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Chapter 20: Problem 97

Calculate the solubility of \(\mathrm{Mg}(\mathrm{OH})_{2}\left(K_{\mathrm{sp}}=8.9 \times 10^{-12}\right)\) in an aqueous solution buffered at \(\mathrm{pH}=9.42 .\)

Short Answer

Expert verified
The solubility of Mg(OH)₂ in an aqueous solution buffered at pH 9.42 can be calculated using the given Ksp value (8.9 x 10⁻¹²) and the calculated OH⁻ concentration (10^{-4.58}). The Mg²⁺ concentration, which is equal to the solubility of Mg(OH)₂, is found to be: \[ [\mathrm{Mg(OH)_2}] = \frac{8.9 \times 10^{-12}}{(10^{-4.58})^2} \]

Step by step solution

01

Write the balanced chemical equation

To determine the solubility, we need to start by writing the balanced chemical equation for the dissolution of Mg(OH)₂: \[ \mathrm{Mg(OH)_2(s)} \rightleftharpoons \mathrm{Mg^{2+}(aq)} + 2\ \mathrm{OH^-(aq)} \]
02

Write the Ksp expression

The solubility product constant, Ksp, can be found by writing an expression for the equilibrium of the dissociation of Mg(OH)₂. Since Mg(OH)₂ dissociates into one Mg²⁺ ion and two OH⁻ ions, the Ksp expression will be: \[ K_{sp} = [\mathrm{Mg^{2+}}][\mathrm{OH^-}]^2 \] The given Ksp value for Mg(OH)₂ is 8.9 x 10⁻¹².
03

Calculate the concentration of OH⁻ ions

We are given the pH of the solution as 9.42. Since pH + pOH = 14, we can find the pOH by subtracting the given pH from 14: \[ pOH = 14 - pH = 14 - 9.42 = 4.58 \] Now, we can calculate the concentration of OH⁻ ions using the pOH value: \[ [\mathrm{OH^-}] = 10^{-pOH} = 10^{-4.58} \]
04

Determine the solubility of Mg(OH)₂

We can rearrange the Ksp expression to solve for the concentration of Mg²⁺ ions: \[ [\mathrm{Mg^{2+}}] = \frac{K_{sp}}{[\mathrm{OH^-}]^2} \] Plugging in the known values of Ksp and OH⁻ concentration, we can solve for the concentration of Mg²⁺ ions: \[ [\mathrm{Mg^{2+}}] = \frac{8.9 \times 10^{-12}}{(10^{-4.58})^2} \] Once we have the concentration of Mg²⁺ ions, we can say that it is equal to the solubility of Mg(OH)₂ in the solution, since the amount of Mg²⁺ ions came from the dissolution of Mg(OH)₂. Calculate the solubility of Mg(OH)₂ in this solution: \[ [\mathrm{Mg(OH)_2}] = [\mathrm{Mg^{2+}}] \]

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