See image — Isomerism and Stereochemistry Chemistry Question

💡 Solution & Explanation

Concept: Geometrical (cis-trans) isomerism in cycloalkanes arises when substituents on the ring can be placed either above (up) or below (down) the plane of the ring. For a trisubstituted cyclohexane like 1,3,5-trimethylcyclohexane, we assign each methyl group an 'up' (U) or 'down' (D) orientation relative to the ring plane. Step 1: List all possible up/down combinations for three methyl groups at C1, C3, C5. The possible arrangements are: - All up: UUU - Two up, one down: UUD, UDU, DUU (these three are related by the symmetry of the 1,3,5 positions) - One up, two down: UDD, DUD, DDU (these three are related by the symmetry) - All down: DDD Step 2: Apply the symmetry of the molecule. Because the 1,3,5-trimethylcyclohexane has a C3 rotation axis (the three positions are equivalent), UUU and DDD are identical (one is just the mirror flip of the other and they superimpose — they are the same compound: all-cis or all-trans relative arrangement). Similarly, UUD, UDU, and DUU are all the same compound by rotation, and UDD, DUD, DDU are all the same compound by rotation. Step 3: Check if the two distinct arrangements are mirror images (enantiomers) or identical. - Arrangement 1: all three methyls on the same face (UUU = DDD): this molecule has a plane of symmetry and is achiral (meso-like). It is the all-cis isomer (each pair is cis to each other): cis-1,3,5-trimethylcyclohexane. - Arrangement 2: two methyls on one face, one on the other (UUD type): this gives the 1,3-cis, 1,5-cis, 3,5-trans pattern (or equivalent). This molecule also has a plane of symmetry and is achiral. Step 4: Count distinct geometrical isomers. The two distinct configurations are: 1. All three methyls on the same side (all-cis): one isomer. 2. Two methyls on one side, one on the other: one isomer. Thus there are 2 theoretical geometrical isomers of 1,3,5-trimethylcyclohexane. Why not more: The C3v symmetry of the molecule reduces all 8 formal combinations to only 2 distinct structures. Neither is chiral, so there are no enantiomeric pairs to add. Therefore, the correct answer is 2.