Which one of the following complexes shows optical isomerism?

cis\([Co(en)_2Cl_2]Cl\)

The correct answer is option 2. cis\([Co(en)_2Cl_2]Cl\).

Optical isomerism is a form of stereoisomerism where molecules exist as non-superimposable mirror images, known as enantiomers. For a complex to exhibit optical isomerism, it must lack any plane of symmetry or a center of symmetry, meaning that the molecule and its mirror image cannot be superimposed on each other.

Let us analyze each of the given complexes in detail:

1. \([Co(NH_3)_3Cl_3]\)

This is an octahedral complex with three ammonia (NH\(_3\)) and three chloride (Cl\(^-\)) ligands. In an octahedral arrangement, the ligands can be arranged in a fac (facial) or mer (meridional) configuration:

Facial isomer: All three chloride ions are adjacent to each other, forming a face of the octahedron.

Meridional isomer: The chlorides are arranged such that two chlorides are trans to each other, and the third one is in the equatorial position.

Isomerism: While this complex can show geometric isomerism (fac and mer forms), neither form lacks symmetry. Both isomers are superimposable on their mirror images, meaning that this complex does not exhibit optical isomerism.

2. cis-\([Co(en)_2Cl_2]Cl\)

Here, "en" stands for ethylenediamine, a bidentate ligand that can bind to the metal ion at two sites. The complex is octahedral, with two ethylenediamine (en) ligands and two chloride (Cl\(^-\)) ligands. In the cis form, the two chloride ligands are adjacent (90° apart), and the two "en" ligands are also adjacent.

Isomerism: The arrangement of the ligands in the cis form creates a chiral center, as there is no plane or center of symmetry. As a result, the complex can exist in two non-superimposable mirror images, meaning it does exhibit optical isomerism.

3. trans-\([Co(en)_2Cl_2]Cl\)

Similar to the cis form, this complex is also octahedral, but the chloride ligands are opposite each other (180° apart). The trans configuration has a plane of symmetry passing through the metal ion and the chloride ligands.

Isomerism: Because of this plane of symmetry, the complex is superimposable on its mirror image and cannot exhibit optical isomerism.

4. \([Co(NH_3)_4Cl_2]Cl\)

This is an octahedral complex with four ammonia (NH\(_3\)) and two chloride (Cl\(^-\)) ligands. The complex can exist in two geometric forms:

Cis isomer: The chloride ions are adjacent.

Trans isomer: The chloride ions are opposite each other.

Isomerism: In both cis and trans forms, the complex has symmetry that allows it to be superimposable on its mirror image, meaning it does not exhibit optical isomerism.

Conclusion:
The only complex among the options that can show optical isomerism is cis-\([Co(en)_2Cl_2]Cl\), which is option 2. This is because the arrangement of ligands in the cis form creates a chiral environment, allowing for non-superimposable mirror images (enantiomers).