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

The maximum uptake is \( \frac{1}{412} \) moles per gram of resin.

Step 1: Understanding the Problem

We need to find the maximum uptake of \( \mathrm{Ca}^{2+} \) ions by the resin. The resin exchanges \( \mathrm{Na}^+ \) ions with \( \mathrm{Ca}^{2+} \) ions. The molecular weight of \( \mathrm{C}_8\mathrm{H}_7\mathrm{SO}_3\mathrm{Na} \) is 206 g/mol. To solve the problem, we must understand how ion exchange occurs.

Step 2: Ion Exchange Mechanism

Since one \( \mathrm{Ca}^{2+} \) ion replaces two \( \mathrm{Na}^+ \) ions in the resin, each exchange corresponds to two molecular units of the resin, because the resin unit is originally balanced with one \( \mathrm{Na}^+ \).

Step 3: Calculating Maximum Uptake

For each \( \mathrm{Ca}^{2+} \) ion uptake, we need two resin units. Therefore, the molar mass for the exchange is 2 times 206 g, i.e., 412 g. The maximum moles of \( \mathrm{Ca}^{2+} \) ions per gram of resin is the inverse of this mass.

Step 4: Computing Final Answer

Calculate the maximum uptake in moles per gram: \( \frac{1}{412} \) moles of \( \mathrm{Ca}^{2+} \) ions per gram of resin.

Key Concepts

Ca2+ ion uptake

Ion exchange resins, such as the commercial resin with a molecular formula of \( \mathrm{C}_{8} \mathrm{H}_{7} \mathrm{SO}_{3} \mathrm{Na} \), are specifically designed to facilitate the exchange of ions. During this process, \( \mathrm{Na}^+ \) ions on the resin are swapped with \( \mathrm{Ca}^{2+} \) ions in the water, which is a common method in water softening. To understand the uptake of \( \mathrm{Ca}^{2+} \) ions, we need to recognize that \( \mathrm{Ca}^{2+} \) has a +2 charge, meaning each \( \mathrm{Ca}^{2+} \) displaces two \( \mathrm{Na}^+ \) ions, which only have a +1 charge each. This two-for-one exchange occurs because the resin maintains a balance in charge, ensuring the same total charge is retained after the ion swap.

• This means for each \( \mathrm{Ca}^{2+} \) ion taken up by the resin, two molecular units of the resin are required.
• This is why the molecular weight consideration is crucial in calculating maximum ion uptake.

Understanding this underwater balancing act is key to harnessing the potential of ion exchange resins effectively.

molecular weight calculation

Molecular weight plays a pivotal role in determining the capacity of ion exchange resins. Let's consider the molecular formula \( \mathrm{C}_{8} \mathrm{H}_{7} \mathrm{SO}_{3} \mathrm{Na} \). This complex combination results in a resin with a molecular weight of 206 g/mol. The calculation involves breaking down the formula into its basic components:

• Carbon (C): 8 atoms, each 12 g/mol, resulting in 96 g/mol
• Hydrogen (H): 7 atoms, each 1 g/mol, totaling 7 g/mol
• Sulfur (S): 1 atom at 32 g/mol
• Oxygen (O): 3 atoms, each 16 g/mol, giving 48 g/mol
• Sodium (Na): 1 atom at 23 g/mol

Adding these together, you arrive at a total of 206 g/mol. For ion exchange, when a \( \mathrm{Ca}^{2+} \) ion replaces two \( \mathrm{Na}^+ \) ions, the molecular weight of two resin units becomes crucial, because each \( \mathrm{Ca}^{2+} \) ion interacts with 2 units of resin, making 412 g (i.e., 2 x 206 g/mol). Understanding this helps in accurately calculating the capacity of the resin to absorb specific ions.

ion exchange mechanism

Ion exchange is a fascinating chemical process where ions are swapped between a solution and the resin. Here's how it happens: an ion exchange resin is composed of cross-linked polymer beads that have active sites. These sites initially hold \( \mathrm{Na}^+ \) ions. When exposed to water containing \( \mathrm{Ca}^{2+} \) ions, an exchange process occurs.

• Each \( \mathrm{Ca}^{2+} \) ion replaces two \( \mathrm{Na}^+ \) ions due to the charge difference.
• The resin beads are structured to allow easy passage and binding of \( \mathrm{Ca}^{2+} \) ions while releasing \( \mathrm{Na}^+ \).
• This process is crucial because it softens water by removing \( \mathrm{Ca}^{2+} \) ions, which are responsible for hardness.

The efficiency of this exchange is what determines how much calcium the resin can absorb per gram. By using resins effectively, the hardness of the water can be significantly reduced, making it softer and preventing issues that arise from hard water usage.