Which of the following orbitals are degenerate?
`3d_(xy), 4d_(xy), 3d_(z)^(2), 3d_(yz), 5d_(z)^(2)`

Options

A. `3d_(xy), 3d_(z)^(2), 3d_(yz)`
B. `4d_(xy), 3d_(z)^(2), 3d_(yz)`
C. `3d_(z)^(2), 3d_(yz), 5d_(z)^(2)`
D. None

Answer

To determine which of the given orbitals are degenerate, we will use the n + l rule. The n + l rule helps us understand the relative energy levels of orbitals in multi-electron atoms.

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number (n) and Azimuthal Quantum Number (l)**:
- For each orbital, identify the values of n and l.
- For the d orbitals, l = 2.

2. **Calculate n + l for Each Orbital**:
- **3d_(xy)**:
- n = 3, l = 2
- n + l = 3 + 2 = 5
- **4d_(xy)**:
- n = 4, l = 2
- n + l = 4 + 2 = 6
- **3d_(z)^(2)**:
- n = 3, l = 2
- n + l = 3 + 2 = 5
- **3d_(yz)**:
- n = 3, l = 2
- n + l = 3 + 2 = 5
- **5d_(z)^(2)**:
- n = 5, l = 2
- n + l = 5 + 2 = 7

3. **Compare the n + l Values**:
- The orbitals with the same n + l value are considered degenerate.
- From our calculations:
- 3d_(xy): n + l = 5
- 3d_(z)^(2): n + l = 5
- 3d_(yz): n + l = 5
- 4d_(xy): n + l = 6
- 5d_(z)^(2): n + l = 7

4. **Identify Degenerate Orbitals**:
- The orbitals 3d_(xy), 3d_(z)^(2), and 3d_(yz) all have the same n + l value of 5, which means they are degenerate.
- The orbitals 4d_(xy) and 5d_(z)^(2) are not degenerate with the others as their n + l values are different.

### Final Answer:
The degenerate orbitals are: **3d_(xy), 3d_(z)^(2), and 3d_(yz)**.

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