For cyclic systems with a potential plane of symmetry, if the substituents on opposite symmetric positions have different stereochemical configurations (i.e., one is a wedge and the other is a dash), the plane of symmetry is broken, which typically makes the molecule chiral.
Step 1: Understanding the Concept:
A molecule is chiral if it lacks any element of symmetry (specifically a plane of symmetry $\sigma$ or a center of inversion $i$) and is non-superimposable on its mirror image. Step 2: Detailed Explanation:
We analyze the symmetry of the substituted 1,3-dioxane isomers by looking for a vertical plane of symmetry bisecting the ring.
$\bullet$ Isomer (b): Both methyl groups are on "wedges" (pointing up). A vertical plane bisecting the molecule reflects one wedge onto the other. It has a plane of symmetry and is achiral (meso).
$\bullet$ Isomer (c): Both methyl groups are on "dashes" (pointing down). Similar to (b), it has a vertical plane of symmetry and is achiral.
$\bullet$ Isomer (d): Both methyl groups are on dashes. This structure also possesses a plane of symmetry and is achiral.
$\bullet$ Isomer (a): One methyl group is on a wedge and the other is on a dash. When you try to bisect the molecule, the wedge does not reflect into a dash. There is no plane of symmetry, making this molecule chiral. Step 3: Final Answer:
Isomer (a) is the only chiral molecule.
This corresponds to option (A).
