The correct order for the wavelength of absorption in the visible region is

Options

A. `[Ni(NO_(2))_(6)]^(2+)lt[Ni(NH_(3))_(6)]^(2+)lt[Ni(H_(2)O)_(6)]^(2+)`
B. `[Ni(NO_(2))_(6)]^(4-)lt[Ni(H_(2)O)_(6)]^(2+)lt[Ni(NH_(3))_(6)]^(2+)`
C. `[Ni(H_(2)O)_(6)]^(2+)lt[Ni(NH_(3))_(6)]^(2+)lt[Ni(NO_(2))_(6)]^(4-)`
D. `[Ni(NH_(3))_(6)]^(2+)lt[Ni(H_(2)O)_(6)]^(2+)lt[Ni(NO_(2))_(6)]^(4-)`

Explanation

To determine the correct order for the wavelength of absorption in the visible region for the given complexes, we need to analyze the factors affecting the crystal field splitting energy (Δo) and how they relate to the wavelength (λ) of absorption.

### Step-by-Step Solution:

1. **Understand the Relationship Between Energy and Wavelength**:
- The energy (E) of light is inversely proportional to its wavelength (λ). This can be expressed by the equation:
\[ E = \frac{hc}{\lambda} \]
- Where \(h\) is Planck's constant and \(c\) is the speed of light. Therefore, as energy increases, wavelength decreases, and vice versa.

2. **Identify the Complexes and Their Metal Oxidation States**:
- The complexes in question involve nickel (Ni) with an oxidation state of +2, which has the electronic configuration of \(d^8\).

3. **Determine the Crystal Field Splitting Energy (Δo)**:
- In octahedral complexes, the \(d\) orbitals split into two sets: \(t_{2g}\) (lower energy) and \(e_g\) (higher energy). The energy difference between these two sets is known as Δo.

4. **Factors Affecting Δo**:
- Δo is influenced by:
- The oxidation state of the metal: Higher oxidation states generally lead to higher Δo.
- The nature of the ligands: Strong field ligands increase Δo, while weak field ligands decrease Δo.

5. **Analyze the Ligands**:
- The ligands in the question are:
- NO2⁻ (strong field ligand)
- NH3 (moderate field ligand)
- H2O (weak field ligand)
- According to the spectrochemical series, the order of ligand strength is:
\[ \text{NO2}^- > \text{NH3} > \text{H2O} \]

6. **Determine the Order of Δo**:
- Since NO2⁻ is the strongest ligand, it will give the highest Δo, followed by NH3, and then H2O with the lowest Δo.
- Therefore, the order of Δo is:
\[ \Delta o (\text{NO2}^-) > \Delta o (\text{NH3}) > \Delta o (\text{H2O}) \]

7. **Relate Δo to Wavelength (λ)**:
- Since Δo is inversely proportional to λ, the order of wavelengths will be the opposite of the order of Δo:
\[ \lambda (\text{H2O}) > \lambda (\text{NH3}) > \lambda (\text{NO2}^-) \]

8. **Final Order of Wavelengths**:
- Thus, the correct order for the wavelength of absorption in the visible region is:
\[ \text{Ni(H2O)}_6^{2+} > \text{Ni(NH3)}_6^{2+} > \text{Ni(NO2)}_6^{2+} \]

### Conclusion:
The correct option is **A**: Ni(H2O)₆²⁺ > Ni(NH₃)₆²⁺ > Ni(NO₂)₆²⁺.