Problem 95 - Chemistry The Molecular Science

Question

In a home, hard water containing \(500 . \mathrm{mg} \mathrm{Ca}^{2+} / \mathrm{gal}\) passed through the \(\mathrm{Na}^{+}\) -based ion-exchange water softener. If there was 200 gal. water and the ion-exchange resin operates at \(100 \%\) efficiency, calculate the mass of \(\mathrm{Na}^{+}\) ions displaced from the resin.

Step-by-Step Solution

Answer

114.73 g of Na\(^+\) ions are displaced.

Step 1: Understand the Ion Exchange Process

When hard water passes through an Na\(^+\)-based ion-exchange water softener, the calcium ions \(\text{Ca}^{2+}\) are exchanged for sodium ions \(\text{Na}^+\). Each \(\text{Ca}^{2+}\) ion displaces two \(\text{Na}^+\) ions.

Step 2: Calculate Total Mass of Calcium Ions

The concentration of \(\text{Ca}^{2+}\) ions is 500 mg/gal. Therefore, for 200 gallons of water, the total mass of \(\text{Ca}^{2+}\) ions is: \[ 200 \, \text{gal} \times 500 \, \text{mg/gal} = 100,000 \, \text{mg} \] which is 100 g.

Step 3: Convert Mass of Calcium Ions to Moles

The molar mass of \(\text{Ca}^{2+}\) is approximately 40.08 g/mol. Calculate the moles of \(\text{Ca}^{2+}\): \[ \frac{100 \, \text{g}}{40.08 \, \text{g/mol}} = 2.495 \, \text{mol} \]

Step 4: Calculate Moles of Sodium Ions Displaced

Since each \(\text{Ca}^{2+}\) ion displaces two \(\text{Na}^+\) ions, the moles of \(\text{Na}^+\) displaced is: \[2.495 \, \text{mol \(\text{Ca}^{2+}\)} \times 2 = 4.99 \, \text{mol \(\text{Na}^+\)} \]

Step 5: Calculate Mass of Sodium Ions Displaced

The molar mass of Na\(^+\) is approximately 22.99 g/mol. Thus, the mass of \(\text{Na}^+\) displaced is: \[4.99 \, \text{mol} \times 22.99 \, \text{g/mol} = 114.73 \, \text{g} \]

Key Concepts

Water Softening

Water softening is a process used to remove calcium and magnesium ions from hard water, transforming it into soft water. Hard water is notorious for causing issues due to the presence of high levels of calcium \(\text{Ca}^{2+}\) and magnesium \(\text{Mg}^{2+}\) ions. These ions contribute to the formation of scale in pipes and on surfaces where water is used. This can lead to lower efficiency of heating systems and other household chores becoming more difficult. The most common method for softening water involves using an ion-exchange system. In these systems, hard water flows through a resin bed where \(\text{Ca}^{2+}\) and \(\text{Mg}^{2+}\) ions are exchanged with \(\text{Na}^+\) or \(\text{K}^+\) ions that are innately part of the resin structure.

• These devices are efficient and result in water that lathers easily with soap and helps in extending the life of plumbing and appliances.
• By removing hard ions, they prevent mineral buildup that can cause clogs and malfunction in systems over time, maintaining proper efficiency.

Once the resin's capacity to exchange ions is exhausted, regeneration is needed to replace the \(\text{Na}^+\) or \(\text{K}^+\) ions, usually involving a saline solution.

Hard Water

Hard water contains a high concentration of calcium and magnesium ions. These are minerals picked up as water moves from its source, through rocks and soil, and finally to homes. Although hard water isn't harmful to one's health, it can cause several inconveniences and maintenance issues in household plumbing and appliances.

• Limescale build-up on fixtures and in heating elements diminishes efficiency and sometimes results in damage.
• Laundry may feel scratchy or look dingy because detergents are less effective in hard water.
• Dishes may come out of the dishwasher spotted or cloudy.

Understanding the impact of hard water motivates the need for softening devices. Knowing the local water hardness level by testing and then choosing an appropriate water softening solution can save significant maintenance costs and improve daily living conditions.

Chemical Calculations

Performing chemical calculations involves a step-by-step approach to solving quantitative problems in chemistry. The solution to any calculation begins by understanding the principles of the chemical process involved, such as an ion exchange in water softening systems. The first step in calculations is determining the total mass of the ions involved. In the given exercise, for instance, determining the total mass of \(\text{Ca}^{2+}\) ions in hard water involves multiplying the concentration per gallon by the volume of water. Next, conversion from mass to moles is performed using the molar mass of the ions, a critical step which allows further stoichiometric calculations. For \(\text{Ca}^{2+}\), this involves dividing the mass by its molar mass, yielding the number of moles.Following this, calculations can ascertain the moles of exchanged ions. Since each \(\text{Ca}^{2+}\) ion displaces two \(\text{Na}^+\) ions, the moles of sodium ions displaced are twice that of calcium.Finally, to determine the mass of sodium ions displaced, multiplying the moles by the molar mass of sodium completes the calculation.

• Ensuring accuracy in these steps is crucial for correct results.
• This structured problem-solving method is applicable across various chemical calculations.