How many orbitals are present in the subshells with (a) `n = 3, l = 2` (b) n = 4, l = 2 (c) n = 5, l = 2?

Explanation

To determine how many orbitals are present in the subshells for the given quantum numbers, we can use the following steps:

### Step-by-Step Solution:

1. **Understand Quantum Numbers**:
- The principal quantum number \( n \) indicates the energy level of the electron.
- The azimuthal quantum number \( l \) determines the shape of the orbital and can take values from \( 0 \) to \( n-1 \).
- The magnetic quantum number \( m \) can take values from \( -l \) to \( +l \).

2. **Determine the Number of Orbitals**:
- The number of orbitals in a subshell is given by the formula: \( 2l + 1 \).
- This formula accounts for all the possible values of the magnetic quantum number \( m \).

3. **Calculate for Each Case**:

**(a) For \( n = 3, l = 2 \)**:
- Here, \( l = 2 \) corresponds to the d subshell.
- Calculate the number of orbitals:
\[ \text{Number of orbitals} = 2l + 1 = 2(2) + 1 = 4 + 1 = 5 \]
- Therefore, there are **5 orbitals** in the 3d subshell.

**(b) For \( n = 4, l = 2 \)**:
- Again, \( l = 2 \) corresponds to the d subshell.
- Calculate the number of orbitals:
\[ \text{Number of orbitals} = 2l + 1 = 2(2) + 1 = 4 + 1 = 5 \]
- Therefore, there are **5 orbitals** in the 4d subshell.

**(c) For \( n = 5, l = 2 \)**:
- Once more, \( l = 2 \) corresponds to the d subshell.
- Calculate the number of orbitals:
\[ \text{Number of orbitals} = 2l + 1 = 2(2) + 1 = 4 + 1 = 5 \]
- Therefore, there are **5 orbitals** in the 5d subshell.

### Final Answers:
- (a) 5 orbitals for \( n = 3, l = 2 \)
- (b) 5 orbitals for \( n = 4, l = 2 \)
- (c) 5 orbitals for \( n = 5, l = 2 \)