Question

Water "softeners" remove metal ions such as \(\mathrm{Ca}^{2+}\) and \(\mathrm{Fe}^{3+}\) by replacing them with enough \(\mathrm{Na}^{+}\) ions to maintain the same number of positive charges in the solution. If \(1.0 \times 10^{3} \mathrm{~L}\) of "hard" water is \(0.015 \mathrm{M} \mathrm{Ca}^{2+}\) and \(0.0010 \mathrm{M} \mathrm{Fe}^{3+},\) how many moles of \(\mathrm{Na}^{+}\) are needed to replace these ions?

Short answer

33 moles of \(\text{Na}^{+}\) are needed.

Step-by-step solution

Calculate the Moles of \(\text{Ca}^{2+}\)

First, determine the number of moles of \(\text{Ca}^{2+}\) in the solution. Use the molarity equation \(n = M \times V\). For \(Ca^{2+}\), where \(M = 0.015 \text{ M}\) and \(V = 1.0 \times 10^{3} \text{ L}\): \[ n_{\text{Ca}^{2+}} = 0.015 \text{ M} \times 1.0 \times 10^{3} \text{ L} = 15 \text{ moles} \]

Calculate the Moles of \(\text{Fe}^{3+}\)

Next, determine the number of moles of \(\text{Fe}^{3+}\) in the solution. Again, use the molarity equation. For \(Fe^{3+}\), where \(M = 0.0010 \text{ M}\) and \(V = 1.0 \times 10^{3} \text{ L}\): \[ n_{\text{Fe}^{3+}} = 0.0010 \text{ M} \times 1.0 \times 10^{3} \text{ L} = 1 \text{ mole} \]

Calculate Moles of \(\text{Na}^{+}\) Needed for \(\text{Ca}^{2+}\)

To replace \(\text{Ca}^{2+}\) with \(\text{Na}^{+}\), use the charge balance principle. Each \(\text{Ca}^{2+}\) ion requires 2 \(\text{Na}^{+}\) ions. Therefore, the moles of \(\text{Na}^{+}\) needed for \(\text{Ca}^{2+}\) is: \[ n_{\text{Na}^{+} (\text{for } \text{Ca}^{2+})} = 2 \times 15 = 30 \text{ moles} \]

Calculate Moles of \(\text{Na}^{+}\) Needed for \(\text{Fe}^{3+}\)

To replace \(\text{Fe}^{3+}\) with \(\text{Na}^{+}\), use the charge balance principle. Each \(\text{Fe}^{3+}\) ion requires 3 \(\text{Na}^{+}\) ions. Therefore, the moles of \(\text{Na}^{+}\) needed for \(\text{Fe}^{3+}\) is: \[ n_{\text{Na}^{+} (\text{for } \text{Fe}^{3+})} = 3 \times 1 = 3 \text{ moles} \]

Total Moles of \(\text{Na}^{+}\) Needed

Finally, add the moles of \(\text{Na}^{+}\) required for \(\text{Ca}^{2+}\) and \(\text{Fe}^{3+}\) to find the total moles of \(\text{Na}^{+}\) needed: \[ n_{\text{Na}^{+} (\text{total})} = 30 \text{ moles} + 3 \text{ moles} = 33 \text{ moles} \]

Key concepts

Molarity Equations

Molarity is a way to express the concentration of a solution. It is defined as the number of moles of solute per liter of solution, usually denoted by the symbol \(M\), and calculated using the formula: \[ M = \frac{n}{V} \] Here, \(n\) represents the number of moles of the solute, and \(V\) is the volume of the solution in liters. For instance, in this exercise, we calculated the number of moles of \(\text{Ca}^{2+}\) and \(\text{Fe}^{3+}\) by multiplying their molarities by the volume of the solution: \[ n_{\text{Ca}^{2+}} = 0.015 \text{ M} \times 1.0 \times 10^{3} \text{ L} = 15 \text{ moles} \] \[ n_{\text{Fe}^{3+}} = 0.0010 \text{ M} \times 1.0 \times 10^{3} \text{ L} = 1 \text{ mole} \] Understanding molarity equations is crucial for accurate solution preparation and analysis in chemistry.

Ion Exchange

Ion exchange is a process where unwanted ions in a solution are replaced by preferred ions using an ion-exchange resin. This principle is fundamental in water softening systems, where ions like \(\text{Ca}^{2+}\) and \(\text{Fe}^{3+}\) from hard water are replaced with \(\text{Na}^{+}\) ions to prevent scale formation and improve water quality. In our exercise, we replaced both \(\text{Ca}^{2+}\) and \(\text{Fe}^{3+}\) ions with \(\text{Na}^{+}\) ions. Specifically, 15 moles of \(\text{Ca}^{2+}\) were replaced by 30 moles of \(\text{Na}^{+}\), and 1 mole of \(\text{Fe}^{3+}\) was replaced by 3 moles of \(\text{Na}^{+}\). This maintenance of overall ion balance is what ensures effective water softening.

Charge Balance

Charge balance is a principle that ensures the total charge of ions in a solution remains constant during chemical reactions. This is essential for maintaining electrical neutrality. In water softening, when \(\text{Ca}^{2+}\) and \(\text{Fe}^{3+}\) ions are replaced by \(\text{Na}^{+}\) ions, it is crucial to add a sufficient number of \(\text{Na}^{+}\) ions to balance the charges. For instance:

• \(15\) moles of \(\text{Ca}^{2+}\) (each with a \(+2\) charge) require \(30\) moles of \(\text{Na}^{+}\) ions (each with a \(+1\) charge) to maintain charge balance.
• Similarly, \(1\) mole of \(\text{Fe}^{3+}\) (with a \(+3\) charge) requires \(3\) moles of \(\text{Na}^{+}\).

Adding these up, the total number of \(\text{Na}^{+}\) moles needed to replace all \(\text{Ca}^{2+}\) and \(\text{Fe}^{3+}\) ions is \(33\) moles. Understanding charge balance ensures the solution remains electrically neutral and chemically stable.