Problem 37 - Chemistry and Chemical Reactivity

Question

Explain briefly why each of the following is not a possible set of quantum numbers for an electron in an atom. In each case, change the incorrect value (or values) to make the set valid. (a) \(n=4, \ell=2, m_{\ell}=0, m_{\mathrm{s}}=0\) (b) \(n=3, \ell=1, m_{\ell}=-3, m_{\mathrm{s}}=-1 / 2\) (c) \(n=3, \ell=3, m_{\ell}=-1, m_{\mathrm{s}}=+1 / 2\)

Step-by-Step Solution

Answer

(a) Change \(m_{\mathrm{s}}\) to \(\pm \frac{1}{2}\); (b) Change \(m_{\ell}\) to -1, 0, or 1; (c) Change \(\ell\) to 2.

Step 1: Check Quantum Numbers Validity for Case (a)

For the quantum numbers set: \(n=4, \ell=2, m_{\ell}=0, m_{\mathrm{s}}=0\), the magnetic spin quantum number \(m_{\mathrm{s}}\) must be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). The provided value \(m_{\mathrm{s}} = 0\) is not valid. To correct this set, change \(m_{\mathrm{s}}\) to either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Step 2: Check Quantum Numbers Validity for Case (b)

For the quantum numbers set: \(n=3, \ell=1, m_{\ell}=-3, m_{\mathrm{s}}=-\frac{1}{2}\), the magnetic quantum number \(m_{\ell}\) must be in the range \(-\ell \leq m_{\ell} \leq \ell\). Given \(\ell=1\), valid \(m_{\ell}\) values are -1, 0, or 1. Change \(m_{\ell} = -3\) to any of the valid values, such as \(m_{\ell} = 0\).

Step 3: Check Quantum Numbers Validity for Case (c)

For the quantum numbers set: \(n=3, \ell=3, m_{\ell}=-1, m_{\mathrm{s}}=+\frac{1}{2}\), the azimuthal quantum number \(\ell\) must be less than \(n\). For \(n=3\), \(\ell\) can only be 0, 1, or 2. Change \(\ell=3\) to a valid value like \(\ell = 2\). Adjust \(m_{\ell}\) to fit the new \(\ell\) value as long as it is within \(-\ell \leq m_{\ell} \leq \ell\).

Key Concepts

Quantum Mechanics

Quantum mechanics is a fundamental branch of physics that describes the physical properties of nature at the smallest scales, such as atoms and subatomic particles. It provides a mathematical framework to understand and predict the behavior of particles at quantum levels, where traditional Newtonian mechanics fails to explain the phenomena.
In the context of quantum mechanics, electrons are described by quantum numbers that specify their properties in an atom. These quantum numbers arise naturally out of the solutions of the Schrödinger equation, which is central to quantum mechanics. Each electron in an atom is described by a unique set of four quantum numbers:

• The principal quantum number ( ) determines the energy level of the electron.
• The azimuthal quantum number ( ) dictates the subshell and shape of the electron's orbital.
• The magnetic quantum number ( ) specifies the orientation of the orbital.
• The spin quantum number ( ) describes the intrinsic spin of the electron.

These quantum numbers follow specific rules and constraints, such as Pauli's exclusion principle, which states that no two electrons in an atom can have the same set of quantum numbers. This principle is pivotal to understanding electron configuration and atomic structure.

Electron Configuration

Electron configuration refers to the distribution of electrons among the atomic orbitals of an atom. This configuration is determined using the principles of quantum mechanics, reflecting the arrangement of electrons in orbitals according to increasing energy levels.
When writing electron configurations, each principal energy level (determined by the principal quantum number ) can hold a certain number of electrons based on their possible orbitals:

• The s sublevel can hold up to 2 electrons.
• The p sublevel can hold up to 6 electrons.
• The d sublevel can hold up to 10 electrons.
• The f sublevel can hold up to 14 electrons.

Moreover, the electron configuration is also influenced by the Aufbau principle, which states that electrons fill orbitals starting from the lowest energy level. Hund's rule further guides this fill by indicating that electrons will equally occupy orbitals of the same energy before doubling up in any orbital. The configuration not only provides insight into the electronic structure but also sheds light on chemical reactivity and periodic properties of elements.

Atomic Structure

Atomic structure is the composition and arrangement of electrons, protons, and neutrons in an atom, forming the basis for the general behavior and interaction of the atom with others. Electrons spin and orbit a dense nucleus made up of protons and neutrons, following the fundamental wave-particle duality explained by quantum mechanics.
The atom's nucleus is positively charged, with protons contributing to this charge while neutrons generally retain no charge. The electron cloud surrounding the nucleus is negatively charged, keeping the atom electrically neutral overall as the charges balance. The arrangement of electrons within this cloud is dictated by their electron configuration, which is influenced by their quantum numbers:

• Electrons fill orbitals in a way that minimizes the atom's energy.
• Electron interactions with each other and their orbitals affect atomic properties and chemical bonding.
• Changes in electron configurations can occur during chemical reactions, affecting atomic structure and resulting in the release or absorption of energy.

Understanding atomic structure gives critical insight into how atoms interact and form bonds, crucial for studying chemical elements and compounds. By using the systematic approach grounded in quantum mechanics, we develop greater precision in predicting atomic behavior and properties.