Question

The \(\mathrm{Ca}^{2+}\) ion in hard water can be precipitated as \(\mathrm{CaCO}_{3}\) by adding soda ash, \(\mathrm{Na}_{2} \mathrm{CO}_{3} .\) If the calcium ion concentration in hard water is 0.010 \(\mathrm{M}\) and if the \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) is added until the carbonate ion concentration is 0.050 \(\mathrm{M},\) what percentage of the calcium ions has been removed from the water? (You may neglect carbonate ion hydrolysis.

Short answer

Approximately 99.999% of calcium ions are removed as \( \mathrm{CaCO}_3 \).

Step-by-step solution

Write the Chemical Equation

The reaction between calcium ions \( \mathrm{Ca}^{2+} \) and carbonate ions \( \mathrm{CO}_3^{2-} \) forms calcium carbonate in the following way: \[ \mathrm{Ca}^{2+} (aq) + \mathrm{CO}_3^{2-} (aq) \rightarrow \mathrm{CaCO}_3 (s) \]. This equation shows how calcium ions are removed from the water by precipitating as calcium carbonate.

Apply the Solubility Product Constant (Ksp)

Calcium carbonate \( \mathrm{CaCO}_3 \) has a solubility product constant \( K_{sp} \). For \( \mathrm{CaCO}_3 \), \( K_{sp} = 4.8 \times 10^{-9} \) at room temperature. The \( K_{sp} \) expression is \[ K_{sp} = [\mathrm{Ca}^{2+}][\mathrm{CO}_3^{2-}] \].

Calculate the Remaining Calcium Ion Concentration

Given \([\mathrm{CO}_3^{2-}] = 0.050 \ \mathrm{M}\), solve for \([\mathrm{Ca}^{2+}]\) using \( K_{sp} \): \[ [\mathrm{Ca}^{2+}] = \frac{K_{sp}}{[\mathrm{CO}_3^{2-}]} = \frac{4.8 \times 10^{-9}}{0.050} \]. Calculating this gives \([\mathrm{Ca}^{2+}] = 9.6 \times 10^{-8} \ \mathrm{M}\).

Calculate the Calcium Ion Removal Percentage

Calculate the initial concentration of calcium ions removed by subtracting the concentration of remaining calcium ions: \[ \mathrm{Initial} - \mathrm{Final} = 0.010 \ \mathrm{M} - 9.6 \times 10^{-8} \ \mathrm{M} \approx 0.010 \ \mathrm{M}\]. The percentage removed is \[ \left( \frac{0.010 - 9.6 \times 10^{-8}}{0.010} \right) \times 100\% \approx 99.999\% \].

Key concepts

Calcium Ion Precipitation

In the world of chemistry, precipitation is a process where a solid is formed in a solution during a chemical reaction. This solid, known as a precipitate, separates from the liquid. In the context of calcium and hard water, precipitation plays a critical role. When calcium ions (\(\mathrm{Ca}^{2+}\)) are present in water, they can be problematic, making the water 'hard.' This is where the process of precipitation comes into play.

By adding a substance like soda ash (\(\mathrm{Na}_2 \mathrm{CO}_{3}\)), which provides carbonate ions (\(\mathrm{CO}_3^{2-}\)), calcium ions can be encouraged to leave the water by forming a solid precipitate known as calcium carbonate (\(\mathrm{CaCO}_3\)). This effectively removes calcium from the liquid, making the water less hard. It's a simple, yet highly efficient way to treat hard water.

• Calcium ions are denoted as \(\mathrm{Ca}^{2+}\)
• This precipitation process forms solid \(\mathrm{CaCO}_3\), which is easy to remove
• Important for industries and households to manage water hardness

Chemical Reaction Equation

Understanding the chemical reaction equation is key to grasping how calcium ions are removed from water. The reaction involves calcium ions (\(\mathrm{Ca}^{2+}\)) reacting with carbonate ions (\(\mathrm{CO}_3^{2-}\)) to form calcium carbonate (\(\mathrm{CaCO}_3\)), as shown below:

\[\mathrm{Ca}^{2+} (aq) + \mathrm{CO}_3^{2-} (aq) \rightarrow \mathrm{CaCO}_3 (s)\]

This equation shows the simple one-to-one reaction where one calcium ion reacts with one carbonate ion to produce one molecule of calcium carbonate. It's crucial to remember that the reaction moves to completion, meaning all the calcium ions will attempt to react until one of the reactants is completely used up. This is how nearly all calcium is effectively precipitated out of the solution.

• The reaction is crucial for removing calcium ions from water
• It outlines a straightforward process with specific inputs and outputs
• Balance in the equation ensures completeness of the reaction

Calcium Carbonate Formation

Calcium carbonate (\(\mathrm{CaCO}_3\)) is the solid product formed when calcium ions precipitate out of solution with carbonate ions. This compound is found naturally in limestone and is a major component in the shells of marine organisms, snails, pearls, and eggshells. In water treatment processes, forming and removing calcium carbonate is a way to tackle water hardness.

When calcium carbonate forms, it has low solubility in water, meaning it readily precipitates out of the solution, which makes it convenient for separation techniques. Understanding the solubility product constant, or (\(K_{sp}\)), is particularly useful in predicting the conditions under which calcium carbonate will form. The (\(K_{sp}\)) is a safeguard guiding how much of the product can dissolve in a solution before it starts to precipitate. For (\(\mathrm{CaCO}_3\)), the (\(K_{sp}\)) is specifically important in calculations for water softening.

• Most common form of precipitate in hard water solutions
• Plays an integral role in nature and industry
• Low solubility helps in effective removal from solutions