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Chapter 2: Problem 154

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

Expert verified
In the ground state of arsenic, there are 15 electrons with \(\ell=1\), 5 electrons with \(m_{\ell}=0\), and 5 electrons with \(m_{\ell}=+1\).

Step by step solution

01

Determining the electron configuration of arsenic.

Arsenic has an atomic number of 33, which means it has 33 electrons in its ground state. The electron configuration of arsenic is determined by filling up the orbitals in the order of increasing energy level. The electron configuration for arsenic can be represented as: \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^3\]
02

Finding the number of electrons with \(\ell=1\).

The quantum number \(\ell\) represents the angular momentum of the electron, and \(\ell=1\) corresponds to the p orbital. In the electron configuration of arsenic, the p orbitals are filled as follows: \[2p^6, 3p^6\text{, and } 4p^3\] There are 6 + 6 + 3 = 15 electrons in the p orbitals, which means 15 electrons have \(\ell=1\).
03

Finding the number of electrons with \(m_{\ell}=0\).

The quantum number \(m_{\ell}\) represents the magnetic quantum number, which can take values between \(-\ell\) and \(+\ell\). For the p orbitals (\(\ell=1\)), \(m_{\ell}\) can have values -1, 0, and +1. Each p orbital contains 3 sub-orbitals that correspond to \(m_{\ell}=-1\), \(m_{\ell}=0\), and \(m_{\ell}=+1\). Here, the electron configuration of the p orbitals is: \[2p^6, 3p^6 \text{, and } 4p^3\] All the 2p and 3p orbitals are completely filled, so they have two electrons each for \(m_{\ell}=0\). For the 4p orbital with \(3\) electrons, there will be one electron for \(m_{\ell}=0\). So, the total number of electrons with \(m_{\ell}=0\) is 2 + 2 + 1 = 5.
04

Finding the number of electrons with \(m_{\ell}=+1\).

Using the same reasoning as in Step 3, we know that the completely filled 2p and 3p orbitals also have two electrons each for \(m_{\ell}=+1\). For the 4p orbital with \(3\) electrons, there will also be one electron for \(m_{\ell}=+1\). So, the total number of electrons with \(m_{\ell}=+1\) is 2 + 2 + 1 = 5. In summary, for the ground state of arsenic, there are 15 electrons with \(\ell=1\), 5 electrons with \(m_{\ell}=0\), and 5 electrons with \(m_{\ell}=+1\).

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Most popular questions from this chapter

The elements \(\mathrm{Cu}, \mathrm{O}, \mathrm{La}, \mathrm{Y}, \mathrm{Ba}, \mathrm{Tl},\) and \(\mathrm{Bi}\) are all found in high-temperature ceramic superconductors. Write the expected electron configuration for these atoms.

Determine the maximum number of electrons that can have each of the following designations: \(2 f, 2 d_{x y}, 3 p, 5 d_{y z},\) and \(4 p\).

X rays have wavelengths on the order of \(1 \times 10^{-10} \mathrm{m}\). Calculate the energy of \(1.0 \times 10^{-10} \mathrm{m}\) X rays in units of kilojoules per mole of X rays. (1 mol X rays \(=6.022 \times 10^{23}\) X rays.) AM radio waves have wavelengths on the order of \(1 \times 10^{4} \mathrm{m}\). Calculate the energy of \(1.0 \times 10^{4} \mathrm{m}\) radio waves in units of kilojoules per mole of radio waves. Consider that the bond energy of a carbon- carbon single bond found in organic compounds is 347 kJ/mol. Would X rays and/or radio waves be able to disrupt organic compounds by breaking carbon- carbon single bonds?

Identify the following elements. a. An excited state of this element has the electron configuration \(1 s^{2} 2 s^{2} 2 p^{5} 3 s^{1}\). b. The ground-state electron configuration is \([\mathrm{Ne}] 3 s^{2} 3 p^{4}\). c. An excited state of this element has the electron configuration \([\mathrm{Kr}] 5 s^{2} 4 d^{6} 5 p^{2} 6 s^{1}\). d. The ground-state electron configuration contains three unpaired \(6 p\) electrons.

Give the maximum number of electrons in an atom that can have these quantum numbers: a. \(n=4\) b. \(n=5, m_{\ell}=+1\) c. \(n=5, m_{s}=+\frac{1}{2}\) d. \(n=3, \ell=2\) e. \(n=2, \ell=1\)
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