Question

The solubility product of \(\mathrm{Mg}(\mathrm{OH})_{2}\) is \(9.0 \times 10^{-12}\). The \(\mathrm{pH}\) of an aqueous saturated solution of \(\mathrm{Mg}(\mathrm{OH})_{2}\) is \((\log 1.8=0.26, \log 3=0.48)\) (a) \(3.58\) (b) \(10.42\) (c) \(3.88\) (d) \(6.76\)

Short answer

The pH of an aqueous saturated solution of \(\mathrm{Mg}(\mathrm{OH})_{2}\) is approximately 10.42.

Step-by-step solution

- Determine the expression for solubility product (Ksp)

The solubility product of \(\mathrm{Mg}(\mathrm{OH})_{2}\) can be expressed as \[K_{sp} = \left[ \mathrm{Mg}^{2+} \right] \left[ \mathrm{OH}^- \right]^2\], assuming \(s\) is the solubility of \(\mathrm{Mg}(\mathrm{OH})_{2}\) in moles per liter.

- Calculate the concentration of hydroxide ions

Given that the Ksp of \(\mathrm{Mg}(\mathrm{OH})_{2}\) is \(9.0 \times 10^{-12}\), we can set up the equation \[9.0 \times 10^{-12} = (s)(2s)^2\] to find \(s\), which will give us \(s = \sqrt[3]{9.0 \times 10^{-12} / 4}\).

- Calculate the solubility \(s\)

Using the approximate logarithms provided, we can estimate the cube root in the equation to find the value of \(s\) which will then be used to calculate \([\mathrm{OH}^-]\), as \(2s\).

- Calculate the pOH

The concentration of \(\mathrm{OH}^-\) ions from the previous steps can then be used to find the pOH using the formula: \[\text{pOH} = -\log \left[ \mathrm{OH}^- \right]\]

- Calculate the pH

Once we have the pOH, the pH can be calculated by using the relation: \[\text{pH} + \text{pOH} = 14\] to find the pH of the solution.

Key concepts

pH Calculation

Understanding the pH scale is crucial when learning about acidity and basicity in chemistry. The pH value is a measure of the hydrogen ion concentration in a solution. It ranges from 0 to 14, where a pH of 7 is neutral, values below 7 represent acidic solutions, and values above 7 represent basic or alkaline solutions.

To calculate the pH of a solution, one typically needs the concentration of the hydrogen ions (a[\text{H}^+]a). However, in the case of a[\text{Mg}(\text{OH})_2]a, which is a base that dissociates to give hydroxide ions (a[\text{OH}^-]a), we calculate the pOH first and then use it to find the pH. The pOH is calculated as the negative logarithm of the hydroxide ion concentration:\[\text{pOH} = -a \text{log}a [\text{OH}^-]a\]Since the sum of pH and pOH is constant at 14 in pure water at 25°C, you can find the pH by subtracting the pOH from 14:\[\text{pH} = 14 - \text{pOH}\]This relationship allows for the translation from the concentration of hydroxide ions in a basic solution to the more common pH scale.

Concentration of Hydroxide Ions

In the context of solubility equilibrium, the concentration of hydroxide ions (a[\text{OH}^-]a) is directly linked to the solubility product (Ksp) of a sparingly soluble base. Given that a[\text{Mg}(\text{OH})_2]a dissociates into a[\text{Mg}^{2+}]a and a[\text{OH}^-]a ions, the Ksp expression for this equilibrium would be:\[K_{sp} = [\text{Mg}^{2+}] a[\text{OH}^-]a^2\]By determining the solubility (s), which represents how much of the compound can dissolve to produce ions, we can find the concentration of hydroxide ions. Typically for a compound like a[\text{Mg}(\text{OH})_2]a, which dissociates to form one a[\text{Mg}^{2+}]a ion and two a[\text{OH}^-]a ions, the concentration of a[\text{OH}^-]a ions will be twice the solubility (2s).

It's important to recognize that the concentration of hydroxide ions will influence the pH of the solution. As the a[\text{OH}^-]a concentration increases, the solution becomes more basic, and the pH will increase accordingly.

Solubility Calculations

Solubility calculations involve finding the amount of a substance that can dissolve in a solvent at equilibrium. The solubility product constant (Ksp) is an essential concept here, representing the maximum product of the ion concentrations that can exist in a saturated solution without precipitating. For a[\text{Mg}(\text{OH})_2]a, the Ksp expression can be interpreted in terms of solubility (s), leading to a cubic equation when considering its dissociation into a[\text{Mg}^{2+}]a and a[\text{OH}^-]a ions in a 1:2 molar ratio:\[K_{sp} = (s)(2s)^2\]Solving this equation for s (the solubility) and then using the concentration of hydroxide ions (2s) allows us to perform the necessary calculations to determine pH, as discussed in the previous sections. The ability to estimate logarithms, as in the use of a[a \text{log}a 1.8 = 0.26, a \text{log}a 3 = 0.48]a provided, helps in solving the roots of the equations and finding the solubility without the need for complex calculators or computational tools.

This integration of solubility with pH and pOH calculations exemplifies the interconnected nature of chemical concepts, demonstrating the holistic approach to understanding chemical equilibria.