To determine which set of quantum numbers is not allowed, we need to understand the properties and restrictions of quantum numbers in atomic structure:
Principal Quantum Number (\(n\)): Can be any positive integer (\(n = 1, 2, 3, \ldots\)). It indicates the main energy level and orbital size.
Azimuthal Quantum Number (\(l\)): Can be an integer ranging from 0 to \(n-1\). It determines the shape of the orbital. For example, if \(n=3\), \(l\) can be 0, 1, or 2.
Magnetic Quantum Number (\(m_l\)): Can be integers between \(-l\) and \(+l\) (inclusive). It determines the orientation of the orbital in space.
Spin Quantum Number (\(s\)): Can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). It specifies the direction of the electron's spin.
Let's evaluate the given options:
Option: \(n=3, l=2, m_l=0, s=+\frac{1}{2}\)
For \(n=3\), \(l\) can be 0, 1, or 2; \(m_l\) can be -2, -1, 0, 1, or 2. This set is valid.
Option: \(n=3, l=2, m_l=-2, s=+\frac{1}{2}\)
For \(n=3\), \(l=2\) is valid, and \(m_l\) can be -2, -1, 0, 1, or 2. This set is valid.
Option: \(n=3, l=3, m_l=-3, s=-\frac{1}{2}\)
For \(n=3\), the possible values of \(l\) can only be 0, 1, or 2. The value of \(l=3\) is not allowed for \(n=3\). Hence, this set is invalid.
Option: \(n=3, l=0, m_l=0, s=-\frac{1}{2}\)
For \(n=3\), \(l\) can be 0, 1, or 2. If \(l=0\), \(m_l\) can only be 0. This set is valid.
The correct answer is: \(n=3, l=3, m_{l}=-3, s=-\frac{1}{2}\) as it is not allowed because \(l = 3\) is outside the permissible range for \(n = 3\).
