• \([\text{OH}^-] = 2s = 6.4 \times 10^{-4} \ \text{M} \)

This calculation shows that the concentration of hydroxide ions is twice the solubility of the compound, reinforcing the point that each molecule of magnesium hydroxide contributes two hydroxide ions.

Determining the concentration of hydronium ions \((\text{H}_3\text{O}^+)\) in a solution is essential to gauge its acidity. This can be calculated using the water ionization constant \( K_w \), which is given by the equation:

• \( K_w = [\text{H}_3\text{O}^+][\text{OH}^-] \)

For water at \( 25^{\circ}C \), \( K_w \) equals \( 1.0 \times 10^{-14} \ \text{M}^2 \). In a solution where the hydroxide ion concentration \([\text{OH}^-]\) is known, the hydronium ion concentration can be solved as:

• \[ [\text{H}_3\text{O}^+] = \frac{1.0 \times 10^{-14}}{6.4 \times 10^{-4}} \]
• This yields \([\text{H}_3\text{O}^+] \approx 1.56 \times 10^{-11} \ \text{M} \)

This calculation highlights the inverse relationship between \([\text{OH}^-]\) and \([\text{H}_3\text{O}^+]\) in maintaining the equilibrium constant of water.

The dissociation equation of a compound defines how it breaks into its component ions in solution. For magnesium hydroxide \( \text{Mg(OH)}_2 \), it dissociates in water according to:

• \[ \text{Mg(OH)}_2 \rightleftharpoons \text{Mg}^{2+} + 2\text{OH}^- \]

This equation is pivotal in studying chemical equilibrium as it tells us about the stoichiometric relationships between the ions produced. When \( \text{Mg(OH)}_2 \) dissolves, the balance between its solid and its ions determines its solubility.
This relationship also explains why in equilibrium states, the concentration of \( \text{Mg}^{2+} \) is directly \( s \), while \( \text{OH}^- \) is \( 2s \). Understanding the dissociation process is key to solving problems involving solubility and equilibrium in chemical reactions.