If x and y are total number of electrons which are present in non-axial and axial set off d-orbitals respectively in Ni cation of `[Ni(DMG)_(2)]`, then calculate the value of `(2x^(2))/(y)`

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) for the coordination compound \([Ni(DMG)_{2}]\) and then calculate the expression \(\frac{2x^{2}}{y}\).

### Step-by-Step Solution:

1. **Identify the Oxidation State of Nickel:**
- The ligand DMG (dimethylglyoxime) has a charge of -1 and there are two DMG ligands in the complex.
- Let the oxidation state of Ni be \( x \).
- The overall charge of the complex is neutral (0).
- Therefore, the equation is:
\[ x + 2(-1) = 0 \implies x - 2 = 0 \implies x = +2 \]
- Thus, the oxidation state of Nickel in \([Ni(DMG)_{2}]\) is +2.

2. **Determine the Electron Configuration of Ni²⁺:**
- The atomic number of Nickel (Ni) is 28.
- The ground state electron configuration of Ni is:
\[ [Ar] 4s^{2} 3d^{8} \]
- For Ni²⁺, we remove two electrons from the 4s orbital:
\[ Ni^{2+} : [Ar] 3d^{8} \]

3. **Distribute the Electrons in d-Orbitals:**
- The 3d orbitals can be represented as follows:
- \( d_{xy}, d_{yz}, d_{zx} \) (non-axial orbitals)
- \( d_{x^{2}-y^{2}}, d_{z^{2}} \) (axial orbitals)
- The distribution of 8 electrons in the 3d orbitals is:
- Non-axial (3 orbitals): \( d_{xy}, d_{yz}, d_{zx} \) can hold 6 electrons (2 in each).
- Axial (2 orbitals): \( d_{x^{2}-y^{2}}, d_{z^{2}} \) can hold 2 electrons (1 in each).

4. **Calculate \( x \) and \( y \):**
- \( x \) (total number of electrons in non-axial orbitals):
\[ x = 6 \quad (\text{2 in } d_{xy} + 2 \text{ in } d_{yz} + 2 \text{ in } d_{zx}) \]
- \( y \) (total number of electrons in axial orbitals):
\[ y = 2 \quad (\text{1 in } d_{x^{2}-y^{2}} + 1 \text{ in } d_{z^{2}}) \]

5. **Substitute \( x \) and \( y \) into the Expression:**
- We need to calculate:
\[ \frac{2x^{2}}{y} \]
- Substitute \( x = 6 \) and \( y = 2 \):
\[ \frac{2(6)^{2}}{2} = \frac{2 \times 36}{2} = 36 \]

### Final Answer:
The value of \(\frac{2x^{2}}{y}\) is **36**.