Which of the following orbitals are degenerate for `[Cr(H_(2)O)_(6)]^(3-)`

Options

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

Answer

To determine which orbitals are degenerate in the complex ion \([Cr(H_2O)_6]^{3-}\), we will follow these steps:

### Step 1: Determine the oxidation state of Chromium
The complex \([Cr(H_2O)_6]^{3-}\) contains water as a neutral ligand. Let the oxidation state of chromium be \(X\). The equation for the oxidation state can be set up as follows:

\[ X + 6(0) = -3 \]

This simplifies to:

\[ X = -3 \]

Thus, the oxidation state of chromium in this complex is +3.

### Step 2: Identify the electron configuration of \(Cr^{3+}\)
Chromium has an atomic number of 24, and its electron configuration in the neutral state is:

\[ [Ar] 3d^5 4s^1 \]

When chromium loses three electrons to form \(Cr^{3+}\), it loses the 4s electron and two 3d electrons:

\[ Cr^{3+}: [Ar] 3d^3 \]

### Step 3: Understand the splitting of d-orbitals in an octahedral field
In an octahedral field, the five d-orbitals split into two sets of orbitals due to the presence of ligands. The two sets are:

- **Higher energy orbitals (eg)**: \(d_{x^2-y^2}\) and \(d_{z^2}\)
- **Lower energy orbitals (t2g)**: \(d_{xy}\), \(d_{yz}\), and \(d_{zx}\)

### Step 4: Determine the electron filling in the d-orbitals
For \(Cr^{3+}\) with a \(3d^3\) configuration, the three electrons will occupy the lower energy \(t_{2g}\) orbitals:

- \(d_{xy}\)
- \(d_{yz}\)
- \(d_{zx}\)

These three orbitals are degenerate, meaning they have the same energy.

### Step 5: Identify the degenerate orbitals
The degenerate orbitals for \(Cr^{3+}\) in the octahedral field are:

- \(d_{xy}\)
- \(d_{yz}\)
- \(d_{zx}\)

### Conclusion
The degenerate orbitals for the complex \([Cr(H_2O)_6]^{3-}\) are \(d_{xy}\), \(d_{yz}\), and \(d_{zx}\).