Step 1: Understanding the Question:
The question asks us to identify the optically inactive compound from the given choices.
Optical activity depends strictly on the presence of a chiral center, which is an asymmetric carbon atom bonded to four completely different groups.
Step 2: Detailed Explanation:
Let's analyze the structure of each option to find chiral centers:
(a) 3-chlorohexane ($CH_3-CH_2-CH(Cl)-CH_2-CH_2-CH_3$): The carbon atom at position 3 is bonded to a hydrogen atom, a chlorine atom, an ethyl group, and a propyl group. All four groups are different, so it is chiral and optically active.
(b) 2-chloro-2-methylbutane ($CH_3-C(Cl)(CH_3)-CH_2-CH_3$): The carbon atom at position 2 is bonded to a chlorine atom, an ethyl group, and two identical methyl groups. Because two groups are the same, it lacks a chiral center and is optically inactive (achiral).
(c) 2-chloropentane ($CH_3-CH(Cl)-CH_2-CH_2-CH_3$): The carbon atom at position 2 is bonded to H, Cl, a methyl group, and a propyl group. It is chiral and optically active.
(d) 2-chloro-3-methylbutane ($CH_3-CH(Cl)-CH(CH_3)-CH_3$): The carbon atom at position 2 is chiral (bonded to H, Cl, methyl, and an isopropyl group). It is optically active.
Step 3: Final Answer:
Since 2-chloro-2-methylbutane does not have a chiral center, it is optically inactive, corresponding to option (b).