To determine which compound of the molecular formula $C_7H_{16}$ shows optical isomerism, we need to understand the concept of chirality in organic chemistry. Optical isomerism occurs when a compound has at least one chiral center, which is a carbon atom bonded to four different groups.
Let's analyze each option given:
- 2, 3-dimethyl pentane: This compound has the structural formula where the carbon chain has methyl groups attached to the second and third carbon atoms. The third carbon atom becomes a chiral center because it is bonded to four different groups: a hydrogen atom, a methyl group, an ethyl group, and a butyl group. Therefore, 2, 3-dimethyl pentane can show optical isomerism.
- 2, 2-dimethyl butane: In this compound, the second carbon atom has two identical methyl groups attached, which means there is no chiral center in this molecule.
- 2-methyl hexane: This compound does not have a chiral center, as each carbon atom is not bonded to four different groups.
- None of the above: This option is incorrect because we have identified that 2, 3-dimethyl pentane does show optical isomerism.
Therefore, the correct answer is 2, 3-dimethyl pentane as it has a chiral center, allowing it to exhibit optical isomerism.