Question

The concentration of \(\mathrm{Mg}^{2+}\) in seawater is \(0.052 M\). At what \(\mathrm{pH}\) will \(99 \%\) of the \(\mathrm{Mg}^{2+}\) be precipitated as the hydroxide salt? \(\left[K_{\text {sp }}\right.\) for \(\left.\mathrm{Mg}(\mathrm{OH})_{2}=8.9 \times 10^{-12} .\right]\)

Short answer

At pH ≈ 10.14, 99% of the Mg²⁺ will be precipitated as the hydroxide salt.

Step-by-step solution

Calculate the concentration of Mg²⁺ after 99% precipitation

If 99% of Mg²⁺ is precipitated, that means 1% of Mg²⁺ is still in solution. So, we need to find the concentration of Mg²⁺ remaining in solution: (1% of 0.052 M) = 0.01 x 0.052 M = 0.00052 M

Write the balanced equation and the solubility product expression for Mg(OH)₂

Mg(OH)₂(s) ⇌ Mg²⁺(aq) + 2OH⁻(aq) Ksp = [Mg²⁺] [OH⁻]² = 8.9 x 10⁻¹²

Use the solubility product expression to find the concentration of OH⁻ ions

Since we know the concentration of Mg²⁺ (0.00052 M) and the Ksp value (8.9 x 10⁻¹²), we can find the concentration of OH⁻ ions: \( 8.9 \times 10^{-12} = (0.00052) [OH^-]^2 \) \[ [OH^-] = \sqrt{\frac{8.9 \times 10^{-12}}{0.00052}} \] Calculate the concentration of OH⁻ ions: \[ [OH^-] \approx 1.37 \times 10^{-4} M \]

Calculate the pH from the OH⁻ concentration

To find the pH, we first need to determine the pOH using the OH⁻ concentration: pOH = -log₁₀[OH⁻] = -log₁₀(1.37 x 10⁻⁴) After calculating the pOH, we can find the pH using the relation between pH and pOH: pH + pOH = 14 pH = 14 - pOH Calculating the pH: pH ≈ 10.14 So, at pH ≈ 10.14, 99% of the Mg²⁺ will be precipitated as the hydroxide salt.

Key concepts

Solubility Product Constant

The solubility product constant, denoted as Ksp, is a crucial concept in precipitation chemistry. It represents the degree to which a compound can dissolve in water. At equilibrium, the product of the concentrations of the ions of a sparingly soluble salt, each raised to the power of their respective stoichiometric coefficients, equals the Ksp. This equilibrium is for the dissolution of the solid salt into its constituent ions in a saturated solution.

For example, the equation for the solubility product constant of magnesium hydroxide, Mg(OH)2, is written as:
\[ Ksp = [Mg^{2+}][OH^-]^2 \]
When the ionic concentrations are plugged into this expression, Ksp remains constant at a given temperature. This allows us to calculate the solubility of the salt under different conditions, like changes in pH, or to determine the remaining concentration of ions in a solution after a certain percentage of precipitation, as illustrated in the exercise.

pH Calculation

The pH of a solution is a numerical scale used to specify the acidity or basicity of an aqueous solution. It is defined as the negative logarithm (base 10) of the hydrogen ion concentration. The pH scale typically ranges from 0 (very acidic) to 14 (very basic), with 7 being neutral.

To calculate the pH, we can use the formula:
\[ pH = -\text{log}_{10}[H^+] \]
In the given exercise, pH calculation is linked to the solubility of magnesium hydroxide. Precipitation of Mg(OH)2 is influenced by the pH of the solution because the solubility of hydroxides change with pH. Determining the pH at which a certain percentage of a compound precipitates can be crucial for processes such as water treatment or in chemical synthesis.

Ionic Product of Water

The ionic product of water, Kw, is the product of the concentrations of hydrogen ions and hydroxide ions in water at a particular temperature and is a constant. For pure water and at 25°C, this value is \(1.0 \times 10^{-14}\). It ensures that if the concentration of one ion is known, the other can be calculated.

The relation is represented as:
\[ Kw = [H^+][OH^-] \]
When applied to the exercise, after calculating the concentration of hydroxide ions needed for precipitation, we can derive the concentration of hydrogen ions and thus determine the resulting pH of the solution.

Hydroxide Concentration

In an aqueous solution, the concentration of hydroxide ions (OH⁻) is a direct measure of how basic the solution is. The more hydroxide ions present, the higher the pH and the more basic the solution. Conversely, fewer hydroxide ions mean a lower pH and a more acidic solution.

To calculate hydroxide ion concentration from pH, you can reverse the earlier mentioned kW relation and use:
\[ [OH^-] = \frac{Kw}{[H^+]} \]
Thus, once pOH is obtained from pH (using \(pOH = 14 - pH\)), concentration of OH⁻ can be calculated as \( [OH^-] = 10^{-pOH} \). In the context of the given exercise, it helps to determine at what pH level a certain concentration of hydroxide ions will result in the precipitation of 99% of Mg²⁺ in the form of Mg(OH)₂.