Animated Solution for Chemistry - Basic Concepts in Chemistry: The molecular formula of a commercial resin used for exchanging ions in water softening is ${\text{C}}_8{\text{H}}_7{\text{SO}}_3\text{Na}$ (mol. wt. = 206). What would be the maximum uptake of ${\text{Ca}}^{2+}$ ions by the resin when expressed in mole per gram resin?

Have you ever wondered how hard water, which leaves stubborn white stains on your taps and makes it difficult to form a lather with soap, is magically transformed into soft water? The secret lies in a fascinating chemical process called ion exchange, and at the heart of this process are commercial resins.

Imagine a tiny, porous bead of resin. This bead is essentially a large, complex organic molecule with active sites. In our problem, the resin is represented by the formula ${\text{C}}_8{\text{H}}_7{\text{SO}}_3\text{Na}$. Notice the sodium (${\text{Na}}^{+}$) at the end? That is the key. The resin is pre-loaded with sodium ions. When hard water, which is rich in calcium (${\text{Ca}}^{2+}$) and magnesium (${\text{Mg}}^{2+}$) ions, flows over these resin beads, a chemical swap occurs. The resin prefers the divalent calcium ions over the monovalent sodium ions, so it grabs the calcium and releases the sodium into the water.

To solve our problem, we first need to understand the exact stoichiometry of this swap. It is not a simple one-to-one exchange. Why? Because nature demands that electrical charge must be conserved.

A calcium ion has a charge of $+2$ (${\text{Ca}}^{2+}$), while a sodium ion has a charge of $+1$ (${\text{Na}}^{+}$). Therefore, to maintain electrical neutrality, one calcium ion must displace exactly two sodium ions from the resin. We can write this as a balanced chemical equation:

This equation is our master key. It tells us that 2 moles of the resin are required to capture 1 mole of calcium ions.

The question asks for the maximum uptake of ${\text{Ca}}^{2+}$ ions per gram of the resin. This means we need to figure out how much calcium can be absorbed by exactly $1\text{ g}$ of the resin.

First, let's find out how many moles of resin are present in $1\text{ g}$. We are given the molar mass of the resin as $206\text{ g/mol}$. Using the fundamental mole concept formula:

Now, we bring back our stoichiometric ratio. We established that $2\text{ moles}$ of resin combine with $1\text{ mole}$ of ${\text{Ca}}^{2+}$. This implies that $1\text{ mole}$ of resin will combine with $\frac{1}{2}\text{ mole}$ of ${\text{Ca}}^{2+}$.

So, if we have $\frac{1}{206}\text{ moles}$ of resin, the amount of calcium it can take up is:

Therefore, the maximum uptake of ${\text{Ca}}^{2+}$ ions by the resin is $\frac{1}{412}\text{ mole per gram}$. This elegant calculation shows how macroscopic properties, like the mass of a resin, are directly linked to the atomic-level dance of exchanging ions.