Question

Which of the following sets of quantum numbers is not allowed?

• A. \(n=1, l=0, m_l=0, m_s=−  \frac{1}{2}\)
• B. \(n=3,l=2,m_l=−1,m_s= \frac{1}{2}\)
• C. \(n=5,l=2,m_l=−3,m_s=−\frac{1}{2}\)
• D. \(n=4,l=3,m_l=−2,m_s=\frac{1}{2}\)

Answer 1

The Correct Option is C

 To determine which set of quantum numbers is not allowed, we must understand the rules governing quantum numbers:

1. Principal Quantum Number (\(n\)): Specifies the energy level and size of the orbital. It can be any positive integer: \(n = 1, 2, 3, \ldots\)
2. Azimuthal Quantum Number (\(l\)): Specifies the shape of the orbital. It ranges from 0 to \(n-1\).
3. Magnetic Quantum Number (\(m_l\)): Specifies the orientation of the orbital. It ranges from \(-l\) to \(+l\), including zero.
4. 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 \(l=2\).