For principle quantum number n=4,the total number of orbitals having l=3 is

Options

A. 3
B. 7
C. 5
D. 9

Explanation

To solve the question regarding the total number of orbitals for a principal quantum number \( n = 4 \) and azimuthal quantum number \( l = 3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Quantum Numbers**:
- The principal quantum number \( n \) indicates the energy level of an electron in an atom and can take positive integer values (1, 2, 3, ...).
- The azimuthal quantum number \( l \) defines the shape of the orbital and can take values from \( 0 \) to \( n-1 \). For \( n = 4 \), \( l \) can take values \( 0, 1, 2, 3 \).

2. **Identify the Magnetic Quantum Number**:
- The magnetic quantum number \( m \) determines the orientation of the orbital and can take values from \( -l \) to \( +l \).
- Therefore, for \( l = 3 \), the possible values of \( m \) are:
\[ m = -3, -2, -1, 0, +1, +2, +3 \]

3. **Count the Number of Orbitals**:
- The total number of orbitals corresponding to a specific \( l \) value can be calculated using the formula:
\[ \text{Number of orbitals} = 2l + 1 \]
- Substituting \( l = 3 \):
\[ \text{Number of orbitals} = 2(3) + 1 = 6 + 1 = 7 \]

4. **Conclusion**:
- Therefore, the total number of orbitals for \( n = 4 \) and \( l = 3 \) is \( 7 \).

### Final Answer:
The total number of orbitals having \( l = 3 \) for \( n = 4 \) is **7**.

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