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 \(\mathrm{Na}^{+}\) is 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

To replace the calcium and magnesium cations in 1500 L of hard water containing 0.020 M Ca⁺² and 0.0040 M Mg⁺², 72 moles of Na⁺ are required. To provide the required sodium ions, 4203.68 grams of NaCl should be added as a water softener.

Step-by-step solution

Calculate the moles of calcium and magnesium cations in the hard water

As the volume and molar concentrations of calcium and magnesium cations are provided, we can use the relation between volume, molarity, and moles to compute this. Number of moles = Molarity * Volume For calcium ions (\(\mathrm{Ca}^{2+}\)): Number of moles = 0.020 M * 1500 L = 30 moles For magnesium ions (\(\mathrm{Mg}^{2+}\)): Number of moles = 0.0040 M * 1500 L = 6 moles

Determine the moles of sodium cations needed to replace calcium and magnesium cations

In order to retain the charge balance, for each divalent cation (\(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\)), we need to substitute it with 2 monovalent sodium cations (\(\mathrm{Na}^{+}\)). Thus, the total sodium cations required would be twice the sum of moles of the divalent cations: Moles of sodium cations required = 2 * (moles of \(\mathrm{Ca}^{2+}\) + moles of \(\mathrm{Mg}^{2+}\)) Moles of sodium cations required = 2 * (30 moles + 6 moles) = 72 moles

Calculate the mass of sodium chloride needed to provide the required sodium cations

The sodium cations are to be supplied in the form of sodium chloride (\(\mathrm{NaCl}\)). To find out the mass of sodium chloride required, we need to convert the moles of sodium cations to moles of sodium chloride, which is a simple 1:1 ratio, and then use the molar mass of sodium chloride. Moles of \(\mathrm{NaCl}\) required = moles of \(\mathrm{Na}^{+}\) = 72 moles Mass of sodium chloride = moles of \(\mathrm{NaCl}\) * molar mass of \(\mathrm{NaCl}\) The molar mass of \(\mathrm{NaCl}\) is the sum of the atomic masses of sodium (Na, 22.99 g/mol) and chlorine (Cl, 35.45 g/mol): Molar mass of \(\mathrm{NaCl}\) = 22.99 g/mol + 35.45 g/mol = 58.44 g/mol Mass of sodium chloride = 72 moles * 58.44 g/mol = 4203.68 g So, 4203.68 grams of sodium chloride are needed to be added as a water softener.

Key concepts

Hard Water

Hard water is a type of water that contains high levels of dissolved minerals, specifically calcium (\( \mathrm{Ca}^{2+} \)) and magnesium (\( \mathrm{Mg}^{2+} \)) ions. These ions are the primary reason why hard water causes problems in daily water usage. For example, when heated, hard water leads to the formation of scale in pipes and containers. Scale is an insoluble solid that disrupts the flow of water and decreases the efficiency of heating appliances. Moreover, the calcium and magnesium ions in hard water interfere with soap's action, leading to less lather production and a necessity for more soap to clean effectively. To resolve these issues, water softeners are often used to replace these troublesome ions.

Calcium Ions

Calcium ions, denoted as \( \mathrm{Ca}^{2+} \), are commonly found in hard water. These ions originate from the dissolution of calcium-rich minerals like limestone and gypsum. In the context of water softening, calcium ions must be removed or replaced in order to prevent the negative effects they have on water systems. \( \mathrm{Ca}^{2+} \) ions are divalent, which means they carry a double positive charge. This property is essential when considering ion replacement because it requires two monovalent (single positive charged) ions like sodium (\( \mathrm{Na}^{+} \)) to replace one \( \mathrm{Ca}^{2+} \) ion in a water softener system.

Magnesium Ions

Magnesium ions (\( \mathrm{Mg}^{2+} \)) are another pervasive mineral component of hard water. Like calcium, \( \mathrm{Mg}^{2+} \) ions are divalent, meaning each magnesium ion carries two positive charges. The source of magnesium ions in water is often related to the erosion of rocks and minerals that contain magnesium, such as dolomite. Similar to calcium, magnesium ions also contribute to scale formation and soap inefficiency. In water softening processes, they are replaced by sodium ions to eliminate these issues, preserving pipelines and improving the ability for soap to clean effectively. The replacement is crucial for maintaining the functionality and longevity of water-related systems.

Sodium Chloride

Sodium chloride (\( \mathrm{NaCl} \)), commonly known as table salt, plays a pivotal role in the water softening process. It is used in water softeners to replace the hard water minerals, specifically calcium and magnesium ions, with sodium ions. This process occurs through ion exchange. In this mechanism, \( \mathrm{NaCl} \) dissolves in water and dissociates into sodium (\( \mathrm{Na}^{+} \)) and chloride (\( \mathrm{Cl}^{-} \)) ions. The sodium ions then replace the calcium and magnesium ions, softening the water. The chloride ions, being harmless, remain in the softened water and cause no further issues.

Molarity

Molarity is a measurement of concentration represented by the number of moles of a solute per liter of solution. It is denoted by the symbol \( M \). In the context of water softening, molarity helps in calculating how much of a particular ion, like calcium or magnesium, is present in a given volume of water. For example, if the molarity of calcium ions in water is given as 0.020 \( M \), it indicates that there are 0.020 moles of calcium ions in every liter of that water. This measure is crucial for determining the required amount of sodium ions necessary to replace the hard water ions through the water softening process. This ensures the precise alteration of water quality, making it suitable for household and industrial purposes.