Step 1: Understanding the Concept:
Isomers are compounds with the same chemical formula but different arrangements of atoms. In coordination chemistry, several types of structural and stereoisomerism are possible. We need to analyze the complex \([Co(NH_3)_4(NO_2)_2]Cl\) for each type.
Step 2: Detailed Explanation:
(A) Ionisation Isomerism: This occurs when a ligand within the coordination sphere and a counter-ion outside the sphere exchange places. The given complex has a NO\(_2^-\) ligand and a Cl\(^-\) counter-ion. They can swap positions to form the isomer \([Co(NH_3)_4(NO_2)Cl]NO_2\). Thus, ionisation isomerism is possible.
(B) Linkage Isomerism: This occurs when the complex contains an ambidentate ligand, which can bind to the central metal atom through different atoms. The nitro ligand (NO\(_2^-\)) is a classic example. It can coordinate through the nitrogen atom (-NO\(_2\), nitro) or through an oxygen atom (-ONO, nitrito). Therefore, an isomer like \([Co(NH_3)_4(ONO)_2]Cl\) can exist. Thus, linkage isomerism is possible.
(C) Geometrical Isomerism: This is a type of stereoisomerism. The complex has an octahedral geometry with the formula MA\(_4\)B\(_2\) (where M=Co, A=NH\(_3\), B=NO\(_2\)). In such complexes, the two 'B' ligands can be positioned adjacent to each other (at 90\(^\circ\)), forming the cis isomer, or opposite to each other (at 180\(^\circ\)), forming the trans isomer. Thus, geometrical isomerism is possible.
(D) Coordination Isomerism: This occurs when both the cation and the anion of a salt are complex ions, and ligands are exchanged between them. Here, the anion is a simple ion (Cl\(^-\)), so this type of isomerism is not possible.
(E) Solvate Isomerism: This is a form of ionisation isomerism where the exchanged ligand is a solvent molecule. No solvent is indicated in the formula, so this is not applicable.
Step 3: Final Answer:
The complex can exhibit ionisation (A), linkage (B), and geometrical (C) isomerism. Therefore, the correct option is (3).