The total number of orbitals present for principle quantum number, n=4 is :

Options

A. 30
B. 12
C. 15
D. 16

Explanation

To determine the total number of orbitals present for a principal quantum number \( n = 4 \), we can follow these steps:

### Step-by-Step Solution:

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

2. **Calculate the Total Number of Orbitals**:
The total number of orbitals in a given energy level is given by the formula:
\[ \text{Total number of orbitals} = n^2 \]
For \( n = 4 \):
\[ \text{Total number of orbitals} = 4^2 = 16 \]

3. **Identify the Types of Subshells**:
The azimuthal quantum number \( l \) can take values from 0 to \( n-1 \). For \( n = 4 \), the possible values of \( l \) are:
- \( l = 0 \) (s subshell)
- \( l = 1 \) (p subshell)
- \( l = 2 \) (d subshell)
- \( l = 3 \) (f subshell)

4. **Determine the Number of Orbitals in Each Subshell**:
- **s subshell** (\( l = 0 \)): 1 orbital
- **p subshell** (\( l = 1 \)): 3 orbitals
- **d subshell** (\( l = 2 \)): 5 orbitals
- **f subshell** (\( l = 3 \)): 7 orbitals

5. **Add the Number of Orbitals**:
Now, we can sum the number of orbitals from each subshell:
\[ \text{Total orbitals} = 1 \, (\text{s}) + 3 \, (\text{p}) + 5 \, (\text{d}) + 7 \, (\text{f}) = 16 \]

6. **Conclusion**:
Therefore, the total number of orbitals present for the principal quantum number \( n = 4 \) is:
\[ \boxed{16} \]