Consider the complex ions, trans-$$[\text{Co(en)}_2\text{Cl}_2]^+$$ (A) and cis-$$[\text{Co(en)}_2\text{Cl}_2]^+$$ (B). The correct statement regarding them is:

First of all, $$\text{en}$$ denotes ethylenediamine, $$H_{2}N-CH_{2}-CH_{2}-NH_{2}$$, which is a bidentate ligand. In the octahedral complex $$[\text{Co(en)}_2\text{Cl}_2]^+$$ the cobalt(III) centre is surrounded by two chelating $$\text{en}$$ ligands and two monodentate chloride ligands.

Optical activity arises when a molecule or ion is chiral, that is, when it is not super-imposable on its mirror image. The necessary and sufficient condition for chirality in an octahedral complex is the absence of any improper symmetry element (plane of symmetry $$\sigma$$, centre of inversion $$i$$ or an $$S_n$$ axis). Whenever such a symmetry element is present, the complex is achiral and therefore cannot show optical activity.

We now examine the two geometrical isomers one by one.

Trans-$$[\text{Co(en)}_2\text{Cl}_2]^+$$ (A): In the trans isomer, the two chloride ions occupy positions opposite each other (i.e. a 180° angle). The two $$\text{en}$$ chelate rings are therefore arranged so that the entire ion contains a vertical mirror plane passing through cobalt, cutting the octahedron into two identical halves: one half contains one $$\text{en}$$ arm from the first ligand and one from the second ligand together with a chloride, and the other half is its mirror image with the second chloride. Because this mirror plane $$\sigma_v$$ exists, the complex possesses an improper symmetry element, making the ion achiral. Consequently,

$$\text{trans-}[\text{Co(en)}_2\text{Cl}_2]^+ \;\text{is optically inactive.}$$

Cis-$$[\text{Co(en)}_2\text{Cl}_2]^+$$ (B): In the cis isomer, the two chloride ligands are adjacent (90° apart). Now try to locate any mirror plane or centre of inversion:

• A plane that would pass through cobalt and both chlorides fails to bisect the two $$\text{en}$$ rings symmetrically because each $$\text{en}$$ ligand wraps around the metal in a fixed sense (left-handed or right-handed twist).
• There is no centre of inversion; the positions of donor nitrogen atoms from $$\text{en}$$ do not come in opposite pairs across cobalt.
• No $$S_4$$ (improper fourfold) or higher improper axis is present because of the asymmetrical chelate rings.

Therefore no improper symmetry element exists. The ion is chiral and occurs as two non-super-imposable mirror images (Δ and Λ forms). Hence,

$$\text{cis-}[\text{Co(en)}_2\text{Cl}_2]^+ \;\text{is optically active.}$$

Putting the results together, we have:

$$\begin{aligned} \text{trans-}[\text{Co(en)}_2\text{Cl}_2]^+ &: \text{optically inactive},\\ \text{cis-}[\text{Co(en)}_2\text{Cl}_2]^+ &: \text{optically active}. \end{aligned}$$

So the trans isomer cannot be optically active, while the cis isomer can be optically active.

Hence, the correct answer is Option D.