The total number of orbitals in a shell having principal quantum n is

Options

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

Answer

To determine the total number of orbitals in a shell with a principal quantum number \( n \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand Principal Quantum Number \( n \)**:
- The principal quantum number \( n \) indicates the shell level of an electron in an atom. It can take positive integer values (1, 2, 3, ...).

2. **Identify Azimuthal Quantum Number \( l \)**:
- The azimuthal quantum number \( l \) defines the subshells within a shell and can take values from \( 0 \) to \( n-1 \).
- For each value of \( l \):
- \( l = 0 \) corresponds to the s-subshell.
- \( l = 1 \) corresponds to the p-subshell.
- \( l = 2 \) corresponds to the d-subshell.
- \( l = 3 \) corresponds to the f-subshell, and so on.

3. **Count Orbitals in Each Subshell**:
- Each subshell has a specific number of orbitals:
- s-subshell (\( l = 0 \)): 1 orbital
- p-subshell (\( l = 1 \)): 3 orbitals
- d-subshell (\( l = 2 \)): 5 orbitals
- f-subshell (\( l = 3 \)): 7 orbitals
- The number of orbitals in a subshell is given by the formula \( 2l + 1 \).

4. **Calculate Total Number of Orbitals**:
- For a given principal quantum number \( n \), the total number of orbitals can be calculated by summing the number of orbitals in all subshells from \( l = 0 \) to \( l = n-1 \).
- This can be expressed mathematically as:
\[ \text{Total orbitals} = \sum_{l=0}^{n-1} (2l + 1) \]
- This simplifies to \( n^2 \) because:
- For \( n = 1 \): \( 1^2 = 1 \)
- For \( n = 2 \): \( 2^2 = 4 \)
- For \( n = 3 \): \( 3^2 = 9 \)
- And so on...

5. **Conclusion**:
- Therefore, the total number of orbitals in a shell with principal quantum number \( n \) is \( n^2 \).

### Final Answer:
The total number of orbitals in a shell having principal quantum number \( n \) is \( n^2 \).

---