Which one of the following will exhibit maximum paramagnetic character?

\([Mn(H_2O)_6]^{2+}\)

The correct answer is option 4. \([Mn(H_2O)_6]^{2+}\).

The paramagnetic character of a transition metal complex is determined by the presence of unpaired electrons in the metal's d orbitals. More unpaired electrons result in higher paramagnetic character. Let's examine each complex in detail:

1. \([Fe(H_2O)_6]^{2+}\):

Iron (Fe) in the +2 oxidation state has an electronic configuration of \([Ar] 3d^6\). In this complex, all six ligands are water molecules, and they form coordinate bonds with iron. The \(3d\) orbital can accommodate a total of 10 electrons. Since \(Fe^{2+}\) has 6 electrons in the \(3d\) orbital, there are 4 unpaired electrons.

2. \([Cr(H_2O)_6]^{2+}\):

Chromium (Cr) in the +2 oxidation state has an electronic configuration of \([Ar] 3d^4\). In this complex, all six ligands are water molecules. There are 4 unpaired electrons in the \(3d\) orbital of chromium.

3. \([Co(H_2O)_6]^{2+}\):

Cobalt (Co) in the +2 oxidation state has an electronic configuration of \([Ar] 3d^7\). In this complex, all six ligands are water molecules. There are 3 unpaired electrons in the \(3d\) orbital of cobalt.

4. \([Mn(H_2O)_6]^{2+}\):

Manganese (Mn) in the +2 oxidation state has an electronic configuration of \([Ar] 3d^5\). In this complex, all six ligands are water molecules. There are 5 unpaired electrons in the \(3d\) orbital of manganese.

Comparison of Unpaired Electrons:
\([Fe(H_2O)_6]^{2+}\): 4 unpaired electrons
\([Cr(H_2O)_6]^{2+}\): 4 unpaired electrons
\([Co(H_2O)_6]^{2+}\): 3 unpaired electrons
\([Mn(H_2O)_6]^{2+}\): 5 unpaired electrons

Therefore, the complex with the maximum paramagnetic character is \([Mn(H_2O)_6]^{2+}\) as it has the highest number of unpaired electrons (5).