To determine the maximum number of orbitals that can be identified with the given quantum numbers \( n = 3 \), \( l = 1 \), and \( m_l = 0 \), we can follow these steps:
### Step-by-Step Solution:
1. **Identify the Principal Quantum Number (n)**:
- The principal quantum number \( n \) indicates the energy level of the electron. Here, \( n = 3 \) means we are dealing with the third energy level.
2. **Identify the Azimuthal Quantum Number (l)**:
- The azimuthal quantum number \( l \) defines the shape of the orbital. The value of \( l \) can range from \( 0 \) to \( n-1 \).
- For \( l = 1 \), we are dealing with a **p** subshell (since \( l = 0 \) corresponds to s, \( l = 1 \) corresponds to p, \( l = 2 \) corresponds to d, and so on).
3. **Determine the Magnetic Quantum Number (m_l)**:
- The magnetic quantum number \( m_l \) can take values from \( -l \) to \( +l \), including zero. Thus, for \( l = 1 \), \( m_l \) can be \( -1, 0, \) or \( +1 \).
- This means that the possible orbitals in the 3p subshell are:
- \( m_l = -1 \) corresponds to the \( 3p_x \) orbital,
- \( m_l = 0 \) corresponds to the \( 3p_y \) orbital,
- \( m_l = +1 \) corresponds to the \( 3p_z \) orbital.
4. **Identify the Specific Orbital**:
- Since we are given \( m_l = 0 \), this specifically refers to the \( 3p_y \) orbital.
- Therefore, with \( m_l = 0 \), we can only identify **one specific orbital**.
### Conclusion:
The maximum number of orbitals that can be identified with the quantum numbers \( n = 3 \), \( l = 1 \), and \( m_l = 0 \) is **1** (specifically the \( 3p_y \) orbital).
