Problem 91 - Chemistry and Chemical Reactivity

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.) (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

100% of the calcium ions are removed from the water.

Step 1: Write the Reaction Equation

The reaction between calcium ions and carbonate ions forms calcium carbonate as a precipitate. The balanced chemical equation is \( \text{Ca}^{2+} + \text{CO}_3^{2-} \rightarrow \text{CaCO}_3 \downarrow \). This shows that one mole of \( \text{Ca}^{2+} \) reacts with one mole of \( \text{CO}_3^{2-} \).

Step 2: Determine Initial Moles in Solution

Calculate the initial moles of calcium ions ([\text{Ca}^{2+}]) in 1 liter of solution and the added carbonate ions ([\text{CO}_3^{2-}]) using their molar concentrations: \[ n_{\text{Ca}^{2+}} = 0.010 \text{ mol/L} \] and \[ n_{\text{CO}_3^{2-}} = 0.050 \text{ mol/L} \]. Since we are assuming these concentrations in 1 L of water, the initial moles are equal to the concentration values.

Step 3: Finding the Limiting Reagent

Compare the initial moles of calcium and carbonate ions to find the limiting reagent. \( n_{\text{Ca}^{2+}} = 0.010 \) mol and \( n_{\text{CO}_3^{2-}} = 0.050 \) mol. Since each 1 mole of calcium reacts with 1 mole of carbonate, calcium ions are the limiting reagent.

Step 4: Determine Moles of Calcium Precipitated

Since calcium ions are the limiting reagent, all of them will precipitate out as \( \text{CaCO}_3 \). Therefore, \( 0.010 \text{ mol} \) of \( \text{Ca}^{2+} \) ions are precipitated.

Step 5: Calculate Percentage of Calcium Ions Removed

The percentage of calcium ions removed is calculated by the formula: \( \text{Percentage removed} = \left( \frac{\text{moles of Ca}^{2+} \text{ ions reacted}}{\text{initial moles of Ca}^{2+}} \right) \times 100 \). Thus \[ \frac{0.010}{0.010} \times 100 = 100\% \]. All calcium ions are removed.

Key Concepts

Understanding Precipitation Reactions

In chemistry, a precipitation reaction occurs when two soluble substances in a solution react to form an insoluble product, known as a precipitate. It's like mixing paint colors, but instead of a new color, you get small solid particles falling out of the solution. For instance, in the reaction between calcium ions (\( \text{Ca}^{2+} \)) and carbonate ions (\( \text{CO}_3^{2-} \)), we see the formation of calcium carbonate (\( \text{CaCO}_3 \)), which precipitates out of the solution.
When these ions combine, a chemical change takes place leading to a solid product that is no longer soluble in water. This is a classic example of a precipitation reaction, useful for removing unwanted ions from solutions, such as in water softening.

• The balanced equation for the reaction is: \( \text{Ca}^{2+} + \text{CO}_3^{2-} \rightarrow \text{CaCO}_3 \).

This reaction is essential in various industries, particularly water treatment, to reduce water hardness.

The Role of the Limiting Reagent

In stoichiometry, the limiting reagent is the substance that gets used up first, limiting the amount of product formed in a chemical reaction. It's like running out of a key ingredient while baking; the cookies can't be made without all necessary items.
For the reaction between calcium ions and carbonate ions, finding the limiting reagent helps determine how much calcium carbonate can form. Even if more carbonate ions are present, all calcium ions react first because they are present in smaller initial amounts (0.010 mol of \( \text{Ca}^{2+} \) vs. 0.050 mol of \( \text{CO}_3^{2-} \)).
This means calcium ions are the limiting reagent, dictating the maximum amount of \( \text{CaCO}_3 \) produced.
In this example, because the reaction ratio is 1:1, the entire quantity of calcium ions will precipitate when equivalent carbonate ions are added, demonstrating the significance of understanding which reagent will run out first.

Molecule by Mole: Understanding Mole Calculations

A mole in chemistry is a standard unit for measuring large quantities of smaller entities, such as atoms, molecules, or ions. Think of it like a completely full carton of eggs; it holds exactly 12 eggs or, in the case of a mole, Avogadro's number, which is approximately 6.022 x 10^{23} of those particles.
Mole calculations are pivotal in chemical reactions to predict yields and reactant needs. For the exercise, we focus on the concentrations given: 0.010 M for calcium ions and 0.050 M for carbonate ions, both measured in moles per liter. This gives us the amount of substance present in the solution.

• When we say 0.010 mol/L of \( \text{Ca}^{2+} \), we're saying that in one liter of solution, there are 0.010 moles of \( \text{Ca}^{2+} \).

This calculation ensures we can accurately determine what happens when two reactants share a reaction, guiding us on the outcome.

Water Hardness and its Reduction

Water hardness is caused by dissolved minerals, primarily calcium and magnesium ions. These minerals can interfere with soap effectiveness and form limescale in pipes. By precipitating these ions, hardness can be reduced, making water cleaner and more efficient for cleaning.
Using soda ash (\( \text{Na}_2 \text{CO}_3 \)) in our exercise, we see a practical application: the calcium ions in hard water react with carbonate ions to form solid calcium carbonate, removing calcium from the solution. This reduces hardness.
When all calcium ions are precipitated in this manner, 100% have been effectively removed, signifying a complete softening reaction.

• Softening improves water's response to soaps and prevents build-up in plumbing systems.

Understanding how and why we remove such minerals emphasizes the importance of chemical reactions in day-to-day life, illustrating their practicality beyond the lab.