Optical Activity and Symmetry in Geometrical Isomers of Complex

To determine the correct option, let us analyze both statements independently:

1. Analysis of STATEMENT-1:
The given complex is an octahedral complex of the type $[\text{MA}_4\text{B}_2]$, specifically $[\text{M}(\text{NH}_3)_4\text{Cl}_2]$. It exhibits two geometrical isomers:

• $trans$-isomer: The two $\text{Cl}^-$ ligands are positioned opposite to each other ($180^\circ$ apart). This isomer possesses multiple planes of symmetry ($\sigma$) and a center of inversion ($i$). Hence, it is achiral and optically inactive.
• $cis$-isomer: The two $\text{Cl}^-$ ligands are adjacent to each other ($90^\circ$ apart). This isomer possesses planes of symmetry ($\sigma$) that pass through the central metal atom and the ligands. Because it contains a plane of symmetry, its mirror image is superimposable, making it achiral and optically inactive.

Since both the $cis$ and $trans$ isomers are optically inactive, STATEMENT-1 is True.

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2. Analysis of STATEMENT-2:

• $trans$-$[\text{M}(\text{NH}_3)_4\text{Cl}_2]$ belongs to the $D_{4h}$ point group and possesses a four-fold rotational axis ($C_4$) along the $\text{Cl}-\text{M}-\text{Cl}$ axis as well as several two-fold rotational axes ($C_2$).
• $cis$-$[\text{M}(\text{NH}_3)_4\text{Cl}_2]$ belongs to the $C_{2v}$ point group and possesses a two-fold rotational axis ($C_2$) bisecting the $\text{Cl}-\text{M}-\text{Cl}$ angle.

Since both geometrical isomers possess a proper axis of symmetry ($C_n$), STATEMENT-2 is True.

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3. Relationship between STATEMENT-1 and STATEMENT-2:

• The presence or absence of optical activity is governed by the presence or absence of an improper axis of rotation ($S_n$), which practically means a plane of symmetry ($\sigma = S_1$) or a center of inversion ($i = S_2$).
• The presence of a proper axis of symmetry ($C_n$) does not cause optical inactivity. In fact, many chiral molecules possess a proper axis of symmetry (for example, $[\text{M}(\text{en})_3]^{n+}$ has $D_3$ symmetry with a $C_3$ axis and is optically active).
• Thus, the optical inactivity of both isomers is due to the presence of a plane of symmetry/center of inversion, not due to the presence of an axis of symmetry.

Therefore, STATEMENT-2 is NOT the correct explanation for STATEMENT-1.

Correct Option: B