Zeolite \(\mathrm{A}, \mathrm{Na}_{12}\left[\left(\mathrm{AlO}_{2}\right)_{12}\left(\mathrm{SiO}_{2}\right)_{12}\right] \cdot 27 \mathrm{H}_{2} \mathrm{O},\) is used to soften water because it replaces \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) dissolved in the water with \(\mathrm{Na}^{+}\). Hard water from a certain source is \(4.5 \times 10^{-3} \mathrm{M}\) \(\mathrm{Ca}^{2+}\) and \(9.2 \times 10^{-4} \mathrm{M} \mathrm{Mg}^{2+},\) and a pipe delivers \(25,000 \mathrm{~L}\) of this hard water per day. What mass (in kg) of zeolite A is needed to soften a week's supply of the water? (Assume zeolite A loses its capacity to exchange ions when \(85 \mathrm{~mol} \%\) of its \(\mathrm{Na}^{+}\) has been lost.)

Step by step solution

01

Understand the Ion Exchange Process

Zeolite A replaces \( \mathrm{Ca}^{2+} \) and \( \mathrm{Mg}^{2+} \) in water with \( \mathrm{Na}^{+} \). This means each mole of \( \mathrm{Ca}^{2+} \) and \( \mathrm{Mg}^{2+} \) needs two moles of \( \mathrm{Na}^{+} \) for replacement.

02

Calculate the Total Moles of \( \mathrm{Ca}^{2+} \) and \( \mathrm{Mg}^{2+} \) in One Day

Total moles of \( \mathrm{Ca}^{2+} \) per day: \[ 4.5 \times 10^{-3} \, \mathrm{mol/L} \times 25000 \, \mathrm{L} = 112.5 \, \mathrm{mol} \] Total moles of \( \mathrm{Mg}^{2+} \) per day: \[ 9.2 \times 10^{-4} \, \mathrm{mol/L} \times 25000 \, \mathrm{L} = 23 \, \mathrm{mol} \]

03

Calculate the Weekly Moles of \( \mathrm{Ca}^{2+} \) and \( \mathrm{Mg}^{2+} \)

The water supply is for a week, so multiply the daily moles by 7: \[ 112.5 \, \mathrm{mol/day} \times 7 \, \mathrm{days} = 787.5 \, \mathrm{mol} \] \[ 23 \, \mathrm{mol/day} \times 7 \, \mathrm{days} = 161 \, \mathrm{mol} \]

04

Calculate the Total Moles of \( \mathrm{Na}^{+} \) Required for Ion Exchange

Each \( \mathrm{Ca}^{2+} \) and \( \mathrm{Mg}^{2+} \) ion requires 2 \( \mathrm{Na}^{+} \) ions for exchange: \[ (787.5 \,\mathrm{mol} \, \mathrm{Ca}^{2+} + 161 \,\mathrm{mol} \, \mathrm{Mg}^{2+}) \times 2 = 1897 \, \mathrm{mol} \, \mathrm{Na}^{+} \]

05

Determine the Effective Moles of \( \mathrm{Na}^{+} \) from Zeolite \

Zeolite A loses its capacity at 85% of \( \mathrm{Na}^{+} \'s \) total capacity. Total moles of \mathrm{Na}^{+} \ in zeolite A per mole: 12 (as zeolite formula shows 12 \( \mathrm{Na}^{+} \) per formula unit). Effective moles of \( \mathrm{Na}^{+} \) per mole of zeolite A: \[ 12 \, \mathrm{mol} \, \times 0.85 = 10.2 \, \mathrm{mol} \]

06

Calculate the Moles of Zeolite Needed

Divide the total required moles of \( \mathrm{Na}^{+} \) by the effective moles provided by zeolite A: \[ \frac{1897 \, \mathrm{mol} }{10.2 \, \mathrm{mol/mol \, zeolite}} = 186.96 \, \mathrm{mol \, zeolite} \]

07

Convert Moles of Zeolite to Mass

Calculate the molar mass of zeolite A: \[ \mathrm{Na}_{12} \left[ \left( \mathrm{AlO}_{2} \right)_{12} \left( \mathrm{SiO}_{2} \right)_{12} \right] \cdot 27 \mathrm{H}_{2} \mathrm{O} \] \[ (12 \times 22.99) + (12 \times 26.98 + 24 \times 16) + (12 \times 28.09 + 24 \times 16) + (27 \times 18.015) = 2194.2 \, \mathrm{g/mol} \] Multiply moles by molar mass: \[ 186.96 \, \mathrm{mol} \times 2194.2 \, \mathrm{g/mol} = 410,176.632 \, \mathrm{g} \approx 410.177 \, \mathrm{kg} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Water Softening

Water softening is the process of removing hardness-causing minerals from water. Hard water contains high levels of calcium (Ca²⁺) and magnesium (Mg²⁺) ions. These minerals can cause scale buildup in pipes and reduce the effectiveness of soap. Water softening systems often use ion exchange methods to replace Ca²⁺ and Mg²⁺ ions with sodium (Na⁺) ions. This process ensures that water used in homes and industries is more manageable and less likely to cause damage. Without water softening, equipment and plumbing systems would face higher maintenance costs and decreased lifespans.

Ion Exchange

Ion exchange is a reversible chemical reaction where ions are swapped between a solution and an ion exchanger. Zeolites, a type of ion exchanger, are particularly effective in water softening. In this process,

• The hard water runs through a bed of zeolite particles.
• Zeolite particles contain Na⁺ ions.
• Ca²⁺ and Mg²⁺ ions in the water are exchanged with Na⁺ ions in the zeolite.

Each Ca²⁺ and Mg²⁺ ion requires two Na⁺ ions for replacement due to their respective charges. This ensures that the softened water leaving the system has Na⁺ instead of the problematic Ca²⁺ and Mg²⁺.

Molar Calculations

Molar calculations are crucial for determining the amount of substances in chemical reactions. Let’s illustrate this with the exercise: For 25,000 liters of water containing 4.5 x 10^-3 M Ca²⁺ and 9.2 x 10^-4 M Mg²⁺, calculating the total moles per day involves:

• Total moles of Ca²⁺ per day: 4.5 x 10^-3 mol/L x 25,000 L = 112.5 mol
• Total moles of Mg²⁺ per day: 9.2 x 10^-4 mol/L x 25,000 L = 23 mol.

Multiplying by 7 days for a week's worth:
Total weekly moles of Ca²⁺ = 112.5 mol/day x 7 days = 787.5 mol
Total weekly moles of Mg²⁺ = 23 mol/day x 7 days = 161 mol
Summing up the values: (787.5 + 161) mols = 1897 mols of Na⁺ ions needed weekly.

Zeolites

Zeolites are naturally occurring or synthetic minerals with a unique framework structure, making them excellent for ion exchange processes. Zeolite A, for instance, is a synthetic zeolite used in water softening.
It has a chemical formula of Na₁₂[AlO₂]₁₂(SiO₂)₁₂ · 27 H₂O. This means each unit has 12 Na⁺ ions positioned to exchange with hard water ions. However, zeolite A can lose its effectiveness and must be replaced or regenerated. In our calculation, we assume it's 85% effective, meaning each unit gives 10.2 Na⁺ ions for ion exchange.

Hard Water Treatment

Treating hard water is essential for maintaining household and industrial systems. The exercise's problem involved determining how much zeolite A was necessary to soften 25,000 liters of hard water over a week.

• First step: Calculate the total moles of Ca²⁺ and Mg²⁺ per day and for the week.
• Next, determine the total moles of Na⁺ ions required, assuming each divalent ion (Ca²⁺, Mg²⁺) needs two Na⁺ ions.

We found that we need 1897 mols of Na⁺ ions to soften a week's supply.
By knowing zeolite A's molar mass and effective ion exchange capacity, we calculated the required moles needed and converted it to find a weight of approximately 410.177 kg of zeolite A.