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 \(\mathrm{Na}^{+}\). (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 \(\mathrm{Na}^{+}\) 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) 36 moles of \(\mathrm{Na}^{+}\) ions are needed to replace the calcium and magnesium ions in the hard water. (b) 2103.84 g of \(\mathrm{NaCl}\) is needed to be added to the water softener to achieve this.

Step-by-step solution

Calculate the moles of calcium and magnesium ions in the water

Given the volume of the water is 1500 L, and the molarity of calcium ions is 0.020 M, and magnesium ions is 0.0040 M: Moles of calcium ions = Molarity × Volume = 0.020 moles/L × 1500 L = 30 moles Moles of magnesium ions = Molarity × Volume = 0.0040 moles/L × 1500 L = 6 moles

Calculate moles of sodium ions needed

Since the sodium ions replace calcium and magnesium ions on a one-to-one basis (each sodium ion replaces one calcium or magnesium ion), the total moles of sodium ions needed will be the sum of the moles of calcium and magnesium ions: Moles of sodium ions needed = Moles of calcium ions + Moles of magnesium ions = 30 moles + 6 moles = 36 moles

Calculate grams of sodium chloride needed

Now we need to convert the moles of sodium ions to grams of sodium chloride. The molar mass of sodium chloride (NaCl) is 58.44 g/mol (22.99 g/mol for Na and 35.45 g/mol for Cl). Grams of NaCl needed = Moles of sodium ions × Molar mass of NaCl = 36 moles × 58.44 g/mol = 2103.84 g So, (a) 36 moles of \(\mathrm{Na}^{+}\) ions are needed to replace the calcium and magnesium ions in the hard water. (b) 2103.84 g of \(\mathrm{NaCl}\) is needed to be added to the water softener to achieve this.

Key concepts

Chemical Calculations

Understanding chemical calculations is essential for various applications, including water treatment processes. To tackle such practical issues, it is vital to know how to calculate the number of moles of substances involved. The mole concept is a fundamental aspect of chemistry that enables us to count particles on a macroscopic scale. When working with solutions, the concentration (molarity) tells us how many moles of solute are present in a certain volume of solution, usually expressed in moles per liter (M or mol/L).

In the context of hard water, we use chemical calculations to determine the precise amount of sodium ions (\(\mathrm{Na}^{+}\)) needed to replace the hard water ions, such as calcium (\(\mathrm{Ca}^{2+}\)) and magnesium (\(\mathrm{Mg}^{2+}\)). As each mole of sodium replaces one mole of either calcium or magnesium, this direct stoichiometric relationship simplifies the calculations, making it a straight multiplication of the molarity of the ions by the volume of water.

Water Softening Process

Hard water poses problems in household and industrial settings by causing scale buildup and reducing the effectiveness of soaps and detergents. Water softening is a method of removing the divalent cations, such as calcium and magnesium, that contribute to water hardness, and replacing them with monovalent sodium ions.

Water softeners achieve this via a process known as ion exchange. Resins within the softener contain sodium ions that are exchanged with the hard water ions as the water passes through. This process continues until the resin is saturated with calcium and magnesium ions. Periodically, the system must be regenerated or recharged with a solution rich in sodium ions (usually a brine solution), which replenishes the resin with sodium ions by again swapping them for the captured calcium and magnesium ions, thus preparing the softener for further use.

Molarity and Moles

Molarity, one of the core concepts in chemistry, is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution. The concept of molarity is directly connected with the concept of moles, which is the SI unit for the amount of substance. One mole corresponds to \(6.022 \times 10^{23}\) entities, whether they are atoms, molecules, ions, or particles.

For example, a \(0.020 \mathrm{M}\) solution of calcium ions means that there are 0.020 moles of calcium ions in every liter of solution. In the exercise, we were given different molarities for the hard water ions, making it possible to calculate the exact amount of moles present in the water and, consequently, the exact amount of moles of \(\mathrm{Na}^{+}\) ions needed for softening.