The solubility of \(\mathrm{Mg}(\mathrm{OH})_{2}\) in pure water is \(9.57 \times 10^{-3} \mathrm{~g}\) litre \(^{-1}\). Calculate the pH of its saturated solution. Assume \(100 \%\) ionisation.

Molar solubility is a term used to describe the maximum amount of a substance that can dissolve in a liter of solvent to form a saturated solution under specified conditions. For instance, in the context of \(\mathrm{Mg}(\mathrm{OH})_{2}\), determining its molar solubility involves dividing the solubility by the substance's molar mass. The molar mass here is calculated by summing the atomic masses of magnesium, oxygen, and hydrogen atoms within one formula unit of \(\mathrm{Mg}(\mathrm{OH})_{2}\). This concept is fundamental for calculating the concentrations of the ions in the solution which is the starting point for many equilibrium problems including pH calculations.

When discussing hydroxide ion concentration, we refer to the amount of hydroxide ions \(\mathrm{OH}^{-}\) present in a solution. Depending on the degree of ionization of the solute, such as \(\mathrm{Mg}(\mathrm{OH})_{2}\) which ionizes to form \(\mathrm{Mg}^{2+}\) and \(\mathrm{OH}^{-}\) ions, we can calculate the hydroxide ion concentration. Since each unit of \(\mathrm{Mg}(\mathrm{OH})_{2}\) produces two \(\mathrm{OH}^{-}\) ions upon complete ionization, the concentration of \(\mathrm{OH}^{-}\) ions is twice the molar solubility of the salt. This concentration is vital for understanding not only the solute's solubility but also the basic properties of the solution.

pOH is a measure of how alkaline (basic) a solution is. It is calculated by taking the negative logarithm (base 10) of the hydroxide ion concentration. With \(pOH = -\log_{10} [OH^-]\), we are able to quantitatively assess the basicity of the solution. The pOH scale runs from 0 to 14, with lower values being more alkaline. Knowing pOH is also crucial as it relates directly to pH through the relation \(pH + pOH = 14\) at 25°C. This means that if you know either the pH or the pOH of a solution, you can easily determine the other. In the exercise, calculating pOH is a stepping stone to finding the pH, revealing the acid-base nature of the solution.

Acid-base equilibrium involves the balance between acid and base concentrations in a solution. This equilibrium is integral in understanding pH and pOH values. The key principle guiding these equilibria is the autoionization of water where \(H_2O\) dissociates into hydrogen (H+) and hydroxide (OH-) ions. In the exercise, we assume 100% ionisation of \(\mathrm{Mg}(\mathrm{OH})_{2}\), which simplifies our calculations. Equilibrium constants, like \(K_w\) for water, are used in these calculations but weren't needed directly in this example. Instead, we used the relationship between pOH and \(OH^-\) concentration, which derives from this equilibrium concept, to finally arrive at the pH of the solution.