Face-mer isomerism is a type of isomerism found in octahedral coordination compounds that contain two or more identical ligands. Face-mer isomerism occurs when three identical ligands occupy positions in the same face of the octahedron (mer) or on adjacent faces (fac).
- Option (a) Co(en)$_2$Cl$_2$: This complex can show face-mer isomerism because the ethylenediamine (en) ligands can be arranged in such a way that they form either a facial (fac) or meridional (mer) isomer.
- Option (b) [Co(NH$_3$)$_4$Cl]$_2$: This is not a valid complex for face-mer isomerism because it has too many NH$_3$ ligands and only one chloride, making face-mer isomerism impossible.
- Option (c) [Co(H$_2$O)$_6$]$_1$X: This compound also does not show face-mer isomerism as it doesn't have the necessary number of identical ligands to form either type of isomer.
- Option (d) [Co(NH$_3$)$_3$Cl$_3$]: This complex cannot exhibit face-mer isomerism because all three NH$_3$ and three chloride ligands are identical in number and cannot form distinct face-mer isomers.
Thus, option (a) is the correct answer.