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 may be "none." In such cases, explain why "none" is the correct answer. (a) \(n=3\) (b) \(n=3\) and \(\ell=2\) (c) \(n=4, \ell=1, m_{\ell}=-1,\) and \(m_{s}=+1 / 2\) (d) \(n=5, \ell=0, m_{\ell}=-1, m_{\mathrm{s}}=+1 / 2\)

Short answer

(a) 18 electrons; (b) 10 electrons; (c) 1 electron; (d) none.

Step-by-step solution

Analyze Quantum Number n=3 (a)

The principal quantum number \( n \) denotes the energy level of an electron in an atom. For \( n=3 \), the maximum number of electrons that can occupy this energy level is given by the formula \( 2n^2 \). Therefore, for \( n=3 \), the maximum number of electrons is \( 2 \times 3^2 = 18 \).

Consider n=3, l=2 (b)

The quantum number \( \ell \) describes the subshell. For \( n=3 \) and \( \ell=2 \), this corresponds to the 3d subshell. The number of electrons in a subshell is given by \( 2(2\ell + 1) \). Therefore, for \( \ell=2 \), the number of electrons is \( 2 \times 5 = 10 \).

Examine n=4, l=1, m_l=-1, m_s=+1/2 (c)

The magnetic quantum number \( m_\ell \) can take values from \( -\ell \) to \( +\ell \). Here, \( \ell=1 \), so \( m_\ell=-1 \) is valid. The spin quantum number \( m_s \) can be either \( +1/2 \) or \( -1/2 \). Since each combination of \( n \), \( \ell \), \( m_\ell \), and \( m_s \) describes a single electron, there can only be one electron with \( n=4 \), \( \ell=1 \), \( m_\ell=-1 \), and \( m_s=+1/2 \).

Evaluate n=5, l=0, m_l=-1, m_s=+1/2 (d)

The range of values for \( m_\ell \) is from \( -\ell \) to \( +\ell \). Here, \( \ell=0 \), so the only allowed value for \( m_\ell \) is 0. Since \( m_\ell \) is given as -1, which is outside the range for \( \ell=0 \), no electron can exist with this combination of quantum numbers. Hence, the answer is "none."

Key concepts

Electron Configuration

Electron configuration is a method used to distribute electrons among the orbitals of an atom. Atoms have a specific configuration based on their energy levels and sublevels. The distribution is done in a way that minimizes the energy of the atom, filling the lowest energy orbitals first. This method follows the aufbau principle, which states that electrons populate lower energy orbitals before moving to higher energy ones.

The electron configuration is often represented using numbers and letters to denote the energy levels ( ), sublevels ( or p), and the number of electrons in those sublevels. Hence, any given element will possess a unique configuration representing its electron structure, giving insight into the atom's chemical behavior and properties.

Principal Quantum Number

The principal quantum number, denoted by the symbol ( ), is integral in describing the energy level and relative size of an atomic orbital within an atom. represents the shell occupied by the electron and can have positive integer values (e.g., 1, 2, 3, and so on). As ( ) increases, the electrons are found at higher energy levels and further from the nucleus, resulting in an increase in orbital size.

• The principal quantum number defines the overall size and energy of an electron shell.
• For each value of , different sublevels exist, governed by the angular momentum quantum number ( ).

The energy level ( =1) is the closest to the nucleus and the lowest in energy. With each increase in energy level ( ), more sublevel orbitals become accessible.

Magnetic Quantum Number

The magnetic quantum number, represented by ( m_ℓ), is part of the quantum number set used to describe an electron's position within an atom. m_ℓ defines the specific orbital within a subshell where an electron is likely to be found. The range of m_ℓ values is between -ℓ to +ℓ, where ℓ is the azimuthal or angular momentum quantum number.

• For a given subshell with an azimuthal quantum number ( ℓ), m_ℓ can take up to 2ℓ + 1 values, indicating the three-dimensional orientation of the orbitals.
• The magnetic quantum number adds depth to the electron configuration process, giving more detailed insights into how electrons are distributed in multi-electron atoms.

Understanding m_ℓ is crucial for visualizing the spatial arrangement of electrons and contributes to predicting magnetic properties and spectroscopic behavior.

Spin Quantum Number

The spin quantum number, denoted ( m_s), describes the intrinsic spin of an electron, a fundamental property akin to angular momentum. Unlike the other quantum numbers, m_s can only have two possible values: +1/2 and -1/2. These values indicate the two different orientations the electron's spin can adopt in a magnetic field.

• The unique aspect of the spin quantum number is that it determines an electron's magnetic moment.
• The two possible spin states ( +1/2 and -1/2) allow for the Pauli exclusion principle, which states that two electrons cannot occupy the same quantum state simultaneously within an atom.

The concept of electron spin is essential for explaining magnetic properties, chemical bonding, and how electrons arrange themselves within atoms and molecules.