Question

What type of electron orbital (i.e., \(\mathrm{s}, \mathrm{p}, \mathrm{d}\), or \(\mathrm{f}\) ) is designated by (a) \(\mathbf{n}=3, \ell=2, \mathbf{m}_{\ell}=-1\) ? (b) \(\mathbf{n}=6, \ell=3, \mathbf{m}_{\ell}=2 ?\) (c) \(\mathbf{n}=4, \ell=3, \mathbf{m}_{\ell}=3\) ?

Short answer

Question: Identify the type of electron orbital for the given quantum numbers: (a) n=3, l=2, m_l=-1 (b) n=6, l=3, m_l=2 (c) n=4, l=3, m_l=3 Answer: (a) d orbital (b) f orbital (c) f orbital

Step-by-step solution

(a) Identify the orbital with n=3, l=2, m_l=-1

Given the quantum numbers n=3, l=2, m_l=-1, we will focus on the value of l, which is 2. Since l=2, it corresponds to the d orbital.

(b) Identify the orbital with n=6, l=3, m_l=2

Given the quantum numbers n=6, l=3, m_l=2, we will focus on the value of l, which is 3. Since l=3, it corresponds to the f orbital.

(c) Identify the orbital with n=4, l=3, m_l=3

Given the quantum numbers n=4, l=3, m_l=3, we will focus on the value of l, which is 3. Since l=3, it corresponds to the f orbital.

Key concepts

Electron Orbitals

In atomic physics, electron orbitals provide a key for understanding the way electrons arrange themselves around the nucleus of an atom. An orbital is a region in which there is a high probability of finding an electron. These orbitals are defined by a set of quantum numbers: \(n\), \(\ell\), and \(m_\ell\). Each quantum number defines specific properties of the orbital.

• The principal quantum number (\(n\)) describes the energy level or shell of the electron, with higher numbers indicating shells farther from the nucleus.
• The azimuthal quantum number (\(\ell\)) corresponds to the angular momentum and determines the shape of the orbital. For instance, \(\ell = 0\) indicates an s orbital, \(\ell = 1\) indicates a p orbital, \(\ell = 2\) indicates a d orbital, and \(\ell = 3\) indicates an f orbital.
• The magnetic quantum number (\(m_\ell\)) defines the orientation of the orbital in space. It ranges from \(-\ell\) to \(\ell\), including zero.

The correlation between these quantum numbers allows us to identify the type of orbital for a given electron. This classification is crucial for understanding an electron's chemical behavior and interaction with other atoms.

Quantum Mechanics

Quantum mechanics is the fundamental theory that underpins modern atomic and molecular physics, explaining how particles like electrons behave on a microscopic scale. At its core, it describes and predicts the behavior of particles which do not obey classical physics. One of the most intriguing elements of quantum mechanics is the concept of wave-particle duality, where particles exhibit properties of both waves and particles.

• The Schrödinger equation is a central tool in quantum mechanics, used to calculate the probability of finding an electron in a particular region in space, which in essence generates the shape of orbitals.
• Quantum superposition allows electrons to exist in multiple states simultaneously, only collapsing into a single state when an observation is made.
• Heisenberg's uncertainty principle indicates that one cannot simultaneously know the exact position and momentum of an electron, highlighting the probabilistic nature of electron positioning.

Quantum mechanics not only explains the movement and placement of electrons within orbitals, but also their role in chemical bonding and molecular structures. This insight helps chemists and physicists to predict and explain the properties and reactions of substances.

Atomic Structure

The atomic structure of an element comprises a central nucleus surrounded by a cloud of electrons. Understanding atomic structure is essential to grasp how elements are built and how they interact with each other.

• The nucleus of an atom contains protons and neutrons, with protons determining the element's identity through the atomic number.
• Electrons, residing in orbitals around the nucleus, play a pivotal role in defining the chemical properties of an atom.
• Orbitals are grouped into shells and subshells, where electrons are located according to specific rules and principles such as the Pauli exclusion principle, which states that no two electrons can occupy the same quantum state simultaneously.

The arrangement of electrons into various orbitals and shells results in the periodic trends observed in the periodic table. This structure determines an element's electronegativity, ionization energy, atomic radius, and other key characteristics. Hence, a comprehensive understanding of atomic structure provides insights into an element's chemical stability, reactivity, and bonding capacity.