The correct orders of increasing energy of atomic orbitals is

Options

A. `5p lt 4f lt 6s lt 5d`
B. `5p lt 6s lt 4f lt 5d`
C. `4f lt 5p lt 5d lt 6s`
D. `5p lt 5d lt 4f lt 6s`

Explanation

To determine the correct order of increasing energy of atomic orbitals, we will use the n + l rule. This rule states that the energy of an atomic orbital increases with the sum of the principal quantum number (n) and the azimuthal quantum number (l). The orbital with the lowest n + l value has the lowest energy. If two orbitals have the same n + l value, the one with the lower n value has lower energy.

### Step-by-Step Solution:

1. **Identify the orbitals and their quantum numbers**:
- For **5p**:
- n = 5, l = 1 (p orbitals)
- n + l = 5 + 1 = 6
- For **4f**:
- n = 4, l = 3 (f orbitals)
- n + l = 4 + 3 = 7
- For **6s**:
- n = 6, l = 0 (s orbitals)
- n + l = 6 + 0 = 6
- For **5d**:
- n = 5, l = 2 (d orbitals)
- n + l = 5 + 2 = 7

2. **Calculate n + l for each orbital**:
- 5p: n + l = 6
- 4f: n + l = 7
- 6s: n + l = 6
- 5d: n + l = 7

3. **Order the orbitals based on n + l values**:
- The orbitals with the same n + l values are compared based on their n values.
- 5p and 6s both have n + l = 6, but since 6s has a higher n value (n=6), it will be higher in energy than 5p.
- 4f and 5d both have n + l = 7, but since 5d has a lower n value (n=5) than 4f (n=4), it will be lower in energy.

4. **Final order of increasing energy**:
- 5p (n + l = 6)
- 6s (n + l = 6)
- 5d (n + l = 7)
- 4f (n + l = 7)

Thus, the correct order of increasing energy of the atomic orbitals is:
**5p < 6s < 5d < 4f**