To determine which set of quantum numbers is not allowed, we must understand the rules that govern the quantum numbers. Here's a brief overview:
Principal Quantum Number (\(n\)): Indicates the energy level and must be a positive integer (\(n = 1, 2, 3, \ldots\)).
Azimuthal Quantum Number (\(l\)): Specifies the subshell and can take any integer value from 0 to \(n-1\) (\(l = 0, 1, 2, \ldots, n-1\)).
Magnetic Quantum Number (\(m_l\)): Indicates the orientation of an orbital and can take values from \(-l\) to \(+l\) (\(m_l = -l, -(l-1), \ldots, 0, \ldots, (l-1), +l\)).
Spin Quantum Number (\(s\)): Specifies the spin of an electron and can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
Let's evaluate each option using these rules:
\(n=3, l=2, m_l=0, s=+\frac{1}{2}\)
Here, \(l = 2\) is within the allowed range (\(0\) to \(n-1 = 2\)) and \(m_l = 0\), which is within the valid range for \(l = 2\) (\(-2\) to \(+2\)). The spin \(s = +\frac{1}{2}\) is valid. This set is allowed.
\(n=3, l=2, m_l=-2, s=+\frac{1}{2}\)
Here, \(l = 2\) is within range, and \(m_l = -2\), which is also valid for \(l = 2\) range (\(-2\) to \(+2\)). The spin \(s = +\frac{1}{2}\) is valid. This set is allowed.
\(n=3, l=3, m_l=-3, s=-\frac{1}{2}\)
Here, \(l = 3\), which is not allowed because \(l\) must be less than \(n\) (i.e., \(l\) can at most be \(2\) in this case). Therefore, this set is not allowed.
\(n=3, l=0, m_l=0, s=-\frac{1}{2}\)
Here, \(l = 0\) is valid (\(0\) to \(n-1 = 2\)), and \(m_l = 0\) is permitted for \(l = 0\). The spin \(s = -\frac{1}{2}\) is valid. This set is allowed.
The set of quantum numbers that is not allowed is:
