Question

Consider the ground state of arsenic, As. How many electrons have \(\ell=1\) as one of their quantum numbers? How many electrons have \(m_{\ell}=0 ?\) How many electrons have \(m_{\ell}=+1 ?\)

Short answer

In the ground state of arsenic, 15 electrons have \(\ell=1\) as one of their quantum numbers, 6 electrons have \(m_{\ell}=0\), and 6 electrons have \(m_{\ell}=+1\).

Step-by-step solution

Find the electron configuration of arsenic

To find the electron configuration of arsenic, we need to know its atomic number. Arsenic has an atomic number of 33, which means it has 33 electrons in its ground state. We can now fill the orbitals in ascending order of energy, following the Aufbau principle. The electron configuration of arsenic is: \[1s^22s^22p^63s^23p^64s^23d^{10}4p^3\]

Count electrons with \(\ell=1\)

Now, we have to find the number of electrons that have \(\ell =1\). The value \(\ell=1\) corresponds to the p-orbitals. In the electron configuration of arsenic, we have: 2p: 6 electrons 3p: 6 electrons 4p: 3 electrons Total number of electrons with \(\ell=1\) is: \(6+6+3 = 15\)

Count electrons with \(m_{\ell}=0\)

Next, we need to find the number of electrons with \(m_{\ell}=0\). The value \(m_\ell=0\) corresponds to those orbitals which are aligned along the z-axis. To find them, we'll look at the electron configuration. In p-orbitals with \(\ell=1\), \(m_\ell\) can be -1, 0, or +1. Since each p-orbital can hold 6 electrons, there will be 2 electrons in each \(m_\ell\) state. So, there are 2 electrons with \(m_\ell=0\) in each of the following orbitals: 2p, 3p, and 4p. Total number of electrons with \(m_{\ell}=0\): \(2+2+2 = 6\)

Count electrons with \(m_{\ell}=+1\)

Lastly, we need to find the number of electrons with \(m_{\ell}=+1\). As mentioned earlier, in p-orbitals with \(\ell=1\), \(m_\ell\) can be -1, 0, or +1. Similar to the previous step, there will be 2 electrons with \(m_\ell=+1\) in each of the following orbitals: 2p, 3p, and 4p. Total number of electrons with \(m_{\ell}=+1\): \(2+2+2 = 6\) So, in the ground state of arsenic: - 15 electrons have \(\ell=1\) as one of their quantum numbers. - 6 electrons have \(m_{\ell}=0\). - 6 electrons have \(m_{\ell}=+1\).