The degenerate orbitals of `[Cr(H_(2)O)_(6)]^(3+)` are :

Options

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

Answer

To find the degenerate orbitals of the complex \([Cr(H_2O)_6]^{3+}\), we will follow these steps:

### Step 1: Determine the oxidation state of chromium
The complex \([Cr(H_2O)_6]^{3+}\) indicates that chromium is in the +3 oxidation state.

### Step 2: Write the electronic configuration of chromium
Chromium (Cr) has an atomic number of 24. The electronic configuration of neutral chromium is:
\[ [Ar] 3d^5 4s^1 \]
When chromium is in the +3 oxidation state, it loses three electrons. The two electrons are removed from the 4s orbital and one from the 3d orbital:
\[ Cr^{3+}: [Ar] 3d^3 \]

### Step 3: Identify the geometry of the complex
The complex \([Cr(H_2O)_6]^{3+}\) is an octahedral complex because it has six ligands (water molecules) surrounding the chromium ion.

### Step 4: Understand the splitting of d-orbitals in an octahedral field
In an octahedral field, the five d-orbitals split into two sets:
- The lower energy set (t2g): \(d_{xy}, d_{xz}, d_{yz}\)
- The higher energy set (eg): \(d_{x^2-y^2}, d_{z^2}\)

### Step 5: Fill the d-orbitals with the electrons
For \(Cr^{3+}\) with a \(3d^3\) configuration, the three electrons will occupy the lower energy t2g orbitals:
- The three electrons will fill the \(d_{xy}, d_{xz}, d_{yz}\) orbitals.

### Step 6: Identify the degenerate orbitals
The degenerate orbitals for the \(Cr^{3+}\) complex in an octahedral field are:
- \(d_{xy}\)
- \(d_{xz}\)
- \(d_{yz}\)

These orbitals are degenerate because they have the same energy level in the octahedral crystal field.

### Final Answer
The degenerate orbitals of \([Cr(H_2O)_6]^{3+}\) are \(d_{xy}, d_{xz}, d_{yz}\).

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