Question

The fluoride ion concentration in a saturated solution of magnesium fluoride, \(\mathrm{MgF}_{2}\), is \(2.3 \times 10^{-3} \mathrm{M}\). Calculate the value for the equilibrium constant.

Short answer

The equilibrium constant \( K_{sp} \) is approximately \( 6.08 \times 10^{-9} \).

Step-by-step solution

Write the Dissociation Equation

The dissociation of magnesium fluoride in water can be represented by the equation: \[ \mathrm{MgF}_{2 (s)} \rightleftharpoons \mathrm{Mg}^{2+}_{(aq)} + 2 \mathrm{F}^-_{(aq)} \] This equation shows that one molecule of \( \mathrm{MgF}_2 \) dissociates into one magnesium ion and two fluoride ions.

Determine Ion Concentrations

According to the equation, for every 1 mole of \( \mathrm{MgF}_2 \) that dissolves, 1 mole of \( \mathrm{Mg}^{2+} \) ions and 2 moles of \( \mathrm{F}^- \) ions are produced. Given that the fluoride ion concentration \([\mathrm{F}^-]\) is \(2.3 \times 10^{-3} \ \mathrm{M}\), the magnesium ion concentration \([\mathrm{Mg}^{2+}]\) must be \(\frac{1}{2} \times 2.3 \times 10^{-3} \ \mathrm{M}\ = 1.15 \times 10^{-3} \ \mathrm{M}\).

Write the Expression for the Solubility Product Constant

The solubility product constant \( K_{sp} \) for the dissociation can be expressed as: \[ K_{sp} = [\mathrm{Mg}^{2+}][\mathrm{F}^-]^2 \] This expression captures the relationship between the ion concentrations at equilibrium.

Insert the Ion Concentrations into the Expression

Substitute the known concentrations into the \( K_{sp} \) expression: \[ K_{sp} = (1.15 \times 10^{-3}) \times (2.3 \times 10^{-3})^2 \] Calculate this step by step: \((2.3 \times 10^{-3})^2 = 5.29 \times 10^{-6}\), and then \( K_{sp} = (1.15 \times 10^{-3}) \times 5.29 \times 10^{-6}\).

Calculate the Solubility Product Constant

Complete the multiplication to find \( K_{sp} \): \[ K_{sp} = 1.15 \times 5.29 \times 10^{-9} = 6.0835 \times 10^{-9} \] So, the equilibrium constant \( K_{sp} \) is approximately \( 6.08 \times 10^{-9} \).

Key concepts

Dissociation Equation

When a compound dissolves in water, it breaks apart into its individual ions. This process is called dissociation. For magnesium fluoride \( \mathrm{MgF}_2 \), when it dissolves in water, it splits into magnesium ions \( \mathrm{Mg}^{2+} \) and fluoride ions \( \mathrm{F}^- \). The dissociation equation is written as: \[ \mathrm{MgF}_{2 (s)} \rightleftharpoons \mathrm{Mg}^{2+}_{(aq)} + 2 \mathrm{F}^-_{(aq)} \] This equation tells us the ratio in which ions are formed:

• 1 molecule of \( \mathrm{MgF}_2 \) produces 1 \( \mathrm{Mg}^{2+} \) ion.
• 1 molecule of \( \mathrm{MgF}_2 \) produces 2 \( \mathrm{F}^- \) ions.

Understanding this ratio is important for calculating ion concentrations in a solution.

Solubility Product Constant

The Solubility Product Constant, or \( K_{sp} \), is a specific type of equilibrium constant that applies to the solubility of ionic compounds. It tells us how much compound can dissolve in a solution to form its respective ions. For our case with \( \mathrm{MgF}_2 \), the expression for \( K_{sp} \) is given by: \[ K_{sp} = [\mathrm{Mg}^{2+}][\mathrm{F}^-]^2 \] This expression reflects how the solubility product relies on the concentrations of the ions in the solution. The exponent 2 on \( [\mathrm{F}^-] \) comes from two fluoride ions being produced from one formula unit of \( \mathrm{MgF}_2 \). Knowing \( K_{sp} \) helps to predict whether a precipitate will form or dissolve in a given solution.

Ion Concentration

Ion concentration is crucial for understanding the behavior of dissolved ionic compounds. In the dissociation of \( \mathrm{MgF}_2 \), for every 1 mole of \( \mathrm{Mg}^{2+} \) produced, 2 moles of \( \mathrm{F}^- \) are formed. If you know one ion's concentration, you can determine the other using the dissociation ratio. For instance, if the concentration of \( [\mathrm{F}^-] \) is \( 2.3 \times 10^{-3} \ \mathrm{M} \), we find \( [\mathrm{Mg}^{2+}] \) by dividing \( [\mathrm{F}^-] \) by 2 because there are twice as many \( \mathrm{F}^- \) ions.

• \( [\mathrm{Mg}^{2+}] = \frac{1}{2} \times [\mathrm{F}^-] = 1.15 \times 10^{-3} \ \mathrm{M} \)

Understanding these concentrations is key when calculating other aspects of the solution, such as the \( K_{sp} \).