The one that will show optical activity is (en = ethane-1, 2-diamine):

For a coordination compound to be optically active it must be chiral, i.e. the whole ion (or molecule) must lack any element of symmetry such as a plane of symmetry ($$\sigma$$), a centre of symmetry ($$i$$) or an improper rotation axis ($$S_n$$).
In octahedral complexes chirality is usually produced in either of two ways:
  • by having three unsymmetrical bidentate ligands arranged octahedrally, giving the well-known $$\Delta$$ and $$\Lambda$$ mirror-image forms;
  • by having two bidentate ligands in the cis configuration (the trans form possesses a plane of symmetry and is achiral).

The four options (en = ethane-1,2-diamine) are assumed to be

Option A $$[Co(NH_3)_5Cl]^{2+}$$

Option B $$[Co(NH_3)_4(en)]^{3+}$$

Option C $$[Co(en)_3]^{3+}$$

Option D trans-$$[Co(en)_2Cl_2]^{+}$$

Now examine each case.

Case A: $$[Co(NH_3)_5Cl]^{2+}$$ has only monodentate ligands. The octahedron contains a plane of symmetry that passes through Co and the four equatorial NH3 ligands, so the ion is achiral. No optical activity.

Case B: $$[Co(NH_3)_4(en)]^{3+}$$ contains one bidentate ligand (en). The four NH3 ligands occupy the remaining positions in such a way that a mirror plane still exists, joining the two N atoms of en and bisecting the opposite Co-N bonds. Hence the ion is achiral and optically inactive.

Case C: $$[Co(en)_3]^{3+}$$ (tris-(ethane-1,2-diamine)cobalt(III)) has three chelating bidentate ligands. There is no plane of symmetry and no centre of symmetry. Two mirror-image arrangements, the $$\Delta$$ and $$\Lambda$$ forms, are possible and cannot be superposed. Therefore the complex is chiral and will rotate the plane of polarised light—i.e. it is optically active.

Case D: trans-$$[Co(en)_2Cl_2]^{+}$$ contains two bidentate ligands but they are placed opposite each other. A mirror plane passes through Co and the two en chelate rings, so the ion is achiral. (Had the arrangement been cis, the complex would have been chiral.)

Only Case C satisfies the requirement for optical activity.

Option C which is: $$[Co(en)_3]^{3+}$$