The type of isomerism exhibited by the complex \([CoCl_2(en)_2]\) is:

Geometrical isomerism

The correct answer is option 4. Geometrical isomerism.

The complex \([CoCl_2(en)_2]\) (where "en" stands for ethylenediamine) exhibits: 4. Geometrical isomerism.

Geometrical isomerism (also known as cis-trans isomerism) occurs in coordination complexes when the spatial arrangement of ligands around the central metal ion results in different isomers that cannot be interconverted without breaking bonds.

Analysis of \([CoCl_2(en)_2]\):

Structure of \([CoCl_2(en)_2]\):

This complex has a cobalt(III) ion coordinated to two chloride ions and two ethylenediamine (en) ligands. Ethylenediamine is a bidentate ligand, meaning each "en" molecule can form two bonds with the metal center.

Possibility of Geometrical Isomerism:

In a coordination complex where the central metal has a coordination number of 6 and the ligands are arranged in an octahedral geometry, geometrical isomerism can occur. In the case of \([CoCl_2(en)_2]\), the two chloride ions and the two ethylenediamine ligands can be arranged in different spatial configurations around the cobalt ion:

Cis Isomer: Both chloride ions are adjacent to each other (90° apart), and both ethylenediamine ligands are adjacent to each other.

Trans Isomer: The chloride ions are opposite each other (180° apart), and the ethylenediamine ligands are opposite each other.

Other Types of Isomerism (Why Not Applicable):

1. Coordination Isomerism: Coordination isomerism occurs in coordination compounds with more than one metal center where the ligands can exchange between metal centers. This does not apply to \([CoCl_2(en)_2]\), which has only one metal center.

2. Ionization Isomerism: Ionization isomerism occurs when a complex and its isomer have different ions in solution. For \([CoCl_2(en)_2]\), the ions remain the same, so this type of isomerism is not applicable.

3. Linkage Isomerism: Linkage isomerism occurs when a ligand can bind to the metal in more than one way. For example, a ligand like NO₂⁻ can bind through nitrogen or oxygen. However, in \([CoCl_2(en)_2]\), the ligands are ethylenediamine (which binds in a single way) and chloride ions, so linkage isomerism is not relevant.

Conclusion: The complex \([CoCl_2(en)_2]\) exhibits geometrical isomerism due to the different possible spatial arrangements of the chloride and ethylenediamine ligands around the cobalt ion. Thus, the correct answer is: 4. Geometrical isomerism.