Quiz  quantum numbers

If \(n=5\), what are the possible values for \(\ell\)?
\(\ell\) permitted values are \(0,1,2,3\ldots\left(n1\right)\). If \(n=5\), \(\ell\) can be \(0,1,2,3,4\).

For a certain electron \(\ell=0\). What is the shape of the subshell electron cloud of that electron?
The orbital quantum number \(\ell\) indicates the shape of the subshell electron cloud
(orbital shape).\(\ell\) Orbital shape \(0\) \(s\) orbital \(1\) \(p\) orbital \(2\) \(d\) orbital \(3\) \(f\) orbital \(\ell=0\) indicates \(s\) orbital, which has a spherical shape.

What are the \(n\) and \(\ell\) values for \(4d\) electrons?
\(n=4\) and \(\ell=2\).

For an electron, principle quantum number, \(n\) is \(3\). What are the possible values for \(\ell\) and \(m_{\ell}\)?
For \(\ell\) permitted values are \(0,1,2,3\ldots\left(n1\right)\). If \(n=3\), \(\ell\) can be \(0,1,2\).
For \(m_{\ell}\) permitted values are \(0,\pm1,\pm2\ldots\pm\ell.\) The following table shows the permitted values of \(m_{\ell}\) for \(\ell=0,1,2\)\(\ell\) \(m_{\ell}\) \(0\) \(0\) \(1\) \(0,+1,1\) \(2\) \(0,+1,1,+2,2\)
 What are the names of the orbitals with the priciple quantum number \(n=3\)?
For \(\ell\) permitted values are \(0,1,2,3\ldots\left(n1\right)\). If \(n=3\), \(\ell\) can be \(0,1,2\).
For \(m_{\ell}\) permitted values are \(0,\pm1,\pm2\ldots\pm\ell.\)\(\ell\) \(m_{\ell}\) Name of the orbitals \(0\) \(0\) \(3s\) \(1\) \(0,+1,1\) \(3p\) \(2\) \(0,+1,1,+2,2\) \(3d\)
 What are the possible values for \(\ell\), \(m_{\ell}\) and \(m_{s}\) for an electron in \(n=1\) state?
For \(\ell\) permitted values are \(0,1,2,3\ldots\left(n1\right)\). If \(n=1\), \(\ell\) can be \(0\).
For \(m_{\ell}\) permitted values are \(0,\pm1,\pm2\ldots\pm\ell.\) The following table shows the permitted values of \(m_{\ell}\) for \(\ell=0\)\(\ell\) \(m_{\ell}\) \(0\) \(0\) (\(1s\) orbital) \(m_{s}=+\frac{1}{2}\), \(\frac{1}{2}\)
 How many valid combinations of quantum numbers exist for \(2p\) electrons?
Principle quantum number \(\left(n\right)\) and orbital quantum number \(\left(\ell\right)\) are given in the question.
For \(2p\) electrons: \(n=2\) and \(\ell=1\)
What are the possible values for \(m_{\ell}\) and \(m_{s}\)?Quantum numbers \(n\) \(\ell\) \(m_{\ell}\) \(m_{s}\) Possible values \(2\) \(1\) \(0,+1,1\) \(\frac{1}{2},+\frac{1}{2}\) How many possible values \(1\) \(1\) \(3\) \(2\) Total valid combinations of quantum numbers \(=\text{3}\times2=6\)
\(n\) \(\ell\) \(m_{\ell}\) \(m_{s}\) \(2\) \(1\) \(0\) \(\frac{1}{2}\) \(2\) \(1\) \(0\) \(+\frac{1}{2}\) \(2\) \(1\) \(+1\) \(\frac{1}{2}\) \(2\) \(1\) \(+1\) \(+\frac{1}{2}\) \(2\) \(1\) \(1\) \(\frac{1}{2}\) \(2\) \(1\) \(1\) \(+\frac{1}{2}\)
 Can an electron exist with \(n=4\), \(\ell=2\) and \(m_{\ell}=4\) quantum numbers? Explain using possible quantum number combinations.
For \(n=4\), \(\ell\) can be \(0,1,2,3\). Therefore, \(\ell=2\) is valid.
For \(\ell=2\), \(m_{\ell}\) can be \(0,+1,1,+2,2\). The valid combinations are given below.\(n\) \(\ell\) \(m_{\ell}\) \(4\) \(2\) \(0\) \(4\) \(2\) \(+1\) \(4\) \(2\) \(1\) \(4\) \(2\) \(+2\) \(4\) \(2\) \(2\) Therefore, \(n=4\), \(\ell=2\) and \(m_{\ell}=4\) is not a valid combination.

Can two electrons have the following combination of quantum numbers \(n=2\), \(\ell=0\) and \(m_{\ell}=0\) ?
Yes. Because one electron can have the upward spin \(m_{s}\) \(=\) \(+\frac{1}{2}\) and the other can have the downward spin \(m_{s}=\frac{1}{2}\).

Which type of orbital has the following combination of quantum numbers? \(n=4,\ell=2,m_{\ell}=1\).
\(n=4\) means fourth shell
\(\ell=2=d\) orbital
For \(n=4\) and \(\ell=2\), \(m_{\ell}\) can be \(0,+1,1,+2,2\). Therefore, \(n=4,\ell=2,m_{\ell}=1\) indicates one of the \(4d\) orbitals.