Answer
The sets of quantum numbers that are not allowed in the hydrogen atom are:
b. \(n=4, \ell=3, m_{\ell}=4\)
c. \(n=0, \ell=0, m_{\ell}=0\)
d. \(n=2, \ell=-1, m_{\ell}=1\)
Work Step by Step
In the hydrogen atom, the quantum numbers describe the energy levels, orbital shapes, and orientations of the electrons. The allowed values for the quantum numbers are as follows:
1. The principal quantum number (\(n\)) represents the energy level of the electron and can take any positive integer value starting from 1.
2. The azimuthal quantum number (\(\ell\)) represents the shape of the orbital and can take values from 0 to \(n-1\).
3. The magnetic quantum number (\(m_{\ell}\)) represents the orientation of the orbital and can take integer values from \(-\ell\) to \(\ell\).
Let's analyze each set of quantum numbers:
a. \(n=3, \ell=2, m_{\ell}=2\)
This set of quantum numbers is allowed. The principal quantum number (\(n\)) is 3, which is a positive integer. The azimuthal quantum number (\(\ell\)) is 2, which is less than \(n\) (3-1=2). The magnetic quantum number (\(m_{\ell}\)) is 2, which is between \(-\ell\) and \(\ell\) (-2 to 2). Therefore, this set is allowed.
b. \(n=4, \ell=3, m_{\ell}=4\)
This set of quantum numbers is not allowed. The azimuthal quantum number (\(\ell\)) cannot be greater than or equal to \(n\). In this case, \(\ell\) is 3, which is not less than \(n\) (4-1=3). Therefore, this set is incorrect.
c. \(n=0, \ell=0, m_{\ell}=0\)
This set of quantum numbers is not allowed. The principal quantum number (\(n\)) cannot be zero. It must be a positive integer. Therefore, this set is incorrect.
d. \(n=2, \ell=-1, m_{\ell}=1\)
This set of quantum numbers is not allowed. The azimuthal quantum number (\(\ell\)) cannot be negative. It must be a non-negative integer. Therefore, this set is incorrect.
