Step 1: Understanding the Concept:
This question asks about the specific ions involved in water softening using the permutit (or ion-exchange) process. Hardness in water is caused primarily by the presence of dissolved calcium (Ca$^{2+}$) and magnesium (Mg$^{2+}$) ions. The permutit process aims to remove these ions.
Step 2: Key Formula or Approach:
The permutit process uses a material, typically a sodium aluminum silicate (zeolite), which is represented as Na$_2$Z. In this ion-exchange resin, the sodium ions (Na$^+$) are loosely held and can be exchanged for other cations. When hard water passes through the resin, the hardness-causing ions are exchanged for sodium ions.
Step 3: Detailed Explanation:
1. Hardness of Water: Permanent hardness is caused by dissolved chlorides and sulfates of calcium and magnesium. The primary ions responsible are Ca$^{2+}$ and Mg$^{2+}$.
2. Permutit Process: Permutit is an ion-exchange resin containing sodium ions. Its chemical formula can be represented as Na$_2$Al$_2$Si$_2$O$_8 \cdot x$H$_2$O, often simplified as Na$_2$Z.
3. The Exchange Reaction: When hard water containing Ca$^{2+}$ and Mg$^{2+}$ ions is passed over the permutit resin, the following exchange reactions take place:
\[ \text{Na}_2\text{Z(s)} + \text{Ca}^{2+}\text{(aq)} \rightarrow \text{CaZ(s)} + 2\text{Na}^+\text{(aq)} \]
\[ \text{Na}_2\text{Z(s)} + \text{Mg}^{2+}\text{(aq)} \rightarrow \text{MgZ(s)} + 2\text{Na}^+\text{(aq)} \]
The Ca$^{2+}$ and Mg$^{2+}$ ions from the hard water are trapped by the resin, and in their place, sodium ions (Na$^+$) are released into the water.
4. Result: The water is softened because the ions that cause hardness (Ca$^{2+}$, Mg$^{2+}$) have been replaced by sodium ions, which do not cause hardness.
5. The other ions listed in the options (K$^+$, Ba$^{2+}$, Fe$^{2+}$, Zn$^{2+}$, Cu$^{2+}$) are generally not considered the primary cause of water hardness and are not the target of this specific process.
Step 4: Final Answer:
In the permutit process, Na$^+$ ions are exchanged with Ca$^{2+}$ and Mg$^{2+}$ ions to remove the permanent hardness of water. Therefore, option (C) is correct.