To determine which set of quantum numbers is not allowed, we must understand the rules governing quantum numbers:
Principal Quantum Number (\(n\)): Specifies the energy level and size of the orbital. It can be any positive integer: \(n = 1, 2, 3, \ldots\)
Azimuthal Quantum Number (\(l\)): Specifies the shape of the orbital. It ranges from 0 to \(n-1\).
Magnetic Quantum Number (\(m_l\)): Specifies the orientation of the orbital. It ranges from \(-l\) to \(+l\), including zero.
Spin Quantum Number (\(m_s\)): Specifies the spin of the electron. It can be \(\frac{1}{2}\) or \(-\frac{1}{2}\).
Now, let's evaluate each option:
\(n=1, l=0, m_l=0, m_s=-\frac{1}{2}\)
For \(n=1\), \(l\) can be 0 (since \(l=0\) to \(n-1\)).
For \(l=0\), \(m_l\) must be 0 (as valid \(m_l\) values range from \(-l\) to \(+l\)).
Spin \((m_s)\) is either \(-\frac{1}{2}\) or \(\frac{1}{2}\), which is correct here.
Therefore, this set is allowed.
\(n=3, l=2, m_l=-1, m_s= \frac{1}{2}\)
For \(n=3\), \(l\) can be 0, 1, or 2.
For \(l=2\), \(m_l\) can be -2, -1, 0, 1, or 2.
The spin \((m_s)\) is valid at \(\frac{1}{2}\).
Thus, this set is allowed.
\(n=5, l=2, m_l=-3, m_s=-\frac{1}{2}\)
For \(n=5\), \(l\) can be 0 to 4, so \(l=2\) is valid.
However, for \(l=2\), \(m_l\) should be -2, -1, 0, 1, or 2. Here, \(m_l=-3\) is not possible.
This means the given set of quantum numbers is not allowed.
\(n=4, l=3, m_l=-2, m_s=\frac{1}{2}\)
For \(n=4\), \(l\) can be 0, 1, 2, or 3.
For \(l=3\), \(m_l\) can include -3, -2, -1, 0, 1, 2, or 3.
The spin \((m_s)\) is valid at \(\frac{1}{2}\).
Therefore, this set is allowed.
Based on the analysis above, the incorrect set of quantum numbers is \(n=5, l=2, m_l=-3, m_s=-\frac{1}{2}\) because the value of \(m_l\) does not fall within the range defined by \(l=2\).
