To solve the problem of finding the total number of electrons in an atom with quantum numbers \( n = 4 \) and \( |m_l| = 1 \), we can follow these steps:
### Step-by-Step Solution:
1. **Identify the Principal Quantum Number (n)**:
- The principal quantum number \( n \) is given as 4. This indicates that we are looking at the fourth energy level of the atom.
2. **Determine Possible Values of the Azimuthal Quantum Number (l)**:
- The azimuthal quantum number \( l \) can take values from 0 to \( n-1 \). Therefore, for \( n = 4 \), the possible values of \( l \) are:
- \( l = 0 \) (s orbital)
- \( l = 1 \) (p orbital)
- \( l = 2 \) (d orbital)
- \( l = 3 \) (f orbital)
3. **Analyze the Magnetic Quantum Number (m_l)**:
- The magnetic quantum number \( m_l \) can take values from \( -l \) to \( +l \). The absolute value \( |m_l| = 1 \) indicates that we are interested in orbitals where \( m_l \) can be either +1 or -1.
- This means we need to find which values of \( l \) allow \( m_l \) to equal +1 or -1:
- For \( l = 1 \): \( m_l = -1, 0, +1 \) (valid)
- For \( l = 2 \): \( m_l = -2, -1, 0, +1, +2 \) (valid)
- For \( l = 3 \): \( m_l = -3, -2, -1, 0, +1, +2, +3 \) (valid)
- For \( l = 0 \): \( m_l = 0 \) (not valid for \( |m_l| = 1 \))
4. **Count the Orbitals**:
- Now we need to count the orbitals corresponding to the valid \( l \) values:
- For \( l = 1 \): There are 3 orbitals (m_l = -1, 0, +1).
- For \( l = 2 \): There are 5 orbitals (m_l = -2, -1, 0, +1, +2).
- For \( l = 3 \): There are 7 orbitals (m_l = -3, -2, -1, 0, +1, +2, +3).
5. **Identify the Relevant Orbitals for |m_l| = 1**:
- We are specifically interested in the orbitals where \( |m_l| = 1 \):
- For \( l = 1 \): 1 orbital (m_l = +1).
- For \( l = 2 \): 2 orbitals (m_l = +1 and m_l = -1).
- For \( l = 3 \): 2 orbitals (m_l = +1 and m_l = -1).
6. **Calculate Total Electrons**:
- Each orbital can hold a maximum of 2 electrons (due to Pauli's exclusion principle):
- Total orbitals with \( |m_l| = 1 \) = 1 (from l=1) + 2 (from l=2) + 2 (from l=3) = 5 orbitals.
- Therefore, total electrons = \( 5 \text{ orbitals} \times 2 \text{ electrons/orbital} = 10 \text{ electrons} \).
### Final Answer:
The total number of electrons having quantum numbers \( n = 4 \) and \( |m_l| = 1 \) is **10 electrons**.
