To solve the problem, we need to determine the total number of electrons in an atom that have the following quantum numbers:
- Principal quantum number (n) = 4
- Magnetic quantum number (m_l) = 1
- Spin quantum number (m_s) = -1/2
### Step-by-Step Solution:
1. **Identify the Principal Quantum Number (n)**:
- The principal quantum number \( n = 4 \) indicates that we are dealing with the fourth energy level or shell of the atom.
2. **Determine the Azimuthal Quantum Number (l)**:
- The azimuthal quantum number \( l \) can take values from \( 0 \) to \( n-1 \). Therefore, for \( n = 4 \), \( l \) can be \( 0, 1, 2, \) or \( 3 \).
- The value of \( m_l \) (magnetic quantum number) indicates the orientation of the orbital. Since \( m_l = 1 \), we need to find the corresponding \( l \) value.
- For \( m_l = 1 \), the possible value of \( l \) must be at least \( 1 \) (since \( m_l \) can take values from \( -l \) to \( +l \)). Thus, \( l \) can be \( 1 \) or higher.
3. **Determine the Possible Values of m_l**:
- For \( l = 1 \), the possible values of \( m_l \) are \( -1, 0, +1 \). Since \( m_l = 1 \) is one of these values, it is valid.
- For \( l = 2 \), the possible values of \( m_l \) are \( -2, -1, 0, +1, +2 \). Here, \( m_l = 1 \) is also valid.
- For \( l = 3 \), the possible values of \( m_l \) are \( -3, -2, -1, 0, +1, +2, +3 \). Again, \( m_l = 1 \) is valid.
4. **Count the Electrons for Each Valid l**:
- For \( l = 1 \) (p-orbital), there are 3 orbitals corresponding to \( m_l = -1, 0, +1 \). Each orbital can hold 2 electrons (one with \( m_s = +1/2 \) and one with \( m_s = -1/2 \)).
- Total electrons for \( l = 1 \) and \( m_s = -1/2 \): 1 orbital (m_l = 1) can hold 1 electron with \( m_s = -1/2 \).
- For \( l = 2 \) (d-orbital), there are 5 orbitals. The orbitals corresponding to \( m_l = -2, -1, 0, +1, +2 \) can hold 2 electrons each.
- Total electrons for \( l = 2 \) and \( m_s = -1/2 \): 1 orbital (m_l = 1) can hold 1 electron with \( m_s = -1/2 \).
- For \( l = 3 \) (f-orbital), there are 7 orbitals. The orbitals corresponding to \( m_l = -3, -2, -1, 0, +1, +2, +3 \) can hold 2 electrons each.
- Total electrons for \( l = 3 \) and \( m_s = -1/2 \): 1 orbital (m_l = 1) can hold 1 electron with \( m_s = -1/2 \).
5. **Total Count of Electrons**:
- For \( l = 1 \): 1 electron (m_l = 1, m_s = -1/2)
- For \( l = 2 \): 1 electron (m_l = 1, m_s = -1/2)
- For \( l = 3 \): 1 electron (m_l = 1, m_s = -1/2)
- Total = 1 + 1 + 1 = 3 electrons.
### Final Answer:
The total number of electrons having quantum numbers \( n = 4, m_l = 1, \) and \( m_s = -\frac{1}{2} \) is **3**.
