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, 9 electrons have \(\ell=1\) as one of their quantum numbers, 4 electrons have \(m_\ell=0\), and 3 electrons have \(m_\ell=+1\).

Step-by-step solution

1. Determine the Electronic Configuration of Arsenic

Arsenic (As) has an atomic number of 33. Thus, its electron configuration in the ground state is: \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^3\]

2. Count the Number of Electrons with \(\ell=1\)

Now, we want to find how many electrons have \(\ell=1\) as their quantum number. \(\ell=1\) represents the p-subshell of an atom. In the electron configuration of arsenic, we can see two p subshells: 2p and 4p. In the 2p subshell, there are 6 electrons, and in the 4p subshell, there are 3 electrons. Therefore, a total of 6 + 3 = 9 electrons have \(\ell=1\) as one of their quantum numbers.

3. Count the Number of Electrons with \(m_\ell=0\)

To find the number of electrons with \(m_\ell=0\), we'll need to identify the orbitals with \(m_\ell=0\) in the electron configuration. Each p-orbital is represented by three magnetic quantum numbers: \(m_\ell=-1,0,+1\). Since there are two p subshells (2p and 4p), there are two orbitals with \(m_\ell=0\) (one in each p-subshell). Each orbital can hold a maximum of 2 electrons, so in total, 2 * 2 = 4 electrons have \(m_\ell=0\).

4. Count the Number of Electrons with \(m_\ell=+1\)

Similarly, to find the number of electrons with \(m_\ell=+1\), we need to identify the orbitals with \(m_\ell=+1\) in the electron configuration. In the 2p subshell, the orbital with \(m_\ell=+1\) is filled with 2 electrons. In the 4p subshell, there is one more electron to be placed in the orbital with \(m_\ell=+1\), making it a total of 2 + 1 = 3 electrons with \(m_\ell=+1\). So, in the ground state of arsenic: - 9 electrons have \(\ell=1\) as one of their quantum numbers. - 4 electrons have \(m_\ell=0\). - 3 electrons have \(m_\ell=+1\).

Key concepts

Electronic Configuration

The concept of electronic configuration is fundamental to understanding how atoms exhibit their behavior and properties. It refers to the arrangement of electrons in the orbitals around the nucleus of an atom. These configurations follow the principles of quantum mechanics and are often denoted using a notation that encapsulates the energy levels, subshells, and number of electrons in each of these subshells.

For example, arsenic (As), with an atomic number of 33, you can express its electronic configuration as \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^3\]. From this notation, one can see that electrons fill in various energy levels, starting from the closest to the nucleus (1s) to higher ones (like 4p). To determine the behavior and chemical properties of an element, first understanding its electronic configuration is crucial.

Understanding the filling order of electrons, which follows the Aufbau principle, the Pauli exclusion principle, and Hund's rule, is essential for accurately determining the electronic configuration. This helps in predicting the element's reactivity, the types of bonds it can form, and its placement in the periodic table.

P-Subshell

When discussing the p-subshell within the electronic configuration, we're looking at one of the types of subshells that exist within an atom's electron orbitals. The p-subshell is designated by the quantum number \(\ell=1\) and can hold a maximum of six electrons. It is composed of three orbitals, each of which can house two electrons with opposite spins.

These p orbitals are oriented perpendicularly to one another in three-dimensional space and are labeled as px, py, and pz, though this designation has no physical significance in an isolated atom and serves more as a spatial orientation reference. When you see a p-subshell in an element's electronic configuration, such as in arsenic's 2p and 4p subshells, this directly relates to the element's ability to form chemical bonds, as it's the outermost electrons that interact during chemical reactions.

Magnetic Quantum Number

Exploring the concept of the magnetic quantum number, denoted as \(m_{\ell}\), we delve into the orientation of the orbital in space. It can take on integer values from \(-\ell\) to \(\ell\), which for a p-subshell (where \(\ell=1\)) translates to \(m_{\ell}\) values of -1, 0, and +1. Each of these values corresponds to a different p orbital: px, py, and pz respectively.

When we consider the magnetic quantum number in an elemental electronic configuration, it tells us how the orbitals are being filled with electrons. If we're analyzing how many electrons have \(m_{\ell}=0\), we're looking at electrons in the orbital that's aligned on the z-axis, which is often visualized as being oriented along the direction of an applied magnetic field. This specific information can be essential when considering an atom's behavior in magnetic fields or its ability to engage in bonding where the orientation of electron clouds plays a role.

The magnetic quantum number helps explain the behavior of electrons in a magnetic field and provides insight into the electron distribution within a subshell, which, in turn, can have implications for the atom's chemical and physical properties.