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 \(\mathrm{Na}^{+} .\) Keep in mind that charge balance must be maintained. (a) 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 Nat are needed to replace these ions? (b) If the sodium is added to the water softener in the form of \(\mathrm{NaCl}\), how many grams of sodium chloride are needed?

Short answer

(a) The number of moles of Na⁺ ions needed to replace Ca²⁺ and Mg²⁺ ions in the hard water is 72 moles. (b) The mass of sodium chloride (NaCl) needed is 4203.68 g.

Step-by-step solution

Identify the given values

We are given the volume of hard water: - Volume = 1500 L - Concentration of Ca²⁺ ions = 0.020 M - Concentration of Mg²⁺ ions = 0.0040 M

Calculate the number of moles of Ca²⁺ and Mg²⁺ ions in the water

We'll use the formula for moles: Moles = Volume × Concentration Moles of Ca²⁺ = 1500 L × 0.020 M = 30 moles Moles of Mg²⁺ = 1500 L × 0.0040 M = 6 moles

Calculate the total positive charge from the original ions

Total charge of Ca²⁺ = Moles of Ca²⁺ × Charge of Ca²⁺ ion Total charge of Mg²⁺ = Moles of Mg²⁺ × Charge of Mg²⁺ ion Total charge of Ca²⁺ = 30 moles × 2 = 60 Total charge of Mg²⁺ = 6 moles × 2 = 12 Total positive charge = 60 + 12 = 72

Calculate the number of moles of Na⁺ ions

To maintain charge balance, the number of moles of Na⁺ = Total positive charge of original ions (Ca²⁺ and Mg²⁺) / Charge of Na⁺ ion. Number of moles of Na⁺ = 72 / 1 = 72 moles

Calculate the mass of NaCl to add

The molar mass of NaCl = 58.44 g/mol Mass of NaCl = number of moles of Na⁺ × molar mass of NaCl Mass of NaCl = 72 moles × 58.44 g/mol = 4203.68 g (a) The number of moles of Na⁺ ions needed to replace Ca²⁺ and Mg²⁺ ions in the hard water is 72 moles. (b) The mass of sodium chloride (NaCl) needed is 4203.68 g.

Key concepts

Hard Water

Hard water is a common household problem caused by an excess of calcium (Ca²⁺), magnesium (Mg²⁺), and sometimes iron ions in the water supply. These minerals originate from underground aquifers and are picked up by water as it passes through rock formations. Hard water can cause various issues such as resistant soap scum, reduced efficiency of cleaning agents, and annoying mineral deposits that build up on plumbing fixtures and appliances.
Water softening is the general term for removing these minerals, leading to softer water that works better with soaps and detergents. Understanding hard water and its effects is crucial for maintaining household plumbing and ensuring the best performance of cleaning products.

• Soap Scum: Hard water reacts with soap to form a residue known as soap scum.
• Appliance Efficiency: Appliances like dishwashers and washing machines are less efficient when using hard water.
• Plumbing: Mineral deposits can clog pipes, leading to expensive repairs.

Ca²⁺ and Mg²⁺ Ions

Calcium (Ca²⁺) and magnesium (Mg²⁺) ions are primarily responsible for the hardness of water. These ions carry a double positive charge, making them bivalent cations. They enter the water as it passes over and dissolves rocks like limestone (calcium carbonate) and dolomite (calcium magnesium carbonate).
While these minerals are essential for human health in small amounts, high concentrations in water can have adverse effects. In the context of water softening, these ions' replacement is vital as they are not soap-friendly. They combine with soap to create scummy deposits instead of yielding a rich lather.

• Iron Ions: Often present alongside Ca²⁺ and Mg²⁺, also contribute to water hardness.
• Health Benefits: Small amounts are beneficial for human bone health.
• Water Hardness: Measures the concentration of Ca²⁺ and Mg²⁺ in water.

Charge Balance

Charge balance is a vital principle when dealing with ionic exchanges in water softening. This principle ensures that the total positive charge of the water remains constant, preserving electrical neutrality. In the process of softening water, calcium and magnesium ions must be replaced with sodium ions to maintain charge balance.
The given problem exemplifies this process: - Calcium and magnesium ions together contribute a total positive charge that must be matched by an equivalent charge of sodium ions. - Since Ca²⁺ and Mg²⁺ ions have a 2+ charge, twice as many Na⁺ ions (which carry a 1+ charge each) are required for their replacement.
The correct application of charge balance principles ensures that water softening is effective without altering the water's overall charge.

Sodium Chloride (NaCl)

Sodium chloride (NaCl), commonly known as table salt, is a crucial component in water softening processes. In water softeners, sodium ions from NaCl replace calcium and magnesium ions to reduce water hardness.
Here's how NaCl works in softening:

• Sodium Replacement: The sodium ions from NaCl swap places with calcium and magnesium ions on an ion exchange resin.
• Resin Regeneration: Once depleted of sodium ions, the resin is flushed with a salt solution to recharge it.
• Commonly Used: NaCl is cost-effective and readily available, making it ideal for household water softeners.

The process not only maintains charge balance but also prevents the troublesome build-up of mineral deposits in household systems. Sodium chloride stands as an effective agent for creating soft, manageable water suitable for everyday use.