Maximum number of electrons in given shell identified by the principal quantum number `n` is equal to

Options

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

Explanation

To determine the maximum number of electrons that can be accommodated in a given shell identified by the principal quantum number \( n \), we can use the formula:

\[ \text{Maximum number of electrons} = 2n^2 \]

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number \( n \)**:
- The principal quantum number \( n \) indicates the shell level. For example, \( n = 1 \) corresponds to the first shell (K shell), \( n = 2 \) corresponds to the second shell (L shell), and so on.

2. **Apply the Formula**:
- The formula to find the maximum number of electrons in a shell is \( 2n^2 \).

3. **Calculate for Different Values of \( n \)**:
- For \( n = 1 \):
\[ \text{Maximum electrons} = 2 \times 1^2 = 2 \text{ electrons} \]
- For \( n = 2 \):
\[ \text{Maximum electrons} = 2 \times 2^2 = 2 \times 4 = 8 \text{ electrons} \]
- For \( n = 3 \):
\[ \text{Maximum electrons} = 2 \times 3^2 = 2 \times 9 = 18 \text{ electrons} \]
- For \( n = 4 \):
\[ \text{Maximum electrons} = 2 \times 4^2 = 2 \times 16 = 32 \text{ electrons} \]

4. **Conclusion**:
- The maximum number of electrons that can be accommodated in a shell is given by the formula \( 2n^2 \). This means that as \( n \) increases, the capacity of the shell to hold electrons increases significantly.

### Final Answer:
The maximum number of electrons in a given shell identified by the principal quantum number \( n \) is \( 2n^2 \).