Correct increasing order for the wavelengths of absorption in the visible region by the complexes of `Co^(3+)` is:

Options

A. `[Co(en)_(3)]^(3+),[Co(NH_(3))_(6)]^(3+),[Co(H_(2)O)_(6)]^(3+)`
B. `[Co(H_(2)O)_(6)]^(3+),[Co(en)_(6)]^(3+),[Co(NH_(3))_(6)]^(3+)`
C. `[Co(H_(2)O)_(6)]^(3+),[Co(NH_(3))_(6)]^(3+),[Co(en)_(3)]^(3+)`
D. `[Co(NH_(3))_(6)]^(3+),[Co(en)_(3)]^(3+),[Co(H_(2)O)_(6)]^(3+)`

Explanation

To determine the correct increasing order for the wavelengths of absorption in the visible region by the complexes of Co^(3+), we need to consider the ligand field strength of the ligands involved. The ligands mentioned are ethylenediamine (EN), ammonia (NH3), and water (H2O).

### Step-by-Step Solution:

1. **Identify the Ligands**: The ligands we have are:
- Ethylenediamine (EN)
- Ammonia (NH3)
- Water (H2O)

2. **Refer to the Spectrochemical Series**: The strength of ligands can be ranked based on their ability to split the d-orbitals in transition metal complexes. The spectrochemical series ranks ligands from strong field to weak field:
- Strongest: Ethylenediamine (EN)
- Intermediate: Ammonia (NH3)
- Weakest: Water (H2O)

3. **Understand the Relationship between Ligand Strength and Splitting Energy (ΔE)**: The splitting energy (ΔE) is directly proportional to the ligand field strength. Stronger field ligands cause greater splitting of the d-orbitals, leading to higher ΔE.

4. **Relate Splitting Energy to Wavelength**: The relationship between splitting energy and wavelength (λ) is given by the equation:
\[ ΔE = \frac{hc}{λ} \]
where:
- \(h\) is Planck's constant,
- \(c\) is the speed of light,
- \(λ\) is the wavelength.

This implies that as ΔE increases, the wavelength (λ) decreases.

5. **Determine the Order of Splitting**: Based on the ligand strength:
- EN has the highest splitting (smallest λ),
- NH3 has intermediate splitting,
- H2O has the lowest splitting (largest λ).

6. **Establish the Increasing Order of Wavelengths**: Since higher splitting corresponds to lower wavelengths, the order of wavelengths of absorption in the visible region will be:
\[ λ(H2O) > λ(NH3) > λ(EN) \]

7. **Final Answer**: Therefore, the correct increasing order for the wavelengths of absorption in the visible region by the complexes of Co^(3+) is:
\[ H2O > NH3 > EN \]