Question

What is the maximum number of electrons that can be identified with each of the following sets of quantum numbers? In some cases, the answer is "none." Explain why this is true. (a) \(n=4, \ell=3, m_{\ell}=1\) (b) \(n=6, \ell=1, m_{\ell}=-1, m_{\mathrm{s}}=-1 / 2\) (c) \(n=3, \ell=3, m_{\ell}=-3\)

Short answer

(a) 2 electrons, (b) 1 electron, (c) None (invalid \(\ell\)).

Step-by-step solution

Understand Quantum Numbers

Quantum numbers are used to describe the state of an electron in an atom. There are four quantum numbers: \(n\) (principal quantum number), \(\ell\) (azimuthal quantum number), \(m_\ell\) (magnetic quantum number), and \(m_s\) (spin quantum number). Each of these numbers represents a different aspect of the electron's characteristics.

Evaluate set (a)

For \(n=4\), \(\ell=3\), \(m_\ell=1\), we check the validity. Here, \(\ell\) can range from 0 to \(n-1\), so \(\ell=3\) is valid. The \(m_\ell\) can range from \(-\ell\) to \(\ell\), so \(m_\ell=1\) is also valid. Electrons can have values of \(m_s\) of either \(+1/2\) or \(-1/2\). Thus, this set of quantum numbers can describe 2 electrons. Answer: 2 electrons.

Evaluate set (b)

For \(n=6\), \(\ell=1\), \(m_\ell=-1\), \(m_s=-1/2\), we check for validity. \(\ell\) can range from 0 to \(n-1\) and \(\ell=1\) is valid. \(m_\ell=-1\) is valid since it lies between \(-1\) and \(1\). The specified spin \(m_s=-1/2\) is one of the two allowed values for electron spin. Therefore, these quantum numbers describe exactly 1 electron. Answer: 1 electron.

Evaluate set (c)

For \(n=3\), \(\ell=3\), \(m_\ell=-3\), we notice \(\ell\) should be between 0 and \(n-1=2\). Since \(\ell=3\) is greater than the maximum value for \(\ell\), this set of quantum numbers is not possible. Answer: None, because \(\ell=3\) is invalid for \(n=3\).

Key concepts

Principal Quantum Number

The principal quantum number, denoted by \(n\), is key to understanding the energy levels of electrons in an atom. This number tells us the main energy shell to which an electron belongs. The value of \(n\) is always a positive integer: 1, 2, 3, and so on. As \(n\) increases, the electron's average distance from the nucleus, as well as its energy, increases.

• \(n = 1\) is the closest to the nucleus and has the lowest energy.
• Higher \(n\) values (like \(n = 4, 5, 6\)) suggest the electron is further out from the nucleus.

In step (a), with \(n=4\), the electron is located in the fourth energy level, a valid configuration which can potentially hold more electrons per energy level due to its higher energy state.

Azimuthal Quantum Number

The azimuthal quantum number, \(\ell\), provides insight into the shape of the electron's orbital, or its angular momentum. The value of \(\ell\) ranges from 0 to \(n-1\), representing different sublevels within a main energy level.

• \(\ell = 0\) is an s orbital, spherical in shape.
• \(\ell = 1\) is a p orbital, with a dumbbell shape.
• \(\ell = 2\) is a d orbital, more complex in shape.
• \(\ell = 3\) is an f orbital, even more complex.

For value \(\ell = 3\), as in step (a), it's valid for \(n = 4\), consistent with f orbitals. However, in step (c) with \(n = 3\), \(\ell = 3\) is invalid since \(\ell\) cannot surpass \(n-1 = 2\). Hence, no such orbital exists.

Magnetic Quantum Number

The magnetic quantum number, \(m_\ell\), denotes the orientation of the orbital in a magnetic field. The range of \(m_\ell\) is given by \(-\ell \leq m_\ell \leq \ell\). This quantum number defines the specific orbital within a sublevel, where each orbital can hold up to two electrons.

• For \(\ell = 1\), possible \(m_\ell\) values are -1, 0, and +1.
• For \(\ell = 3\), potential \(m_\ell\) values are -3, -2, -1, 0, +1, +2, and +3.

In case (a) \(m_\ell = 1\) is permissible for \(\ell = 3\). However, in step (c), despite \(m_\ell\) being valid, it is irrelevant since \(\ell = 3\) itself isn't valid for \(n = 3\).

Electron Configuration

Electron configuration refers to the arrangement of electrons in an atom's orbitals. It is determined based on rules related to quantum numbers, ensuring electrons occupy orbitals starting from lower to higher energy levels and filling each orbital in a balanced manner.
Important principles include:

• Aufbau principle: Electrons fill lower-energy orbitals first.
• Pauli exclusion principle: No two electrons can have the same set of four quantum numbers.
• Hund's rule: Electrons will fill degenerate orbitals singly before pairing up.

As shown in step (b) with \(n=6\), \(\ell=1\), \(m_{\ell}=-1\), and \(m_s=-1/2\), the configuration accurately reflects the placement of one electron in the \(p\) sublevel of the sixth energy level, adhering to all configuration principles.