Set of d-orbitals which is used by central metal during formation of `MnO_(4)^(-)`?

Options

A. `d_(x^(2)-y^(2)),d_(x^(2)),d_(xy)`
B. `d_(xy),d_(yz),d_(xz)`
C. `d_(x^(2)-y^(2)),d_(xy),d_(xz)`
D. `d_(x^(2)-y^(2)),d_(z^(2)),d_(xz)`

Explanation

To determine the set of d-orbitals used by the central metal atom (manganese) during the formation of the permanganate ion \( \text{MnO}_4^{-} \), we can follow these steps:

### Step 1: Determine the oxidation state of manganese in \( \text{MnO}_4^{-} \)

Let the oxidation state of manganese be \( x \). The oxidation state of oxygen is \(-2\), and there are 4 oxygen atoms in the ion. The overall charge of the ion is \(-1\).

The equation can be set up as follows:
\[ x + 4(-2) = -1 \]
This simplifies to:
\[ x - 8 = -1 \]
Solving for \( x \):
\[ x = +7 \]

### Step 2: Write the electronic configuration of manganese

Manganese has an atomic number of 25. The electronic configuration of manganese is:
\[ \text{Mn}: [\text{Ar}] 4s^2 3d^5 \]

### Step 3: Identify the electron configuration in the +7 oxidation state

In the +7 oxidation state, manganese loses all its valence electrons. Therefore, we remove the two \( 4s \) electrons and all five \( 3d \) electrons:
\[ \text{Mn}^{7+}: [\text{Ar}] \]
This means that in the +7 oxidation state, manganese has no electrons in the \( 4s \) or \( 3d \) orbitals.

### Step 4: Determine the hybridization

In the formation of \( \text{MnO}_4^{-} \), the central metal ion (Mn) will use its available orbitals for hybridization. Since all \( 3d \) orbitals are vacant, the hybridization will involve the \( 3d \) orbitals and the \( 4s \) and \( 4p \) orbitals.

The hybridization that occurs here is \( d^3s \) because we can utilize three \( 3d \) orbitals and one \( 4s \) orbital to form four hybrid orbitals.

### Step 5: Identify the d-orbitals used

The specific d-orbitals that are involved in the hybridization are:
- \( d_{x^2-y^2} \)
- \( d_{xy} \)
- \( d_{xz} \)

These orbitals are used to form the hybrid orbitals that will bond with the oxygen atoms in the permanganate ion.

### Conclusion

The set of d-orbitals used by the central metal (Mn) during the formation of \( \text{MnO}_4^{-} \) is the \( d_{x^2-y^2} \), \( d_{xy} \), and \( d_{xz} \) orbitals, leading to a hybridization type of \( d^3s \).

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