The complex having lowest \(\Delta _O\) value among the given complexes is :

\([CrCl_6]^{3-}\)

The correct answer is option 4. \([CrCl_6]^{3-}\).

To understand which complex has the lowest \(\Delta_O\) (octahedral crystal field splitting energy), let us first break down the factors that influence \(\Delta_O\) and then analyze the complexes step by step.

Crystal Field Splitting Energy (\(\Delta_O\)): In an octahedral crystal field, the interaction between the metal d-orbitals and the surrounding ligands causes the d-orbitals to split into two sets of different energy levels: the higher-energy \(e_g\) orbitals and the lower-energy \(t_{2g}\) orbitals. The energy difference between these two sets is called \(\Delta_O\) or the crystal field splitting energy.

Factors Affecting \(\Delta_O\):

Nature of the ligand: According to the spectrochemical series, different ligands produce different strengths of crystal fields:

\(\text{Weak field ligands: } \text{I}^- < \text{Br}^- < \text{S}^{2-} < \text{Cl}^- < \text{F}^- < \text{OH}^- < \text{H}_2O\)

\(\text{Moderate field ligands: } \text{NH}_3 < \text{en} < \text{phen}\)

\(\text{Strong field ligands: } \text{NO}_2^- < \text{CN}^- < \text{CO}\)

Weak field ligands (e.g., \(Cl^-\)) cause a smaller splitting (\(\Delta_O\)) while strong field ligands (e.g., \(CN^-\)) cause larger splitting.

Oxidation state of the metal ion: A higher oxidation state of the metal ion increases the metal’s effective nuclear charge, drawing the ligands closer and increasing \(\Delta_O\).

Nature of the metal: Transition metals from the second and third rows (e.g., Rh, Ir) tend to have larger \(\Delta_O\) values compared to those in the first row (e.g., Co, Cr).

Analyzing the Complexes:

1. \([Co(NH_3)_6]^{3+}\):

Ligand: Ammonia \((NH_3)\) is a moderately strong field ligand according to the spectrochemical series.

Metal: Cobalt \((Co)\) in a +3 oxidation state.

\(\Delta_O\): Since ammonia is a moderate field ligand, this complex will have a moderate \(\Delta_O\) value.

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

Ligand: Water \((H_2O)\) is a weaker field ligand than ammonia.

Metal: Chromium \((Cr)\) in a +3 oxidation state.

\(\Delta_O\): Water being a weaker ligand than ammonia results in a smaller \(\Delta_O\) than \([Co(NH_3)_6]^{3+}\).

3. \([Co(CN)_6]^{3-}\):

Ligand: Cyanide \((CN^-)\) is a very strong field ligand.

Metal: Cobalt \((Co)\) in a +3 oxidation state.

\(\Delta_O\): Cyanide is one of the strongest field ligands, which means this complex will have the largest \(\Delta_O\) among the given complexes.

4. \([CrCl_6]^{3-}\):

Ligand: Chloride \((Cl^-)\) is a very weak field ligand.

Metal: Chromium (Cr) in a +3 oxidation state.

\(\Delta_O\): Since chloride is a weak field ligand, this complex will have the smallest \(\Delta_O\) among all the given complexes.

Conclusion:

The complex with the smallest crystal field splitting energy (\(\Delta_O\)) is the one containing the weakest field ligand, chloride \((Cl^-)\). Therefore, \([CrCl_6]^{3-}\) has the lowest \(\Delta_O\) value.

Thus, Option (4) \([CrCl_6]^{3-}\) has the lowest \(\Delta_O\) value.