Question

Calcium hydroxide, \(\operatorname{Ca}(\mathrm{OH})_{2},\) is almost insoluble in water \(=\) only \(0.50 \mathrm{g}\) can be dissolved in \(1.0 \mathrm{L}\) of water at \(25^{\circ} \mathrm{C} .\) If the dissolved substance is completely dissociated into its constituent ions, what is the pH of a saturated solution?

Short answer

The pH of the saturated solution is approximately 12.13.

Step-by-step solution

Write the Dissociation Equation

The first step is to write the chemical equation for the dissociation of calcium hydroxide in water. Calcium hydroxide dissociates into calcium ions \( \text{Ca}^{2+} \) and hydroxide ions \( \text{OH}^- \):\[\text{Ca(OH)}_2 (s) \rightarrow \text{Ca}^{2+} (aq) + 2 \text{OH}^- (aq)\]

Calculate Molarity of \(\text{Ca(OH)}_2\)

Next, determine the molarity of calcium hydroxide. The molar mass of \( \text{Ca(OH)}_2 \) is approximately 74.1 g/mol. Since 0.50 g of \( \text{Ca(OH)}_2 \) dissolves in 1.0 L of water:\[\text{Molarity (M)} = \frac{0.50 \, \text{g}}{74.1 \, \text{g/mol}} = 0.00675 \, \text{mol/L}\]

Determine [OH\(^-\)] Concentration

Since \( \text{Ca(OH)}_2 \) dissociates into one \( \text{Ca}^{2+} \) ion and two \( \text{OH}^- \) ions, the concentration of \( \text{OH}^- \) is doubled:\[[\text{OH}^-] = 2 \times 0.00675 = 0.0135 \, \text{mol/L}\]

Calculate pOH

The next step is to calculate the pOH using the concentration of hydroxide ions:\[\text{pOH} = -\log_{10} [\text{OH}^-] = -\log_{10} (0.0135)\approx 1.87\]

Convert pOH to pH

The final step is to convert pOH to pH. Since the relationship between pH and pOH is given by \( \text{pH} + \text{pOH} = 14 \):\[\text{pH} = 14 - \text{pOH} = 14 - 1.87 = 12.13\]

Key concepts

Molarity Calculation

Molarity is a key concept in chemistry that helps us understand the concentration of a solution. It is defined as the number of moles of a solute divided by the volume of the solution in liters. In the case of calcium hydroxide (\(\text{Ca(OH)}_2\)), we are calculating how many moles of this compound are dissolved in a given volume of water. To find the molarity, you first need to know the molar mass of \(\text{Ca(OH)}_2\), which is approximately 74.1 g/mol.

For our example, only 0.50 grams of \(\text{Ca(OH)}_2\) dissolve in 1 liter of water, which gives us the amount in moles:

• Moles of \(\text{Ca(OH)}_2\) = \(\frac{0.50}{74.1}\) = 0.00675 moles

Since this amount is dissolved in 1 liter, the molarity (\(\text{M}\)) is 0.00675 mol/L. This low molarity reflects the low solubility of \(\text{Ca(OH)}_2\) in water at 25°C, demonstrating its status as an "almost insoluble" compound.

Hydroxide Ion Concentration

Once you have the molarity of calcium hydroxide, the next step is to figure out the concentration of hydroxide ions (\([\text{OH}^-]\)) in the solution. When \(\text{Ca(OH)}_2\) dissociates, it releases one calcium ion (\(\text{Ca}^{2+}\)) and two hydroxide ions (\(\text{OH}^-\)). This means that for every mole of calcium hydroxide dissolved, two moles of hydroxide ions are produced.

To calculate the concentration of these ions, you multiply the original molarity of \(\text{Ca(OH)}_2\) by two:

• \([\text{OH}^-]\) = 2 \times 0.00675 = 0.0135 mol/L

This means every liter of the solution contains 0.0135 moles of hydroxide ions. This concentration is crucial when determining the basicity of the solution and its pH level.

pH and pOH Relationship

Understanding the relationship between pH and pOH is crucial to determining the characteristics of any aqueous solution. Both pH and pOH are measures of hydrogen and hydroxide ion concentrations, respectively. Their sum in any solution at 25°C is always 14. This relationship allows us to find one value if the other is known.

For calcium hydroxide, once you calculate the concentration of hydroxide ions, you can determine the pOH using the formula:

• \(\text{pOH} = -\log_{10} [\text{OH}^-]\ = -\log_{10} (0.0135)\) \(\approx 1.87\)

From this pOH, you then compute the pH using the formula:

• \(\text{pH} = 14 - \text{pOH}\ = 14 - 1.87 = 12.13\)

A pH of 12.13 indicates a highly basic solution, as expected since calcium hydroxide is known to produce strong bases.