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 \([Ni(DMG)_{2}]\) and then calculate the expression \(\frac{2x^2}{y}\).

### Step 1: Determine the oxidation state of Nickel in \([Ni(DMG)_{2}]\)

1. **Identify the ligands**: DMG (Dimethylglyoxime) is a neutral ligand.
2. **Calculate the oxidation state**: Since DMG is neutral, the oxidation state of Nickel (Ni) in the complex is \( +2 \).

### Step 2: Determine the electron configuration of Ni²⁺

1. **Nickel (Ni) atomic number**: 28
2. **Electron configuration of Ni**: \([Ar] 3d^8 4s^2\)
3. **For Ni²⁺**: Remove 2 electrons from the 4s orbital.
- Thus, the electron configuration of Ni²⁺ is \([Ar] 3d^8\).

### Step 3: Identify the geometry of the complex

1. **Geometry of \([Ni(DMG)_{2}]\)**: The complex is square planar due to the strong field ligands (DMG).
2. **Splitting of d-orbitals in square planar geometry**: The d-orbitals split into two sets:
- Non-axial: \(d_{xy}, d_{xz}, d_{yz}\)
- Axial: \(d_{x^2-y^2}, d_{z^2}\)

### Step 4: Count the electrons in the d-orbitals

1. **Filling of d-orbitals for Ni²⁺ in square planar geometry**:
- The 8 electrons in \(3d^8\) will fill the orbitals as follows:
- \(d_{xy}\) (1), \(d_{xz}\) (1), \(d_{yz}\) (1), \(d_{x^2-y^2}\) (2), and \(d_{z^2}\) (0).
- Thus, the filling will be:
- \(d_{xy} = 2\)
- \(d_{xz} = 2\)
- \(d_{yz} = 2\)
- \(d_{x^2-y^2} = 2\)
- \(d_{z^2} = 0\)

2. **Count the electrons**:
- Non-axial orbitals (\(d_{xy}, d_{xz}, d_{yz}\)): \(x = 6\) (2+2+2)
- Axial orbitals (\(d_{x^2-y^2}, d_{z^2}\)): \(y = 2\) (2 in \(d_{x^2-y^2}\) and 0 in \(d_{z^2}\))

### Step 5: Calculate the expression \(\frac{2x^2}{y}\)

1. **Substituting the values**:
- \(x = 6\)
- \(y = 2\)

2. **Calculate \(2x^2\)**:
\[ 2x^2 = 2 \times (6^2) = 2 \times 36 = 72 \]

3. **Calculate \(\frac{2x^2}{y}\)**:
\[ \frac{2x^2}{y} = \frac{72}{2} = 36 \]

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

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