A weak field complex of `Ni^(2+)` has magnetic moment of 2.82 BM. The number of electron in the `t_(2g)` level of `Ni^(2+)` will be

Answer

To solve the problem, we need to determine the number of electrons in the \( t_{2g} \) level of the \( Ni^{2+} \) ion, given that it has a magnetic moment of 2.82 Bohr Magnetons (BM).

### Step-by-Step Solution:

1. **Identify the electronic configuration of Nickel (Ni)**:
- Nickel has an atomic number of 28. Its electronic configuration in the ground state is:
\[ \text{Ni: } [Ar] 4s^2 3d^8 \]

2. **Determine the configuration of \( Ni^{2+} \)**:
- When nickel loses two electrons to form \( Ni^{2+} \), the electrons are removed first from the 4s orbital. Therefore, the configuration becomes:
\[ Ni^{2+}: [Ar] 3d^8 \]

3. **Understand the splitting of d-orbitals in a crystal field**:
- In an octahedral field, the \( 3d \) orbitals split into two sets: \( t_{2g} \) (lower energy) and \( e_g \) (higher energy). The \( t_{2g} \) level consists of three orbitals, while the \( e_g \) level consists of two orbitals.

4. **Determine the magnetic moment**:
- The magnetic moment (\( \mu \)) can be calculated using the formula:
\[ \mu = \sqrt{n(n + 2)} \text{ BM} \]
where \( n \) is the number of unpaired electrons.

5. **Calculate the number of unpaired electrons**:
- Given that the magnetic moment is 2.82 BM, we can set up the equation:
\[ 2.82 = \sqrt{n(n + 2)} \]
- Squaring both sides:
\[ 2.82^2 = n(n + 2) \]
\[ 7.9524 = n^2 + 2n \]
- Rearranging gives us a quadratic equation:
\[ n^2 + 2n - 7.9524 = 0 \]

6. **Solve the quadratic equation**:
- Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1, b = 2, c = -7.9524 \):
\[ n = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-7.9524)}}{2 \cdot 1} \]
\[ n = \frac{-2 \pm \sqrt{4 + 31.8096}}{2} \]
\[ n = \frac{-2 \pm \sqrt{35.8096}}{2} \]
\[ n = \frac{-2 \pm 5.983}{2} \]
- This gives two possible values for \( n \):
\[ n = \frac{3.983}{2} \approx 1.991 \quad \text{(approximately 2)} \]
or
\[ n = \frac{-7.983}{2} \quad \text{(not a valid solution)} \]

7. **Determine the electron distribution in \( t_{2g} \)**:
- Since \( Ni^{2+} \) has 8 electrons in the \( 3d \) subshell and we have found that there are approximately 2 unpaired electrons, we can deduce how the electrons are distributed:
- In a weak field ligand situation (high spin), the electrons will fill the \( t_{2g} \) and \( e_g \) levels as follows:
- \( t_{2g} \): 6 electrons (fully filled)
- \( e_g \): 2 electrons (unpaired)

8. **Conclusion**:
- Therefore, the number of electrons in the \( t_{2g} \) level of \( Ni^{2+} \) is:
\[ \text{Number of electrons in } t_{2g} = 6 \]