Geometrical Isomerism and Properties of Cobalt Complex

To determine which statements are correct for the coordination complex $[\text{Co}(\text{en})(\text{NH}_3)_3(\text{H}_2\text{O})]^{3+}$, let us analyze each option individually:

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1. Analysis of Option (A)

The complex $[\text{Co}(\text{en})(\text{NH}_3)_3(\text{H}_2\text{O})]^{3+}$ is of the type $[\text{M}(\text{AA})\text{b}_3\text{c}]$, where:

• $\text{AA} = \text{en}$ (symmetrical bidentate ligand)
• $\text{b} = \text{NH}_3$ (monodentate ligand)
• $\text{c} = \text{H}_2\text{O}$ (monodentate ligand)

Since $\text{en}$ spans adjacent (cis) positions, we can determine the distinct geometrical isomers based on the ligands occupying positions trans to the donor atoms of $\text{en}$:

1. Isomer 1: One donor nitrogen of $\text{en}$ is trans to $\text{H}_2\text{O}$, and the other is trans to $\text{NH}_3$. The remaining two $\text{NH}_3$ ligands are trans to each other. $\text{Trans-pairs: } (\text{N}_{\text{en}}, \text{H}_2\text{O}), \; (\text{N}_{\text{en}}, \text{NH}_3), \; (\text{NH}_3, \text{NH}_3)$

2. Isomer 2: Both donor nitrogens of $\text{en}$ are trans to $\text{NH}_3$ ligands. The $\text{H}_2\text{O}$ ligand is trans to the remaining $\text{NH}_3$ ligand. $\text{Trans-pairs: } (\text{N}_{\text{en}}, \text{NH}_3), \; (\text{N}_{\text{en}}, \text{NH}_3), \; (\text{NH}_3, \text{H}_2\text{O})$

Thus, the complex exhibits two geometrical isomers.

Statement (A) is CORRECT.

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2. Analysis of Option (B)

Replacing the bidentate $\text{en}$ ligand with two $\text{CN}^-$ ligands yields the complex $[\text{Co}(\text{CN})_2(\text{NH}_3)_3(\text{H}_2\text{O})]^+$, which is of the type $[\text{M}\text{a}_2\text{b}_3\text{c}]$, where $\text{a} = \text{CN}^-$, $\text{b} = \text{NH}_3$, and $\text{c} = \text{H}_2\text{O}$.

The possible trans-pair combinations for a general $[\text{M}\text{a}_2\text{b}_3\text{c}]$ complex are:

1. $(\text{a}, \text{a})(\text{b}, \text{b})(\text{b}, \text{c})$
2. $(\text{a}, \text{b})(\text{a}, \text{b})(\text{b}, \text{c})$
3. $(\text{a}, \text{b})(\text{a}, \text{c})(\text{b}, \text{b})$

Thus, it exists as three geometrical isomers.

Statement (B) is CORRECT.

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3. Analysis of Option (C)

In $[\text{Co}(\text{en})(\text{NH}_3)_3(\text{H}_2\text{O})]^{3+}$:

• Oxidation state of Cobalt: $\text{Co}^{3+}$
• Electronic configuration of $\text{Co}^{3+}$: $3d^6$

Due to the high positive charge of $+3$ on the central cobalt ion along with strong-field ligands ($\text{en}$ and $\text{NH}_3$), the crystal field splitting energy ($\Delta_o$) exceeds the electron pairing energy ($P$). Consequently, pairing of $d$-electrons occurs to form a low-spin octahedral complex: $\text{Electronic configuration: } t_{2g}^6 e_g^0$

Since the number of unpaired electrons $n = 0$, the complex is diamagnetic (not paramagnetic).

Statement (C) is INCORRECT.

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4. Analysis of Option (D)

According to the spectrochemical series, the order of ligand field strength is: $\text{H}_2\text{O} < \text{NH}_3 < \text{en}$

Comparing $[\text{Co}(\text{en})(\text{NH}_3)_3(\text{H}_2\text{O})]^{3+}$ with $[\text{Co}(\text{en})(\text{NH}_3)_4]^{3+}$, replacing one $\text{NH}_3$ ligand with a weaker ligand $\text{H}_2\text{O}$ decreases the crystal field splitting energy ($\Delta_o$).

The energy of absorbed light is inversely proportional to its wavelength: $\Delta_o = \frac{hc}{\lambda} \implies \lambda \propto \frac{1}{\Delta_o}$

Because $[\text{Co}(\text{en})(\text{NH}_3)_3(\text{H}_2\text{O})]^{3+}$ has a smaller $\Delta_o$, it absorbs light of a longer wavelength ($\lambda$) relative to $[\text{Co}(\text{en})(\text{NH}_3)_4]^{3+}$.

Statement (D) is CORRECT.

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Conclusion

The correct options are A, B, and D.