Question

Hard water contains \(\mathrm{Ca}^{2+}, \mathrm{Mg}^{2+}\), and \(\mathrm{Fe}^{2+}\), 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?

Short answer

To replace the \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) ions in \(1500 \mathrm{~L}\) of hard water with concentrations \(0.020 \mathrm{M}\) and \(0.0040 \mathrm{M}\), respectively, you would need \(72\) moles of \(\mathrm{Na}^{+}\) ions.

Step-by-step solution

Find moles of Ca2+ and Mg2+

First, we need to find the moles of Ca2+ and Mg2+ in the 1500 L of hard water. We can use the formula: moles = Molarity × Volume For Ca2+ ions: moles of Ca2+ = (0.020 M) × (1500 L) = 30 moles For Mg2+ ions: moles of Mg2+ = (0.0040 M) × (1500 L) = 6 moles

Determine the moles of Na+ needed to replace each ion.

Since each Ca2+ ion has a charge of +2, it takes 2 moles of Na+ ions (with a +1 charge each) to replace one mole of Ca2+ ions. Similarly, two moles of Na+ ions are required to replace one mole of Mg2+ ions. Therefore, we need to multiply the moles of each ion with the ratio 2:1, since 2 moles of Na+ are needed for each mole of Ca2+ or Mg2+. For Ca2+ ions: moles of Na+ required = 2 moles of Na+ per mole of Ca2+ × 30 moles of Ca2+ = 60 moles of Na+ For Mg2+ ions: moles of Na+ required = 2 moles of Na+ per mole of Mg2+ × 6 moles of Mg2+ = 12 moles of Na+

Calculate the total moles of Na+ needed.

Now, we just need to add up the moles of Na+ needed to replace both ions: Total moles of Na+ = moles of Na+ for Ca2+ + moles of Na+ for Mg2+ = 60 moles + 12 moles = 72 moles So, 72 moles of Na+ ions are needed to replace the Ca2+ and Mg2+ ions in 1500 L of hard water.

Key concepts

Water Softeners

A water softener is a device that helps to remove calcium, magnesium, and other metal ions from hard water. Hard water is notorious for causing issues in our daily household activities, like making soaps less effective and leading to mineral buildup inside appliances and pipes. Water softeners are typically installed in the plumbing system to treat this water as it enters a home.
They work by replacing the harsh minerals with more soluble ions, usually sodium ions, which do not create the same issues as calcium and magnesium.

• Improves soap lathering.
• Prevents mineral buildup on fixtures.
• Prolongs the life of appliances by reducing scale deposits.

By understanding the primary function of water softeners, maintenance and usage can greatly enhance household water quality and reduce repair and cleaning efforts.

Ion Exchange

Ion exchange is a chemical process used extensively in water softeners. The process involves replacing undesirable ions in the water with more desirable ones. This is achieved through an ion exchange resin. The resin is typically engineered beads saturated with sodium ions. As the hard water passes through the resin, a swap occurs.

• Calcium and magnesium ions attach to the resin.
• Sodium ions are released into the water.

This exchange is essential to tackle the challenges posed by hard water. Without ion exchange, hard water can leave spots on dishes, clog pipes, and reduce the efficiency of heating elements. Besides water softening, ion exchange has other applications like water purification and metal separation in various industries.

Calcium and Magnesium Ions

Calcium ( ext{Ca}^{2+} ) and magnesium ( ext{Mg}^{2+} ) ions are the primary culprits making water hard. These ions are naturally found in water sources as they dissolve from soil and rock minerals like limestone and dolomite.
Despite being natural, their presence makes detergents less effective because they form insoluble compounds with soap. This is what causes that sticky and dull residue you find on bathrooms and kitchens.

• Calcium ions create lime scale which is harsh on plumbing.
• Magnesium ions also contribute to scale deposits, impacting water flow.

By using water softeners, these ions are swapped out for sodium which is less reactive and does not form such deposits.

Sodium Ions Replacement

The process of sodium ion replacement is fundamental in the working of water softeners. When hard water passes through the ion exchange resin, sodium ions ( ext{Na}^{+} ) are exchanged for the ext{Ca}^{2+} and ext{Mg}^{2+} ions. For every calcium or magnesium ion removed, two sodium ions are released into the water.
This is because of the charge balance technique—calcium and magnesium are divalent, meaning they have a charge of +2, while sodium is monovalent with a charge of +1. To maintain electrical neutrality, the exchange is two sodium ions for every calcium or magnesium ion. This transformation helps prevent the issues associated with hard water, such as:

• Improving the effectiveness of soaps and detergents.
• Reducing scale buildup in pipes and appliances.
• Improving the aesthetic and functional quality of water.

Sodium ion replacement ensures your water is free from troublesome minerals, providing a smoother and more efficient water system.