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 on the same face (cis) or opposite faces (trans) of the ring plane. The compound shown is 1,3,5-trimethylcyclohexane, which has three methyl groups at positions 1, 3, and 5 of the cyclohexane ring. Step 1: Identify the stereocenters. Each of C1, C3, and C5 bears one methyl group and is a stereocenter (each has two different substituents relative to the ring plane: up or down). Step 2: Count possible arrangements. With three stereocenters, there are in principle 2^3 = 8 arrangements, but many are equivalent by symmetry. Step 3: Enumerate distinct geometric isomers: - All three methyls on the same face: all-cis (1,3,5-all-cis). This molecule has a C3v symmetry axis and is achiral (it has a plane of symmetry). This is one isomer. - Two methyls on one face, one on the other face. By the symmetry of the 1,3,5 positions, all arrangements of 'two up, one down' are equivalent to each other (any one of the three can be 'down', but due to the 3-fold symmetry they all give the same compound). This is a second isomer. - Wait, let us reconsider: with positions 1, 3, 5 being equivalent by the C3 axis, we have: (a) All three up (all-cis): one compound. (b) Two up, one down: all such arrangements are related by the C3 rotation, so this is one compound — but we need to check if 'one up, two down' is the same as 'two up, one down' by a ring flip or reflection. In a flat representation (geometric isomerism, not conformational), 'two up, one down' and 'one up, two down' are mirror images. If they are non-superimposable, they are enantiomers and count as separate geometric isomers. Step 4: Check chirality: - All-cis (all three methyls same side): has a C3 axis and three mirror planes — it is achiral. One isomer. - Two up, one down: this isomer lacks a plane of symmetry and is chiral. Its mirror image is 'one up, two down.' These two are enantiomers. Since geometric isomerism counts cis/trans arrangements (including enantiomeric pairs as distinct geometric isomers when they differ in spatial arrangement), these count as two separate geometric isomers. Step 5: Total geometric isomers = 1 (all-cis, achiral) + 2 (the chiral pair: two-up-one-down and one-up-two-down) = 3 geometric isomers. Why other options fail: - (a) 0: Incorrect; there are clearly different spatial arrangements possible. - (b) 2: Incorrect; this would miss counting the enantiomeric chiral pair as two separate isomers. - (d) 4: Incorrect; overcounts because the three-fold symmetry reduces the number of distinct arrangements. Therefore, the correct answer is C.