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Geometric isomerism arises in coordination complexes where ligands can adopt different arrangements around the central metal atom. This typically occurs in octahedral (\[\text{octahedral molarity:} ~ MX_4Y_2\] ) or square planar (\[\text{square planar molarity:} ~ MX_2Y_2\] ) complexes.
In the given complexes, the \[\text{[Co(NH}_3)_3\text{Cl}_3]^{2+}\] complex can exhibit geometric isomerism because it has three ammonia (\[\text{NH}_3\] ) and three chloride (\[\text{Cl}^-\] ) ligands that can arrange in facial or meridional formations.
Facial (fac) isomers have three identical ligands occupying one face of the octahedron, while meridional (mer) isomers have them in a plane passing through the octahedron.
On the other hand, \[\text{[Co(NH}_3)_5\text{SCN}]^{2+}\] has five \[\text{NH}_3\] ligands and one thiocyanate (\[\text{SCN}^-\] ), showing less potential for geometric variety.
Complex \(\text{CoClBr} \cdot 5\text{NH}_3\) is less likely to have distinct geometric isomers due to an unclear structure and less symmetry.

Linkage isomerism occurs when a ligand can bind to a metal center in more than one way. This often involves ligands with multiple potential donor atoms.
A classic example is the thiocyanate ion (\[\text{SCN}^-\]), which can bind through either the sulfur (\[\text{S}\] ) atom, termed thiocyanato, or through the nitrogen (\[\text{N}\] ) atom, termed isothiocyanato.
In the provided complexes, only \[\text{[Co(NH}_3)_5\text{SCN}]^{2+}\] is capable of showing linkage isomerism. This is because of the versatile binding nature of \[\text{SCN}^-\] .
The other complexes like \(\text{[Co(NH}_3)_3 \text{Cl}_3]^{2+}\) and \(\text{CoClBr} \cdot 5\text{NH}_3\) do not demonstrate linkage isomerism since their ligands (\[\text{Cl}^-\] and \[\text{Br}^-\] ) are monodentate and cannot reconfigure in such a manner.

Optical isomerism refers to the ability of chiral compounds to exist as mirror images that cannot be superimposed on one another. This requires the complex to lack a plane of symmetry.
In coordination complexes, optical isomers (enantiomers) are common when bidentate ligands create a chiral environment around the metal center. For instance, an octahedral complex with three bidentate ligands, such as ethylenediamine, can become chiral.
The complex \[\text{[Co(NH}_3)_5\text{SCN}]^{2+}\] could potentially form optical isomers if its configuration is suitably asymmetric. However, due to its simpler ligand structure, this is rare. \[\text{[Co(NH}_3)_3 \text{Cl}_3]^{2+}\] is less likely to exhibit optical isomerism due to potential possession of internal symmetry.
Similarly, \(\text{CoClBr} \cdot 5\text{NH}_3\) would not exhibit optical isomerism without a more complex chiral setup, which is not apparent from its described formula.

Coordination-sphere isomerism is an intriguing form of isomerism. It arises when ligands are exchanged between the coordination sphere and the outer sphere of a complex.
This type of isomerism is more about how ligands are positioned relative to the complex rather than within the coordination sphere itself.
For the complexes given, only \(\text{CoClBr} \cdot 5\text{NH}_3\) suggests a potential for coordination-sphere isomers. This is because its formulation indicates the possibility of ligand interchange between the inner and outer coordination sphere.
Meanwhile, \[\text{[Co(NH}_3)_5\text{SCN}]^{2+}\] and \[\text{[Co(NH}_3)_3 \text{Cl}_3]^{2+}\] do not offer coordination-sphere isomer possibilities as they do not possess distinctly bound external ligands that could alter this arrangement.