Question

Slaked lime, \(\mathrm{Ca}(\mathrm{OH})_{2}\), is used to soften hard water by removing calcium ions from hard water through the reaction \(\mathrm{Ca}(\mathrm{OH})_{2}(a q)+\mathrm{Ca}^{2+}(a q)+2 \mathrm{HCO}_{3}^{-}(a q) \rightarrow\) Although \(\mathrm{CaCO}_{3}(s)\) is considered insoluble, some of it does dissolve in aqueous solutions. Calculate the molar solubility of \(\mathrm{CaCO}_{3}\) in water \(\left(K_{\mathrm{sp}}=8.7 \times 10^{-9}\right)\)

Short answer

The molar solubility of \(\mathrm{CaCO}_{3}\) in water is approximately \(9.33 \times 10^{-5}\) mol/L.

Step-by-step solution

Write the dissolution equation.

First of all, we need to write the dissolution equation for calcium carbonate in water: \[ \ce{CaCO3_{(s)} -> Ca^{2+}_{(aq)} + CO3^{2-}_{(aq)}} \]

Define the molar solubility.

According to the dissolution equation, when one mole of \(\mathrm{CaCO}_{3}\) dissolves, one mole of \(\mathrm{Ca}^{2+}\) ions and one mole of \(\mathrm{CO}_{3}^{2-}\) ions are produced. Let's define the molar solubility of \(\mathrm{CaCO}_{3}\) in water as "x" mol/L. This means that the concentrations of both the \(\mathrm{Ca}^{2+}\) ions and the \(\mathrm{CO}_{3}^{2-}\) ions will also be x mol/L.

Write the expression for Ksp.

Now we write down the expression for the Ksp of calcium carbonate. According to the dissolution equation, Ksp is equal to the product of the concentrations of the ions in the solution raised to their stoichiometric coefficients: \[ K_{sp}=[\ce{Ca^{2+}}][\ce{CO3^{2-}}] \]

Substitute the molar solubility into the Ksp expression.

We substitute the molar solubility value x mol/L into the solubility product constant equation: \[ 8.7 \times 10^{-9}= [x][x] = x^2 \]

Calculate the molar solubility of calcium carbonate.

Now, we solve the equation for x to find the molar solubility of calcium carbonate in water: \[ x = \sqrt{8.7 \times 10^{-9}} \] \[ x = \approximately 9.33 \times 10^{-5} \hspace{4pt} \text{mol/L} \] So, the molar solubility of \(\mathrm{CaCO}_{3}\) in water is approximately \(9.33 \times 10^{-5}\) mol/L.

Key concepts

Calcium Carbonate

Calcium carbonate is a common chemical compound found in rocks as minerals like calcite and aragonite. It is often used in industries like construction and pharmaceuticals due to its abundance and versatility.

In water, calcium carbonate is known to be sparingly soluble. This means it dissolves very little, contributing minimal calcium ions ( ext{Ca}^{2+} ) and carbonate ions ( ext{CO}_3^{2-} ) to the solution.

• The insufficient solubility is beneficial in certain applications, such as in water softening, where hardness (usually caused by calcium ions) needs to be reduced or removed.
• Despite being termed "insoluble", it can indeed dissolve to a small extent, which is necessary to calculate its molar solubility.

Understanding the solubility behavior of calcium carbonate is essential for fields dealing with water purity and mineral deposition.

Dissolution Equation

The dissolution equation of a substance in water describes how it breaks down into its ionic components. For calcium carbonate, the dissolution equation is:\[ ext{CaCO}_3_{( ext{s})} \rightarrow ext{Ca}^{2+}_{( ext{aq})} + ext{CO}_3^{2-}_{( ext{aq})} \]This chemical equation shows:

• One molecule of calcium carbonate solid ( ext{CaCO}_3_{( ext{s})}) dissociates into one calcium ion ( ext{Ca}^{2+}_{( ext{aq})}) and one carbonate ion ( ext{CO}_3^{2-}_{( ext{aq})}) in aqueous solution.
• The stoichiometry of this dissolution is important as it determines the relationship between the amount of the solid that dissolves and the concentration of ions formed.

Considering these elements, the dissolution equation is a foundational tool for determining ion concentrations and solving solubility problems.

Solubility Product Constant (Ksp)

The solubility product constant, Ksp, is a way to quantitatively express the solubility of an ionic compound in water. It reflects the maximum product of the ion concentrations that can exist in steady equilibrium. In the case of calcium carbonate, the expression for Ksp is represented as:\[K_{sp} = [ ext{Ca}^{2+}][ ext{CO}_3^{2-}] \]

• The square brackets denote concentration, and the expression shows that Ksp relies upon the product of the calcium ion and carbonate ion concentrations.
• A higher Ksp value indicates greater solubility, and vice versa.
• In the exercise example, Ksp helps calculate how much calcium carbonate will dissolve in pure water by allowing substitution of the molar solubility into the Ksp equation.

Understanding Ksp is crucial for predicting the behaviors of sparingly soluble salts in solutions.

Hard Water Softening

Hard water contains high levels of minerals, particularly calcium and magnesium ions, which can cause scaling and reduce the effectiveness of soap. To counteract this, water softening techniques are used, often involving chemical reactions that remove these hardness-causing ions.

One such method uses slaked lime ( ext{Ca(OH)}_2 ) in a reaction that results in the precipitation of calcium carbonate, effectively removing calcium ions from the solution.

• By adding slaked lime, calcium ions from hard water react to form insoluble calcium carbonate, which precipitates out.
• This process not only reduces water hardness but also aids in preventing damage to plumbing and appliances caused by mineral deposits.

Thus, calcium carbonate plays a pivotal role in water treatment processes, highlighting its importance beyond simple solubility calculations.