The total number of electrons in a principal energy shell is designated by expression

Options

A. n
B. `2n+1`
C. `n^(2)`
D. `2n^(2)`

Explanation

To determine the total number of electrons that can occupy a principal energy shell, we use the formula:

**Total number of electrons (N) = 2n²**

where **n** is the principal quantum number of the shell.

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number (n)**:
- The principal quantum number (n) indicates the energy level or shell. For example:
- For the K shell, n = 1
- For the L shell, n = 2
- For the M shell, n = 3

2. **Apply the Formula**:
- The formula to calculate the maximum number of electrons in a shell is **N = 2n²**.

3. **Calculate for Each Shell**:
- **For the K shell (n = 1)**:
\[ N = 2(1)² = 2 \text{ electrons} \]
- **For the L shell (n = 2)**:
\[ N = 2(2)² = 2 \times 4 = 8 \text{ electrons} \]
- **For the M shell (n = 3)**:
\[ N = 2(3)² = 2 \times 9 = 18 \text{ electrons} \]

4. **Conclusion**:
- The total number of electrons that can occupy a principal energy shell is determined by the formula \(2n²\), where n is the principal quantum number.
- For the K shell, it can hold 2 electrons; for the L shell, it can hold 8 electrons; and for the M shell, it can hold 18 electrons.