Which series of subshells is arranged in the order of increasing energy for multi-electron atoms?

Options

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

Answer

To determine the order of increasing energy for the given subshells in multi-electron atoms, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Subshells and Their Quantum Numbers**:
We have the following subshells: 6s, 4f, 5d, and 6p. Each subshell has a principal quantum number (n) and an azimuthal quantum number (l):
- For 6s: n = 6, l = 0
- For 4f: n = 4, l = 3
- For 5d: n = 5, l = 2
- For 6p: n = 6, l = 1

2. **Calculate n + l Values**:
We calculate the n + l values for each subshell:
- 6s: n + l = 6 + 0 = 6
- 4f: n + l = 4 + 3 = 7
- 5d: n + l = 5 + 2 = 7
- 6p: n + l = 6 + 1 = 7

3. **Determine the Order Based on n + l Values**:
According to the rules of energy levels in multi-electron atoms:
- The subshell with the lowest n + l value has the lowest energy.
- If two subshells have the same n + l value, the one with the lower n value has lower energy.

From our calculations:
- 6s has the lowest n + l value (6), so it has the lowest energy.
- 4f, 5d, and 6p all have the same n + l value (7). We compare their n values:
- 4f (n = 4) < 5d (n = 5) < 6p (n = 6)

4. **Arrange the Subshells in Increasing Order of Energy**:
Based on the above analysis, we can arrange the subshells in increasing order of energy:
- 6s < 4f < 5d < 6p

### Final Answer:
The increasing order of energy for the subshells is:
**6s < 4f < 5d < 6p**