Problem 12 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

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) \(\frac{2}{309}\) (b) \(\frac{1}{412}\) (c) \(\frac{1}{103}\) (d) \(\frac{1}{206}\)

Step-by-Step Solution

Answer

(d) \( \frac{1}{206} \) moles of \( \mathrm{Ca}^{2+} \) per gram.

Step 1: Understand the Resin's Function

The resin functions as an ion exchange medium to replace sodium ions, \( \mathrm{Na}^{+} \), with divalent calcium ions, \( \mathrm{Ca}^{2+} \). In this exchange, two \( \mathrm{Na}^{+} \) ions are replaced by one \( \mathrm{Ca}^{2+} \) ion.

Step 2: Calculate Moles of \( \mathrm{Ca}^{2+} \) per Gram

Given the molar mass of the resin repeating unit is 206 g/mol, we calculate the moles of \( \mathrm{Ca}^{2+} \) ions that can be exchanged per unit of resin. Since it takes two repeating units to exchange one \( \mathrm{Ca}^{2+} \) ion, the calculation is \( \frac{1}{2 \times 206} \) moles of \( \mathrm{Ca}^{2+} \) per gram.

Key Concepts

Molecular Formula

Molecular formulas are crucial for understanding the makeup of any compound. In the world of chemistry, a molecular formula gives us insight into the types and numbers of each atom present in a molecule. For instance, the molecular formula of the ion exchange resin given in the exercise is \( \mathrm{C}_{8} \mathrm{H}_{7} \mathrm{SO}_{3}^{-} \mathrm{Na}^{+} \). This tells us that each molecule consists of 8 carbon atoms, 7 hydrogen atoms, 3 oxygen atoms, one sulfur atom, and it bears a sodium ion.

Knowing the molecular formula is essential for reactions such as ion exchange, as it reveals the structure and the possible interactions with other ions. In the case of water softening, it's about how the sodium ions in the resin can be replaced by calcium ions. When you understand the components, you understand how the resin interacts within a system.

• The carbon, hydrogen, and oxygen atoms form the backbone of the resin molecule.
• The sulfate \( \mathrm{SO}_{3}^{-} \) contributes to the resin's negative charge, which is crucial for attracting positively charged ions.
• The sodium \( \mathrm{Na}^{+} \) is present in exchange for divalent ions like calcium \( \mathrm{Ca}^{2+} \).

Calcium Ion Uptake

Ion exchange resins are fascinating in how they operate. Specifically, they work by swapping ions: one type of ion for another. In our case, the resin exchanges sodium ions \( \mathrm{Na}^{+} \) for calcium ions \( \mathrm{Ca}^{2+} \).

However, here's a key point to remember: calcium ions have a double positive charge, while sodium ions have a single positive charge. This means that it takes two sodium ions to ensure an equivalent exchange for one calcium ion.

The theoretical maximum uptake for \( \mathrm{Ca}^{2+} \) ions is derived from this balance. Basically, for every mole of \( \mathrm{Ca}^{2+} \) taken up, the resin releases two moles of \( \mathrm{Na}^{+} \), balancing the charges.

• Exchange occurs due to the charge difference between \( \mathrm{Ca}^{2+} \) and \( \mathrm{Na}^{+} \).
• Understanding this uptake helps predict how much calcium can be softened from water.
• Effective water softening relies on knowing these molecular exchanges accurately.

Molar Mass Calculation

Calculating the molar mass of a compound is a straightforward yet essential task. It's about finding out how much one mole of a substance weighs in grams. The molar mass of our resin's formula is stated to be 206 g/mol, which means every mole of this resin weighs 206 grams.

This information ties directly into how we calculate calcium ion uptake per gram of resin. When determining how many moles of \( \mathrm{Ca}^{2+} \) the resin can accommodate, the molar mass helps us convert between the amount of resin and moles of ions exchanged. Remember, to replace a \( \mathrm{Ca}^{2+} \) ion, we need two repeating units of the resin.

The conversion involves dividing the molar mass by the number of units needed in the exchange. So, in our case, to find moles of \( \mathrm{Ca}^{2+} \) per gram of resin, we perform the following calculation: \( \frac{1}{2 \times 206} \), leading to \( \frac{1}{412} \) moles of \( \mathrm{Ca}^{2+} \) per gram.

• Molar mass is the weight of one mole, crucial for conversions.
• Understanding this concept ensures precise calculations in chemistry.
• It's directly applicable to tasks like determining ion uptake and compound calculations.