To determine the correct set of four quantum numbers for a 4d electron, we need to understand the definitions and possible values for each quantum number:
1. **Principal Quantum Number (n)**: This number indicates the main energy level of the electron. For a 4d electron, \( n = 4 \).
2. **Azimuthal Quantum Number (l)**: This number defines the shape of the orbital. The value of \( l \) can range from \( 0 \) to \( n-1 \). For d orbitals, \( l = 2 \). Therefore, for a 4d electron, \( l = 2 \).
3. **Magnetic Quantum Number (m_l)**: This number indicates the orientation of the orbital in space. The values of \( m_l \) range from \( -l \) to \( +l \). For \( l = 2 \) (which corresponds to d orbitals), the possible values of \( m_l \) are \( -2, -1, 0, +1, +2 \).
4. **Spin Quantum Number (s)**: This number describes the spin of the electron. The possible values are \( +\frac{1}{2} \) or \( -\frac{1}{2} \).
Now, let’s analyze the options given in the question:
1. **Option 1**: If the spin quantum number is \( +\frac{1}{4} \), this is incorrect because the spin quantum number can only be \( +\frac{1}{2} \) or \( -\frac{1}{2} \).
2. **Option 2**: If the spin quantum number is \( 0 \), this is also incorrect for the same reason.
3. **Option 3**: Here, \( n = 4 \), \( l = 3 \), \( m_l = -2 \), and \( s = +\frac{1}{2} \). The value of \( l \) is incorrect because for a 4d electron, \( l \) should be \( 2 \) (not \( 3 \)).
4. **Option 4**: Here, \( n = 4 \), \( l = 2 \), \( m_l = 1 \), and \( s = -\frac{1}{2} \). This option is correct because:
- \( n = 4 \) (correct for 4d electron)
- \( l = 2 \) (correct for d orbitals)
- \( m_l = 1 \) (valid value for \( m_l \) when \( l = 2 \))
- \( s = -\frac{1}{2} \) (valid spin quantum number)
Thus, the correct set of quantum numbers for a 4d electron is represented by **Option 4**: \( (n=4, l=2, m_l=1, s=-\frac{1}{2}) \).
