Problem 178 - A Complete Resource Book in Chemistry for JEE Main

Question

The molecular formula of a commercial resin used for exchanging ions in water softening is \(\mathrm{C}_{8} \mathrm{H}_{7} \mathrm{SO}_{3} \mathrm{Na}\) (mol. wt. 206). What would be the maximum uptake of \(\mathrm{Ca}^{2+}\) ions by the resin when expressed in mole per gram resin? (a) \(\frac{1}{103}\) (b) \(\frac{1}{206}\) (c) \(\frac{2}{309}\) (d) \(\frac{1}{412}\)

Step-by-Step Solution

Answer

(d) \(\frac{1}{412}\)

Step 1: Understand the Problem

The problem requires us to calculate the maximum uptake of \( \mathrm{Ca}^{2+} \) ions per gram of the resin, whose molecular formula is \( \mathrm{C}_{8} \mathrm{H}_{7} \mathrm{SO}_{3} \mathrm{Na} \) with a molecular weight of 206 g/mol.

Step 2: Analyze the Resin's Function

The resin acts by exchanging \( \mathrm{Na}^+ \) ions with \( \mathrm{Ca}^{2+} \) ions. Since one \( \mathrm{Ca}^{2+} \) ion replaces two \( \mathrm{Na}^+ \) ions, we need two resin molecules per \( \mathrm{Ca}^{2+} \) ion.

Step 3: Determine Moles of Resin per Gram

The molecular weight of the resin is 206 g/mol. Therefore, 1 gram of the resin corresponds to \( \frac{1}{206} \) moles of resin.

Step 4: Calculate Exchange Ratio

Since two molecules of resin are required to exchange with one \( \mathrm{Ca}^{2+} \) ion, we divide the number of moles in a gram of resin by 2. This gives us \( \frac{1}{2} \times \frac{1}{206} = \frac{1}{412} \) moles of \( \mathrm{Ca}^{2+} \) ions per gram of resin.

Key Concepts

Water Softening

Water softening is an essential process that eliminates unwanted minerals, like calcium and magnesium, which cause water to be 'hard'. Hard water, though not harmful to health, can lead to various issues: clogged pipes, ineffective soap and detergents, and unsightly deposits on fixtures. The water softening process typically involves **ion exchange**, a method that uses resin beads to capture these minerals and replace them with more benign ions like sodium or potassium.

Here's how the process typically works:

• Hard water flows through a bed of resin beads contained in a water-softening system.
• The resin beads, which are charged with sodium ions, attract calcium and magnesium ions from the water.
• As the calcium and magnesium ions bind to the resin beads, sodium ions are released in their place.
• This exchange continues until the resin beads contain an excess of calcium and magnesium ions, at which point the resin must be regenerated.

Understanding how water softening systems function is critical to grasping how ion exchange resins work, especially in applications such as reducing hard water's effects.

Resin Molecular Formula

The resin's molecular formula, \( \mathrm{C}_{8} \mathrm{H}_{7} \mathrm{SO}_{3} \mathrm{Na} \), provides insight into its composition and functionality within the water-softening process. This formula represents a complex organic compound that houses a sodium ion capable of being exchanged with other ions like calcium. The molecular weight of 206 g/mol, noted in the problem, is foundational for calculating how the resin performs within ion exchange.The composition can be broken down as follows:

• **Carbon atoms (C extsubscript{8})** — Form the backbone of the resin, providing structure and stability.
• **Hydrogen atoms (H extsubscript{7})** — Bond with carbon atoms to create the resin framework.
• **Sulfonate group (SO extsubscript{3})** — Essential for allowing charge exchange, as this group holds the key \( \mathrm{Na}^+ \) ion.
• **Sodium ion (Na extsuperscript{+})** — Easily exchangeable with calcium ions, leading to the softening of water.

This molecular formula highlights the **repeated structure** within the resin, a critical aspect of its function in long-term and effective ion exchange. It shows how resins are designed to consistently capture and release ions, ensuring efficient water treatment over time.

Calcium Ion Uptake

Calcium ion uptake is a critical measure of a resin's effectiveness in water softening. This uptake involves the resin's capacity to exchange sodium ions with calcium ions from hard water fully. The concept is vital because the amount of calcium ions a resin can uptake determines its efficacy.The step-by-step solution provides insight into this process:

• The molecular weight of the resin is 206 g/mol, giving a guide to how many molecules are present per gram.
• Since one calcium ion (\( \mathrm{Ca}^{2+} \)) replaces two sodium ions (\( \mathrm{Na}^{+} \)), each calcium ion requires two resin molecules for exchange.
• This exchange results in \( \frac{1}{206} \times \frac{1}{2} \), equating to \( \frac{1}{412} \), calculated as the moles of calcium ions a single gram of resin can uptake.

This calculation is crucial for understanding how much calcium can be removed, as it shows not just the theoretical capacity but also practical limitations in water softening systems. Having this clear understanding allows adjustments in softening systems for maximum efficiency, ensuring optimal operation and maintenance.