Question

A saturated solution of the strong electrolyte \(\mathrm{Ca}(\mathrm{OH})_{2}\) is prepared by adding sufficient water to \(5.0 \times 10^{-4}\) mole of \(\mathrm{Ca}(\mathrm{OH})_{2}\) to form \(100 \mathrm{ml}\) of solution. What is the \(\mathrm{pH}\) of this solution?

Short answer

The pH of the saturated Ca(OH)₂ solution is approximately 12.

Step-by-step solution

Calculate the initial concentration of Ca(OH)₂ in the solution.

We are given the amount (mole) of Ca(OH)₂ in the solution, which is 5.0 x 10⁻⁴ mole. The volume of the solution is 100 ml. First, we convert the volume to liters: \(100 \mathrm{ml} \times \frac{1\ \mathrm{L}}{1000\ \mathrm{ml}} = 0.1\ \mathrm{L}\) Now, we can calculate the initial concentration of Ca(OH)₂ in the solution: \[\frac{5.0 \times 10^{-4}\ \text{mole}}{0.1\ \mathrm{L}} = 0.005\ \text{M}\]

Determine the hydroxide ion concentration from the dissociation of Ca(OH)₂.

Ca(OH)₂ is a strong electrolyte, and it dissociates completely when it dissolves in water: \(\mathrm{Ca(OH)_2\ \to\ Ca^{2+} + 2\ OH^{-}}\) Each mole of Ca(OH)₂ dissociates into 2 moles of hydroxide ions. Therefore, the hydroxide ion concentration in the solution is double the initial concentration of Ca(OH)₂: \(0.005\ \mathrm{M} \times 2= 0.01\ \mathrm{M}\)

Calculate the pOH of the solution from the hydroxide ion concentration.

We can now calculate the pOH of the solution using the relationship: \[\mathrm{pOH} = - \log_{10}[OH^{-}]\] We plug in the hydroxide ion concentration we found in step 2: \(\mathrm{pOH} = - \log_{10}(0.01\ \mathrm{M})\) By calculating: \(\mathrm{pOH} \approx 2\)

Calculate the pH of the solution using the relationship between pH and pOH.

We can find the pH of the solution using the relationship: \(\mathrm{pH} + \mathrm{pOH} = 14\) From step 3, we know that the pOH is approximately 2. Therefore, we can find the pH of the solution: \(\mathrm{pH} = 14 - \mathrm{pOH}\) \(\mathrm{pH} = 14 - 2\) \(\mathrm{pH} = 12\) #Conclusion# The pH of the saturated Ca(OH)₂ solution is approximately 12.

Key concepts

Saturated Solution

A saturated solution is one in which the maximum amount of a solute has been dissolved in a solvent at a given temperature and pressure. Once a solution reaches saturation, any additional solute will not dissolve. Instead, it will remain as another phase, typically as a solid.
For calcium hydroxide, Ca(OH)₂, this means that the solution contains the maximum concentration of Ca(OH)₂ that can dissolve. If more Ca(OH)₂ is added beyond this point, it will remain undissolved.

• Saturation depends on temperature - more solute can usually dissolve at higher temperatures.
• The concept of saturation is crucial for calculating solute concentrations in solutions.

Understanding saturation is particularly important when working with strong electrolytes, as their complete dissociation in water can significantly increase the concentration of ions in the solution.

Strong Electrolyte

Strong electrolytes are substances that completely dissociate into ions when dissolved in water. This means that they split into their constituent ions, which are then free to move in the solution.
In the case of Ca(OH)₂, it fully dissociates into Ca²⁺ and OH⁻ ions. This complete dissociation is what makes it a strong electrolyte. Complete dissociation ensures that no intact Ca(OH)₂ molecules remain in the solution.

• Strong acids, strong bases, and most salts are strong electrolytes.
• The presence of free ions makes these solutions excellent conductors of electricity.

When calculating the properties of a solution containing a strong electrolyte, you must account for the complete abundance of ions, as seen in our calculation for hydroxide ions from Ca(OH)₂.

Hydroxide Ion Concentration

The hydroxide ion concentration is a critical factor in determining the basicity of a solution. For basic solutions, hydroxide ions ( OH⁻ ) are present in higher concentrations.
For example, in our calculation, Ca(OH)₂ dissociates completely, producing two OH⁻ ions for every formula unit dissolved. As a result, the hydroxide ion concentration is twice the initial molarity of Ca(OH)₂.

• The concentration of hydroxide ions determines the solution's pOH, a direct measure of basicity.
• It is important to understand the role of dissociation in calculating ion concentration.

By calculating the hydroxide ion concentration accurately, one can ascertain many properties, including the pH or pOH which indicates the acidity or basicity of the solution.

Dissociation

Dissociation is the process by which compounds split into their ions in solution. This is especially crucial for understanding reactions in aqueous environments and the behavior of strong electrolytes.
For strong electrolytes like Ca(OH)₂, this process is complete, meaning the solute completely dissociates into its Ca²⁺ and OH⁻ components. This full breakdown is pivotal in calculations of ion concentrations.

• Dissociation allows the calculation of individual ion concentrations.
• The extent of dissociation can be different depending on whether the compound is a strong or weak electrolyte.

Accurate prediction of how a compound dissociates helps in calculating pH and pOH, providing insight into whether a solution is acidic or basic.

pOH

The pOH of a solution offers a measure of its basicity, much like pH measures acidity. It's calculated using the concentration of hydroxide ions present in the solution.
Given a strong base like Ca(OH)₂, once you know the hydroxide ion concentration (in our case, 0.01 M), you can find the pOH using the formula: \[\text{pOH} = -\log_{10}[OH^-]\]This value provides insight into how the solution compares to neutral solutions (which have a pOH of about 7).

• pOH is inversely related to pH, where \(\text{pH + pOH} = 14\).
• Knowing the pOH is essential for creating a comprehensive understanding of a solution's properties.

Calculating pOH is a crucial step in deriving the pH of strong electrolyte solutions and understanding their behavior in chemical reactions.