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 nickel cation in the complex \([Ni(DMG)_2]\).

### Step 1: Determine the oxidation state of Nickel
Nickel in the complex \([Ni(DMG)_2]\) is in the +2 oxidation state. This is because DMG (Dimethylglyoxime) is a neutral ligand.

### Step 2: Write the electronic configuration of Ni²⁺
The atomic number of Nickel (Ni) is 28. The electronic configuration of neutral Nickel is:
\[ [Ar] 3d^8 4s^2 \]
When it loses two electrons to form Ni²⁺, the configuration becomes:
\[ Ni^{2+}: [Ar] 3d^8 \]

### Step 3: Identify the d-orbitals and their types
In the d-orbital, there are five orbitals:
- Non-axial orbitals (t2g): \( d_{xy}, d_{yz}, d_{zx} \) (3 orbitals)
- Axial orbitals (eg): \( d_{x^2-y^2}, d_{z^2} \) (2 orbitals)

### Step 4: Determine the distribution of electrons in the d-orbitals
For a strong field ligand like DMG, the electrons will fill the lower energy t2g orbitals first before occupying the eg orbitals.

- The 8 electrons in \( 3d \) will fill as follows:
- t2g (non-axial): 6 electrons (3 orbitals, 2 electrons each)
- eg (axial): 2 electrons (2 orbitals, 1 electron each)

Thus, we have:
- \( x = 6 \) (number of electrons in non-axial orbitals)
- \( y = 2 \) (number of electrons in axial orbitals)

### Step 5: Calculate the value of \( \frac{2x^2}{y} \)
Now we can substitute the values of \( x \) and \( y \) into the expression \( \frac{2x^2}{y} \):
\[ \frac{2x^2}{y} = \frac{2(6^2)}{2} = \frac{2 \times 36}{2} = 36 \]

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

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