Find the approximate \(\mathrm{pH}\) range suitable for separating \(\mathrm{Mg}^{2+}\) and \(\mathrm{Zn}^{2+}\) by the precipitation of \(\mathrm{Zn}(\mathrm{OH})_{2}\) from a solution that is initially \(0.010 M\) in \(\mathrm{Mg}^{2+}\) and \(\mathrm{Zn}^{2+}\)

Understanding the concept of pH calculation is essential when dealing with chemical reactions that are influenced by acidity or basicity of the solution. pH is a logarithmic measure of the hydrogen ion concentration \( [H^+] \) and is calculated using the formula \( pH = -\text{log}([H^+]) \). A key point to note is that pH and pOH (the measure of hydroxide ion concentration \( [OH^-] \) ) are related through the equation \( pH + pOH = 14 \) at 25°C. This means, if you know the concentration of hydroxide ions in a solution, you can find the pOH by taking the negative logarithm of \( [OH^-] \) and subsequently calculating the pH by subtracting pOH from 14.

In our exercise, by calculating the pOH from the solubility product constant (Ksp), we could derive the pH range necessary for selective precipitation. This step is crucial in many fields, from environmental science to medicine, where controlling pH can mean the difference between a successful outcome and failure.

The solubility product constant, abbreviated as Ksp, is a useful indicator of the extent to which a compound will dissolve in water. Specifically, Ksp is the equilibrium constant for the dissolving process of a sparingly soluble ionic compound. For a general compound \( AB \) that dissociates into \( A^+ \) and \( B^- \) ions, the solubility product is expressed as \( Ksp = [A^+][B^-] \). This constant is unique to each compound and depends on temperature and the ionic strength of the solution.

In the provided exercise, the Ksp values help us to determine at which pH magnesium hydroxide \( \text{Mg(OH)}_2 \) and zinc hydroxide \( \text{Zn(OH)}_2 \) begin to precipitate from the solution. By setting up the proper ionic equilibrium expression and taking the cube root, as demonstrated with \( \text{Mg(OH)}_2 \) and \( \text{Zn(OH)}_2 \) in the solution, students can solve for the solubility and thus the pH at which these precipitations occur.

Selective precipitation is a principle widely employed in the separation of ions in a mixture. It relies on the fact that different compounds have different solubilities in water, and therefore can be precipitated out of solution at different pH levels. This technique is commonly used in qualitative chemical analysis and wastewater treatment, among other applications.

In the exercise, we have a mixture of \( \text{Mg}^{2+} \) and \( \text{Zn}^{2+} \) ions and we wish to precipitate only \( \text{Zn}(\text{OH})_2 \) while keeping \( \text{Mg}(\text{OH})_2 \) in solution. We achieve this by adjusting the pH to a point where only \( \text{Zn}(\text{OH})_2 \) is insoluble. The pH range that facilitates this is found between the points where each respective metal hydroxide commences to precipitate, thus leveraging the principle of selective precipitation for effective separation.