Question

How many electrons in an atom can have each of the following quantum number designations? (a) \(\mathbf{n}=2, \ell=1, \mathbf{m}_{\ell}=0\) (b) \(\mathbf{n}=2, \ell=1, \mathbf{m}_{\ell}=-1\) (c) \(\mathbf{n}=3, \ell=1, \mathbf{m}_{\ell}=0, \mathbf{m}_{s}=+\frac{1}{2}\)

Short answer

(a) \(n=2, \ell=1, m_ℓ=0\): 2 electrons (b) \(n=2, \ell=1, m_ℓ=-1\): 2 electrons (c) \(n=3, \ell=1, m_ℓ=0, m_s=+\frac{1}{2}\): 1 electron

Step-by-step solution

Case (a): \(n=2, \ell=1, m_ℓ=0\)

For this case, we are only given the values of \(n\), \(\ell\), and \(m_ℓ\). The spin quantum number \(m_s\) is not provided. Since \(m_s\) can have two values: \(+\frac{1}{2}\) and \(-\frac{1}{2}\), the maximum number of electrons with the given set of quantum numbers (n=2, ℓ=1, \(m_ℓ=0\)) is 2.

Case (b): \(n=2, \ell=1, m_ℓ=-1\)

Similar to the previous case, we are only given the values of \(n\), \(\ell\), and \(m_ℓ\). The spin quantum number \(m_s\) is not provided. As mentioned earlier, since \(m_s\) can have two values: \(+\frac{1}{2}\) and \(-\frac{1}{2}\), the maximum number of electrons with the given set of quantum numbers (n=2, ℓ=1, \(m_ℓ=-1\)) is also 2.

Case (c): \(n=3, \ell=1, m_ℓ=0, m_s=+\frac{1}{2}\)

In this case, we are given all four quantum numbers: \(n\), \(\ell\), \(m_ℓ\), and \(m_s\). As per the Pauli Exclusion Principle, no two electrons in an atom can have the same set of quantum numbers. Therefore, for the given set of quantum numbers (n=3, ℓ=1, \(m_ℓ=0\), \(m_s=+\frac{1}{2}\)), only 1 electron can have these specific values.

Key concepts

Electron Configuration

Electron configuration is a representation of how electrons are distributed among the various atomic orbitals in an atom. Understanding electron configuration is fundamental for students as it explains the arrangement of electrons and gives insight into the chemical behavior of an element.

• Each element has a unique electron configuration that can be determined by its atomic number, which is the number of protons (or electrons in a neutral atom).
• The configuration follows a specific order filling the atomic orbitals from the lowest energy level to the highest, adhering to the Aufbau principle, Hund's rule, and the Pauli Exclusion Principle.
• The simplified notation for electron configuration uses numbers and letters to describe the energy levels (represented by the principal quantum number, n), sublevels (designated by the letters s, p, d, f corresponding to different values of the azimuthal quantum number, l), and number of electrons in that sublevel, in the format 1s² 2s² 2p⁶, etc.

The exercise given provides us with a practical application of these rules to understand how electrons populate specific quantum states. In the cases (a) and (b), we see that two electrons are allowed in each of the stated quantum states because of the two possible values of the spin quantum number, \(m_s\). The last example, (c), illustrates the restriction set by the Pauli Exclusion Principle by showing that only one electron can occupy that particular quantum state with the given quantum numbers.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a foundational concept in quantum mechanics that plays a critical role in determining the structure of the atom.

• It states that no two electrons in an atom can have the same set of four quantum numbers \(n, l, m_l, m_s\), which represent the principal, azimuthal, magnetic, and spin quantum numbers respectively.
• This principle explains why electrons fill into separate orbitals and helps predict the maximum number of electrons that can occupy each energy level.
• As demonstrated in the step-by-step solution for case (c), only one electron can exist with a specific combination of quantum numbers, which in this scenario is \(n=3, l=1, m_l=0, m_s=+1/2\).

For students, realizing the exclusivity that the Pauli Exclusion Principle imposes on electron configurations is essential in understanding not only the structure of an atom but also its chemical properties and the way it will interact with other atoms.

Quantum Mechanical Model of the Atom

The quantum mechanical model of the atom describes the behavior of electrons within the atom, allowing us to understand the complex nature of atoms beyond the simplified planetary model.

• It relies on quantum theory and wave mechanics, with calculations involving probabilities rather than definitive paths for electron movement.
• Electrons are considered to exist in atomic orbitals rather than orbits; these orbitals are regions in space with a high probability of finding an electron, defined by the solutions to the Schrödinger equation.
• The model breaks down the electronic structure of atoms into energy levels (shell), sublevels (subshell) , and orbitals, where each electron is described by a set of quantum numbers.

The exercise we're discussing reflects the quantum mechanical model's approach to identifying how many electrons can share certain quantum attributes, which in turn allows us to predict and visualize the electron arrangement in an atom and how each electron contributes to the atom's properties.

Atomic Orbitals

Atomic orbitals are specific areas around the nucleus of an atom where electrons are most likely to be found. Each orbital can hold a maximum of two electrons, with opposite spins, as dictated by the Pauli Exclusion Principle.

• They come in different shapes (spherical for s orbitals, dumbbell-shaped for p orbitals, etc.) and are defined by three quantum numbers: the principal quantum number (n), azimuthal quantum number (l), and magnetic quantum number (m_l).
• These quantum numbers collectively determine the size, shape, and orientation in space of the orbital.
• In cases (a) and (b) of the exercise, the respective orbitals can accommodate two electrons each because the spin quantum number \(m_s\) can have two possible values, but in case (c), only one electron is allowed in that quantum state shown by the specific combination of all four quantum numbers provided.

For students, grasping the concept of atomic orbitals is imperative as it lays the groundwork for visualizing the arrangement and probability density of electrons in an atom, which is crucial for understanding chemical bonding and reactions.