For $S_N2$ reaction, the increasing order of the reactivity of the following alkyl halides is:

(A) $CH_3CH_2CH_2CH_2Br$
(B) $CH_3CH_2CH(Br)CH_3$
(C) $(CH_3)_3CBr$
(D) $(CH_3)_2CHCH_2Br$

Choose the correct answer from the options given below:

(C) < (B) < (D) < (A)

The correct answer is Option (4) → (C) < (B) < (D) < (A).

Let us delve deeper into the factors influencing the reactivity of alkyl halides in \(S_N2\) reactions, focusing on the specific compounds given.

\(S_N2\) Reaction:

\(S_N2\) (Substitution Nucleophilic Bimolecular) reactions involve a single concerted step where a nucleophile attacks the electrophilic carbon atom, displacing the leaving group (in this case, a halide). The rate of the reaction depends on the concentrations of both the nucleophile and the alkyl halide:

\(\text{Rate} = k[\text{alkyl halide}][\text{nucleophile}]\)

Factors Affecting Reactivity:

Steric Hindrance:

Primary Alkyl Halides: These have only one carbon group attached to the carbon bearing the halogen. They are generally very reactive in \(S_N2\) reactions due to minimal steric hindrance.

Secondary Alkyl Halides: These have two carbon groups attached, leading to moderate steric hindrance. Their reactivity in \(S_N2\) reactions is less than that of primary halides but higher than that of tertiary halides.

Tertiary Alkyl Halides: With three carbon groups attached, these are highly hindered, making them largely unreactive in \(S_N2\) reactions. Instead, they tend to undergo \(S_N1\) reactions, which involve carbocation formation.

Analysis of the Given Alkyl Halides:

(A) \(CH_3CH_2CH_2CH_2Br\) :

A straight-chain primary alkyl halide.

Reactivity: High reactivity due to minimal steric hindrance. The nucleophile can easily access the electrophilic carbon atom.

(B) \(CH_3CH_2CH(Br)CH_3\) :

A secondary alkyl halide.

Reactivity: Moderate reactivity due to some steric hindrance. The nucleophile still has access, but the presence of two carbon groups makes it less reactive than (A).

(C) \((CH_3)_3CBr\):

A tertiary alkyl halide.

Reactivity: Very low reactivity in \(S_N2\) reactions. The bulky groups surrounding the electrophilic carbon prevent the nucleophile from effectively attacking, thus leading to negligible \(S_N2\) reactivity.

(D) \((CH_3)_2CHCH_2Br\):

Another secondary alkyl halide but with a slightly less hindered structure compared to (B).

Reactivity: This compound is somewhat more reactive than (B) due to the positioning of the bromine and the overall sterics involved, but still less than (A).

Summary of Reactivity Order:

Based on the above analysis, we can rank the compounds in terms of their increasing order of reactivity in \(S_N2\) reactions:

(C) \((CH_3)_3CBr\): Least reactive (due to tertiary structure)

B) \(CH_3CH_2CH(Br)CH_3\) : More reactive than (C) but less than (D)

(D) \((CH_3)_2CHCH_2Br\): More reactive than (B) due to less steric hindrance

(A) \(CH_3CH_2CH_2CH_2Br\) : Most reactive (primary structure)

Final Increasing Order of Reactivity:

Thus, the correct increasing order of reactivity for the alkyl halides in \(S_N2\) reactions is:

\((C) < (B) < (D) < (A)\)

Conclusion:

In summary, when considering the reactivity of alkyl halides in SN2 reactions, the structure of the halide significantly impacts how readily it can undergo nucleophilic substitution. The order we derived reflects the steric hindrance that each structure presents, which in turn dictates the accessibility of the electrophilic carbon to the nucleophile.