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GATE 2026 CY – Question 18

Physical Chemistry · Group Theory: Symmetry elements and symmetry operations. Basic postulates of groups and group multiplication tables. Classes, reducible and irreducible representations of symmetry elements. Classification and labelling of molecular point groups. Symmetry based selection rules and their application to electronic and vibrational spectroscopy. Internal coordinates and symmetry labelling of vibrational modes; symmetry adapted linear combination of atomic orbitals (LCAO-MO); construction of hybrid orbitals using molecular point group symmetry. · 1 mark · Multiple choice

The point group of $\mathrm{CH_2=C=CH_2}$ is

  1. $D_{2h}$
  2. $C_{2h}$
  3. $C_{2v}$
  4. $D_{2d}$

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Correct answer: (D) $D_{2d}$

Explanation

**Structure of allene.** $\text{CH}_2=\text{C}=\text{CH}_2$ is a linear $\text{C}=\text{C}=\text{C}$ chain. The central carbon is sp-hybridised, and the two $\text{CH}_2$ groups lie in planes that are **perpendicular** to each other.

**Symmetry elements.**
- A $C_2$ axis along the C=C=C axis (principal axis).
- Two perpendicular $C_2$ axes through the central carbon, bisecting the H–C–H angles.
- Two mirror planes $\sigma_d$, each containing the C=C=C axis and bisecting the angle between the two $\text{CH}_2$ planes.
- An $S_4$ improper axis along the C=C=C axis.

These operations (order 8) make up the point group **$D_{2d}$** (option D). There is no horizontal mirror plane, so it is not $D_{2h}$.