Problem 102 - Chemistry: The Central Science

Question

Hard water contains \(\mathrm{Ca}^{L+}, \mathrm{Mg}^{2+}\), and \(\mathrm{Fe}^{L+}\), which interfere with the action of soap and leave an insoluble coating on the insides of containers and pipes when heated. Water softeners replace these ions with \(\mathrm{Na}^{+}\). If \(1500 \mathrm{~L}\) of hard water contains \(0.020 \mathrm{M} \mathrm{Ca}^{2+}\) and \(0.0040 \mathrm{M} \mathrm{Mg}^{2+}\), how many moles of \(\mathrm{Na}^{+}\) are needed to replace these ions?

Step-by-Step Solution

Answer

To replace the \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) ions in 1500 L of hard water, 72 moles of \(\mathrm{Na}^{+}\) ions are needed.

Step 1: Calculate moles of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) ions

We can find the moles of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) ions using their given molar concentrations and the volume of hard water: Moles of \(\mathrm{Ca}^{2+}\) ions = (Molar concentration of \(\mathrm{Ca}^{2+}\) ions) × (Volume of hard water) Moles of \(\mathrm{Mg}^{2+}\) ions = (Molar concentration of \(\mathrm{Mg}^{2+}\) ions × (Volume of hard water) Moles of \(\mathrm{Ca}^{2+}\) ions = (0.020 M) × (1500 L) = 30 moles Moles of \(\mathrm{Mg}^{2+}\) ions = (0.0040 M) × (1500 L) = 6 moles

Step 2: Calculate moles of \(\mathrm{Na}^{+}\) ions needed

Each \(\mathrm{Ca}^{2+}\) ion and \(\mathrm{Mg}^{2+}\) ion has a charge of 2+. To replace these ions with \(\mathrm{Na}^{+}\) ions (which have a charge of 1+), we'll need two moles of \(\mathrm{Na}^{+}\) ions for every mole of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) ions: Moles of \(\mathrm{Na}^{+}\) ions needed = 2 × (Moles of \(\mathrm{Ca}^{2+}\) ions + Moles of \(\mathrm{Mg}^{2+}\) ions) Moles of \(\mathrm{Na}^{+}\) ions needed = 2 × (30 moles + 6 moles) = 2 × 36 moles = 72 moles Hence, 72 moles of \(\mathrm{Na}^{+}\) ions are needed to replace the \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) ions in 1500 L of hard water.

Key Concepts

Hard Water

Hard water is a common phenomenon where water contains high concentrations of calcium, magnesium, and sometimes iron ions. These ions interfere with cleaning processes by reacting with soap to form an insoluble substance, often referred to as soap scum. This reaction makes soap less effective for washing. Additionally, when hard water is heated, these ions can deposit as solid scales inside pipes and containers, leading to clogs and reduced water flow. Recognizing the presence and problems of hard water is crucial for homeowners and industries alike. Proper management can prevent damage to plumbing systems and reduce soap usage.

Ion Exchange

Ion exchange is a water softening process that involves swapping hard water ions, like calcium and magnesium, with sodium ions. This is usually done using a special medium, often called a resin, which has bound sodium ions ready for exchange. When hard water passes through the resin:

• The calcium (\( \mathrm{Ca}^{2+} \)) and magnesium (\( \mathrm{Mg}^{2+} \)) ions are attracted and held by the resin molecules.
• In replace, sodium ions (\( \mathrm{Na}^{+} \)) are released into the water.

This process effectively softens the water by eliminating the minerals that cause hardness. While ion exchange does remove the problematic aspects of hard water, it introduces sodium, which might not be ideal for sodium-restricted diets. Monitoring sodium levels ensures that the water remains suitable for all uses.

Molar Concentration

Molar concentration, also known as molarity, is a measure of the number of moles of solute present in a liter of solution. It provides a way to express the concentration of ions or molecules in a mixture, fundamental to understanding chemical reactions and processes like ion exchange. For instance, in the original problem:

• The molar concentration of calcium ions (\( \mathrm{Ca}^{2+} \)) in the hard water was given as 0.020 M.
• The molar concentration of magnesium ions (\( \mathrm{Mg}^{2+} \)) was 0.0040 M.

To determine how many moles of \( \mathrm{Na}^{+} \) are needed, it's essential to calculate the total moles of these ions first. By multiplying their molar concentration by the volume of water they are in, you can find the total moles. Understanding molar concentration helps dissect water chemistry and guides in solving various chemical equations and problems effectively.