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

Options

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

Explanation

To find the total number of orbitals for a given principal quantum number \( n \) and azimuthal quantum number \( l \), we can follow these steps:

### Step 1: Understand the Quantum Numbers
The principal quantum number \( n \) indicates the energy level of an electron in an atom, while the azimuthal quantum number \( l \) determines the shape of the orbital. The value of \( l \) can range from \( 0 \) to \( n-1 \).

### Step 2: Identify the Given Values
In this case, we have:
- Principal quantum number \( n = 4 \)
- Azimuthal quantum number \( l = 3 \)

### Step 3: Determine the Possible Values of the Magnetic Quantum Number \( m_l \)
The magnetic quantum number \( m_l \) can take values from \( -l \) to \( +l \). Therefore, for \( l = 3 \):
\[ m_l = -3, -2, -1, 0, +1, +2, +3 \]

### Step 4: Count the Values of \( m_l \)
The total number of values for \( m_l \) can be calculated as follows:
- The values are: \( -3, -2, -1, 0, 1, 2, 3 \)
- This gives us a total of \( 7 \) values.

### Step 5: Conclusion
Thus, the total number of orbitals for \( l = 3 \) is \( 7 \).

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

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