Step 1: Understanding the Concept:
Nucleophilicity is the ability of a species to donate an electron pair. For halide ions in protic solvents, nucleophilicity follows the order: I\(^-\) > Br\(^-\) > Cl\(^-\).
Step 2: Detailed Explanation:
In protic solvents (like water or alcohols), nucleophilicity increases down the group for halide ions. This is because the smaller ions (F\(^-\), Cl\(^-\)) are more strongly solvated by hydrogen bonding, making them less available for nucleophilic attack. I\(^-\) is the least solvated and thus the best nucleophile. The order of nucleophilicity in protic solvents is:
\[ \text{I}^- > \text{Br}^- > \text{Cl}^- \]
Therefore, the increasing order is Cl\(^-\) < Br\(^-\) < I\(^-\).
Step 3: Final Answer:
The correct increasing order is Cl\(^-\) < Br\(^-\) < I\(^-\), which corresponds to option (C).