Answer
The sets of quantum numbers that are not allowed are:
a. \( n=3, \ell=3, m_{\ell}=0, m_{s}=-\frac{1}{2} \) (due to \(\ell\) restriction)
e. \( n=5, \ell=-4, m_{\ell}=2, m_{s}=+\frac{1}{2} \) (due to \(\ell\) restriction)
f. \( n=3, \ell=1, m_{\ell}=2, m_{s}=-\frac{1}{2} \) (due to \(m_{\ell}\) restriction)
Work Step by Step
let's analyze each set of quantum numbers:
a. \( n=3, \ell=3, m_{\ell}=0, m_{s}=-\frac{1}{2} \)
This set is not allowed because the azimuthal quantum number (\(\ell\)) cannot be greater than or equal to the principal quantum number (\(n\)). In this case, \(\ell=3\) is not allowed for \(n=3\).
b. \( n=4, \ell=3, m_{\ell}=2, m_{s}=-\frac{1}{2} \)
This set is allowed because all the quantum numbers are within the allowed ranges. There are no restrictions for this set.
c. \( n=4, \ell=1, m_{\ell}=1, m_{s}=+\frac{1}{2} \)
This set is allowed because all the quantum numbers are within the allowed ranges. There are no restrictions for this set.
d. \( n=2, \ell=1, m_{\ell}=-1, m_{s}=-1 \)
This set is allowed because all the quantum numbers are within the allowed ranges. There are no restrictions for this set.
e. \( n=5, \ell=-4, m_{\ell}=2, m_{s}=+\frac{1}{2} \)
This set is not allowed because the azimuthal quantum number (\(\ell\)) cannot be negative. In this case, \(\ell=-4\) is not allowed.
f. \( n=3, \ell=1, m_{\ell}=2, m_{s}=-\frac{1}{2} \)
This set is not allowed because the magnetic quantum number (\(m_{\ell}\)) cannot be greater than the azimuthal quantum number (\(\ell\)). In this case, \(m_{\ell}=2\) is not allowed for \(\ell=1\).
