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, there are 15 electrons with \(\ell=1\), 5 electrons with \(m_{\ell}=0\), and 5 electrons with \(m_{\ell}=+1\).

Step-by-step solution

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: \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^3\]

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 \(\ell=1\).

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.

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 \(m_{\ell}=+1\).

Key concepts

Electron Configuration

The electron configuration of an atom is a way to describe the arrangement of its electrons in the various atomic orbitals. It provides important information about the electron structure, which is crucial to understanding the atom’s reactivity and properties.
Arsenic, for example, has an atomic number of 33, which means it contains 33 electrons in its ground state. Its electron configuration is expressed as: \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^3\]This can be read as follows:

• The "1s" orbital has two electrons.
• "2s" and "2p" orbitals have 2 and 6 electrons, respectively.
• "3s" and "3p" orbitals also follow with 2 and 6 electrons each.
• The "4s" and "3d" contain 2 and 10 electrons, respectively.
• Finally, the "4p" orbital accommodates 3 electrons.

The sequence shows the filling order of orbitals from lower to higher energy using the Aufbau principle.

Ground State

When electrons occupy the lowest possible energy levels available, the atom is in its ground state. This is the most stable and naturally occurring configuration for an atom, as electrons tend to settle in positions where their energy is minimized.
For arsenic, the ground state implies having all of its 33 electrons distributed across the various orbitals as described in its electron configuration.
In this state, electrons fill each orbital starting from the lowest energy level, which is the "1s" orbital, and continue to fill higher energy orbitals. The concept of the ground state is essential for analyzing the basic properties of an element, such as its chemical behavior, reactivity, and ability to form bonds.

p Orbitals

An essential component of electron configurations, the p orbitals have an angular momentum quantum number enoted as \(\ell=1\). They have a distinct dumbbell shape and can hold a maximum of six electrons.For arsenic, electrons fill p orbitals in several energy levels: \(2p^6\), \(3p^6\), and \(4p^3\).Each level contains:

• Three sub-orbitals corresponding to the magnetic quantum numbers: \(m_{\ell}=-1, 0, +1\).
• These sub-orbitals can each hold up to two electrons, following Hund's rule and the Pauli exclusion principle.

Their capacity to hold six electrons total comes from three possible orientations on a 3D plane, defining the x, y, and z axes. Due to their shape and orientation, p orbitals allow for significant chemical bonding and play a crucial role in determining the geometry of molecules.

Magnetic Quantum Number

The magnetic quantum number, represented by \(m_{\ell}\), is a quantum number that specifies the orientation of an orbital in space relative to the other orbitals. It defines which particular sub-orbital within a main orbital that an electron occupies.
For p orbitals, where \(\ell=1\), \(m_{\ell}\) can be -1, 0, or +1, which are the three possible orientations of a p orbital along the space axes.
In arsenic’s ground state, the distribution of electrons among these orientations is crucial for understanding their magnetic properties. In completely filled \(2p\) and \(3p\) orbitals, \(m_{\ell}\) values are equally occupied meaning:

• Two electrons are present in each of the \(m_{\ell}=-1\), \(m_{\ell}=0\), and \(m_{\ell}=+1\) sub-orbitals.

In the 4p orbitals, however, the third, electron only provides partial filling ("4p^3"), there is one electron for each \(m_{\ell}\) value. Thus, these orbitals play a key role in determining magnetic and chemical character of elements.