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

Answer

To solve the problem of determining the maximum uptake of \( \text{Ca}^{2+} \) ions by the resin \( \text{C}_8\text{H}_7\text{SO}_3\text{Na} \), we will follow these steps:

### Step 1: Understand the Molecular Weight
The molecular weight of the resin \( \text{C}_8\text{H}_7\text{SO}_3\text{Na} \) is given as 206 g/mol. This means that 1 mole of this resin weighs 206 grams.

### Step 2: Calculate Moles of Resin per Gram
To find out how many moles of resin are present in 1 gram, we can use the formula:
\[ \text{Moles of resin} = \frac{1 \text{ gram}}{\text{Molar mass of resin (g/mol)}} \]
Substituting the values:
\[ \text{Moles of resin} = \frac{1 \text{ gram}}{206 \text{ g/mol}} = \frac{1}{206} \text{ moles} \]

### Step 3: Understand the Ion Exchange Reaction
In the ion exchange process, \( \text{Na}^+ \) ions in the resin are exchanged for \( \text{Ca}^{2+} \) ions from hard water. The stoichiometry of the reaction indicates that:
\[ 2 \text{Na}^+ + \text{Ca}^{2+} \rightarrow \text{C}_8\text{H}_7\text{SO}_3\text{Ca} + 2 \text{Na}^+ \]
This means that 2 moles of \( \text{Na}^+ \) are exchanged for 1 mole of \( \text{Ca}^{2+} \).

### Step 4: Calculate Moles of \( \text{Ca}^{2+} \) per Mole of Resin
From the stoichiometry, we can deduce that:
\[ 2 \text{ moles of resin} \rightarrow 1 \text{ mole of } \text{Ca}^{2+} \]
Thus, for 1 mole of resin:
\[ \text{Moles of } \text{Ca}^{2+} = \frac{1}{2} \text{ moles} \]

### Step 5: Calculate Moles of \( \text{Ca}^{2+} \) per Gram of Resin
Now, we can find the moles of \( \text{Ca}^{2+} \) that can be exchanged for the amount of resin we calculated earlier:
\[ \text{Moles of } \text{Ca}^{2+} = \left(\frac{1}{206} \text{ moles of resin}\right) \times \left(\frac{1}{2} \text{ moles of } \text{Ca}^{2+} \text{ per mole of resin}\right) \]
Calculating this gives:
\[ \text{Moles of } \text{Ca}^{2+} = \frac{1}{206} \times \frac{1}{2} = \frac{1}{412} \text{ moles} \]

### Final Answer
Thus, the maximum uptake of \( \text{Ca}^{2+} \) ions by the resin when expressed in moles per gram of resin is:
\[ \frac{1}{412} \text{ moles of } \text{Ca}^{2+} \text{ per gram of resin} \]

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