Problem 101 - Chemistry The Central Science In Si Units

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

Step-by-Step Solution

Answer

72 moles of \( \mathrm{Na}^{+} \) needed; 4207.68 g \( \mathrm{NaCl} \).

Step 1: Calculate moles of hard water ions

First, calculate the moles of \( \mathrm{Ca}^{2+} \) and \( \mathrm{Mg}^{2+} \) in the hard water. Use the formula: \( \text{moles} = \text{molarity} \times \text{volume} \). For \( \mathrm{Ca}^{2+} \):\[ \text{moles of } \mathrm{Ca}^{2+} = 0.020 \; \mathrm{M} \times 1500 \; \mathrm{L} = 30 \; \text{moles}\]For \( \mathrm{Mg}^{2+} \):\[ \text{moles of } \mathrm{Mg}^{2+} = 0.0040 \; \mathrm{M} \times 1500 \; \mathrm{L} = 6 \; \text{moles} \]

Step 2: Relate moles of ions to moles of \( \mathrm{Na}^{+} \)

To replace the \( \mathrm{Ca}^{2+} \) and \( \mathrm{Mg}^{2+} \) ions with \( \mathrm{Na}^{+} \), the sodium ions must balance the charge. Each \( \mathrm{Ca}^{2+} \) requires 2 \( \mathrm{Na}^{+} \), and each \( \mathrm{Mg}^{2+} \) also requires 2 \( \mathrm{Na}^{+} \). Calculate the total moles of \( \mathrm{Na}^{+} \) needed:\[ 2 \times (30 + 6) = 72 \; \text{moles of } \mathrm{Na}^{+} \]

Step 3: Calculate grams of \( \mathrm{NaCl} \) needed

To find out how much \( \mathrm{NaCl} \) is needed, use the molar mass of \( \mathrm{NaCl} \), which is approximately 58.44 g/mol. Calculate the mass:\[ \text{mass of } \mathrm{NaCl} = 72 \; \text{moles} \times 58.44 \; \text{g/mol} = 4207.68 \; \text{g} \]

Key Concepts

hard water

Hard water is water that contains high concentrations of certain metal ions, such as calcium (\( \mathrm{Ca}^{2+} \)), magnesium (\( \mathrm{Mg}^{2+} \)), and sometimes iron (\( \mathrm{Fe}^{2+} \)). These metal ions can come from groundwater sources that flow over or through rocks containing calcium or magnesium minerals.

Hard water is known for causing a variety of issues. One of the main problems is that these ions react with soap to form scum or an insoluble precipitate, which reduces soap's effectiveness. This can be particularly bothersome for household cleaning, as it makes it difficult to form a good lather.

Additionally, when hard water is heated, the calcium and magnesium ions can precipitate out as solid deposits on pipes and appliances. This can lead to issues such as reduced water flow and higher energy costs. To address these issues, water is often treated to become "soft," usually by removing these ions.

calcium ions

Calcium ions, \( \mathrm{Ca}^{2+} \), are commonly found in hard water. These ions enter the water supply mainly through leaching from calcium-rich minerals, like limestone, marble, and chalk. As a divalent cation, calcium plays a significant role in water hardness.

In water softening processes, calcium ions are often replaced by sodium ions (\( \mathrm{Na}^{+} \)) to reduce water hardness. This exchange occurs in systems such as ion-exchange water softeners. The reason for this replacement is that sodium does not form insoluble precipitates with soaps and detergents, thus preventing issues like soap scum.

The calcium ion exchange process helps maintain functionality in plumbing systems and water-using appliances. It also improves the efficiency of soaps and detergents, making washing and cleaning easier and more effective.

magnesium ions

Magnesium ions, \( \mathrm{Mg}^{2+} \), are another contributor to the hardness of water. Like calcium ions, magnesium ions are often released into the water from mineral deposits. These ions can interfere with the cleaning action of soaps and detergents.

Magnesium can create similar issues to calcium in terms of leaving mineral deposits in pipes and appliances, often referred to as scale. These deposits can reduce the lifespan of plumbing systems and increase the need for maintenance.

In water softening, magnesium ions are also exchanged for sodium ions, helping in mitigating the hardness of the water. The replacement process allows for more effective cleaning and reduces the potential for costly maintenance of water systems due to mineral buildup.

sodium chloride

Sodium chloride (\( \mathrm{NaCl} \)), commonly known as table salt, is often used in water softening processes. In these systems, sodium chloride provides the sodium ions needed for the ion-exchange process. The sodium ions replace the calcium and magnesium ions present in hard water.

The choice of sodium chloride is convenient because it is readily available and relatively inexpensive. It dissolves well in water, allowing sodium ions to easily exchange places with the hardness ions. These systems usually involve a resin bed where the sodium ions are housed, and as hard water passes through, the exchange occurs, effectively softening the water.

The softened water, now carrying sodium ions, does not form deposits with household soaps making them more effective. Regular replenishment of sodium chloride is necessary to ensure the continued effectiveness of the water softening system.

molarity calculations

Molarity is a key concept used in calculating the concentration of ions or molecules in a solution. It is defined as the number of moles of solute per liter of solution, expressed as \( ext{M} \) (molar).

The formula for molarity is:\[ \text{Molarity} = \frac{\text{moles of solute}}{\text{liters of solution}} \]

In the context of water softening, molarity calculations help determine the number of moles of ions present in a given volume of hard water. For example, given a certain molarity and volume, you can compute the moles of \( \mathrm{Ca}^{2+} \) or \( \mathrm{Mg}^{2+} \) ions that need to be replaced by sodium ions.

Correct calculations are essential to ensure that the right amount of sodium chloride is used in the softening process. Knowing how to perform these calculations is vital for maintaining an effective and efficient water softening system.