Concept:
The given reaction is an SN2 reaction, as methyl bromide (\(\text{CH}_3\text{Br}\)) undergoes nucleophilic substitution via a single-step backside attack mechanism.
For nucleophilicity in SN2 reactions:
Higher electron density increases nucleophilicity
Resonance stabilization decreases nucleophilicity
Weakly basic and highly stabilized anions are poor nucleophiles
Step 1: Analyze each nucleophile.
\(HO^-\):
Small size, high charge density, and no resonance stabilization. Hence, it is a very strong nucleophile.
\(PhO^-\) (Phenoxide ion):
The negative charge is delocalized over the aromatic ring through resonance, reducing its availability for nucleophilic attack. Thus, it is weaker than \(\text{HO}^-\).
\(CH_3COO^-\) (Acetate ion):
The negative charge is delocalized over two oxygen atoms due to resonance, making it less nucleophilic than phenoxide.
\(ClO_4^-\) (Perchlorate ion):
Highly resonance-stabilized with extensive charge delocalization. It is an extremely weak nucleophile.
Step 2: Compare nucleophilic strength based on availability of lone pair.
\[
\text{HO}^→ \text{PhO}^→ \text{CH}_3\text{COO}^→ \text{ClO}_4^-
\]
Step 3: Match the order with the given options.
This order corresponds to Option (1).