The molecular formula of a commercial resin used for exchanging ions in water softening is `C_(8)H_(7)SO_(3)Na(mol. Wt. 206)` . What would be the maximum uptake of `Ca^(2+)` ions by the resin when expressed in mole per gram resin?

Options

A. `(2)/(309)`
B. `(1)/(412)`
C. `(1)/(103)`
D. `(1)/(206)`

Explanation

To find the maximum uptake of \( \text{Ca}^{2+} \) ions by the resin expressed in moles per gram of resin, we can follow these steps:

### Step 1: Understand the Ion Exchange Reaction
The resin has the formula \( \text{C}_8\text{H}_7\text{SO}_3\text{Na} \). In the ion exchange process, one sodium ion (\( \text{Na}^+ \)) from the resin is exchanged for one calcium ion (\( \text{Ca}^{2+} \)). However, since \( \text{Ca}^{2+} \) has a +2 charge, it will replace two sodium ions.

### Step 2: Write the Reaction
The ion exchange reaction can be represented as:
\[ \text{C}_8\text{H}_7\text{SO}_3\text{Na} + \text{Ca}^{2+} \rightarrow \text{C}_8\text{H}_7\text{SO}_3\text{Ca} + 2\text{Na}^+ \]
From this reaction, we can see that 1 mole of \( \text{Ca}^{2+} \) replaces 2 moles of \( \text{Na}^+ \).

### Step 3: Determine Moles of Resin
The molecular weight of the resin \( \text{C}_8\text{H}_7\text{SO}_3\text{Na} \) is given as 206 g/mol.

### Step 4: Calculate Moles of Resin in 1 Gram
To find the number of moles of resin in 1 gram, we use the formula:
\[ \text{Moles of resin} = \frac{\text{mass of resin}}{\text{molar mass of resin}} = \frac{1 \text{ g}}{206 \text{ g/mol}} \approx 0.00485 \text{ mol} \]

### Step 5: Calculate Maximum Uptake of \( \text{Ca}^{2+} \)
From the stoichiometry of the reaction, we know that 2 moles of resin can exchange for 1 mole of \( \text{Ca}^{2+} \). Therefore, the amount of \( \text{Ca}^{2+} \) that can be exchanged by 1 mole of resin is:
\[ \text{Moles of } \text{Ca}^{2+} = \frac{1 \text{ mol } \text{Ca}^{2+}}{2 \text{ mol resin}} = 0.5 \text{ mol } \text{Ca}^{2+} \text{ per mol resin} \]
Thus, for 0.00485 moles of resin:
\[ \text{Moles of } \text{Ca}^{2+} = 0.5 \times 0.00485 \approx 0.002425 \text{ mol} \]

### Step 6: Express in Moles per Gram of Resin
Since we calculated the uptake for 1 gram of resin, the maximum uptake of \( \text{Ca}^{2+} \) ions by the resin expressed in moles per gram of resin is:
\[ \text{Maximum uptake of } \text{Ca}^{2+} = 0.002425 \text{ mol/g} \]

### Final Answer
The maximum uptake of \( \text{Ca}^{2+} \) ions by the resin when expressed in moles per gram of resin is approximately \( 0.002425 \text{ mol/g} \).

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