The molecular formula of a commercial resin used for exchanging ions in water softening is $$C_8H_7SO_3Na$$ (molecular weight = 206). What would be the maximum uptake of $$Ca^{2+}$$ ions by the resin if expressed in mol per gm?

We start with the information that one empirical unit of the resin has the molecular formula $$C_8H_7SO_3Na$$ and a molecular weight of $$206\ \text{g mol}^{-1}$$. This means that $$206\ \text{g}$$ of the dry resin correspond to exactly $$1\ \text{mol}$$ of the formula units.

The functional group responsible for ion exchange is the sulphonate group written in the formula as $$-SO_3Na$$. Inside this group, the sodium is present as a monovalent cation $$Na^+$$. Thus, every single formula unit is capable of donating exactly one $$Na^+$$ ion to the solution and, in exchange, accepting another positive ion of equal charge.

Water-softening requires the removal of divalent calcium ions $$Ca^{2+}$$. Because calcium carries a double positive charge while sodium carries only a single positive charge, the principle of charge balance tells us that one calcium ion can replace two sodium ions:

$$2\bigl(R{-}SO_3^-Na^+\bigr)+Ca^{2+} \;\;\rightarrow\;\; \bigl(R{-}SO_3^-\bigr)_2Ca^{2+}+2\,Na^+$$

In words, two resin formula units are required for the uptake of one $$Ca^{2+}$$ ion. Stating this as a ratio:

Number of resin units per calcium ion $$=2$$. Equivalently, the number of calcium ions taken up per resin unit is

$$\frac{1\ \text{mol }Ca^{2+}}{2\ \text{mol resin}}=\frac{1}{2}\ \text{mol }Ca^{2+}\text{ per mol resin}.$$

Now we convert this result into a capacity expressed in moles of calcium per gram of resin. By definition,

$$1\ \text{mol resin}=206\ \text{g}.$$

Hence, $$\frac{1}{2}\ \text{mol }Ca^{2+}$$ are taken up by $$206\ \text{g}$$ of resin. The uptake per gram is therefore

$$ \frac{\frac{1}{2}\ \text{mol }Ca^{2+}}{206\ \text{g}} =\frac{1}{2\times206}\ \text{mol g}^{-1} =\frac{1}{412}\ \text{mol g}^{-1}. $$

This fraction exactly matches the value given in Option A.

Hence, the correct answer is Option A.