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30.8 Quantum Numbers and Rules

4 min read•june 18, 2024

are the building blocks of atomic structure. They describe an 's energy, angular momentum, orientation, and spin within an atom. Understanding these numbers is crucial for grasping how electrons behave and interact in atoms.

Electron configurations show how electrons are distributed in an atom's orbitals. They follow key principles like the and . These concepts help explain atomic properties and chemical behavior, making them essential for understanding the periodic table and chemical bonding.

Quantum Numbers and Atomic States

Quantum numbers in atomic states

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  • uniquely describe the state of an electron in an atom
    • Each electron characterized by a unique set of four quantum numbers
    • No two electrons in an atom can have the same set of four quantum numbers ()
  • The four quantum numbers:
    • [n](https://www.fiveableKeyTerm:N)[n](https://www.fiveableKeyTerm:N)
    • [l](https://www.fiveableKeyTerm:L)[l](https://www.fiveableKeyTerm:L)
    • mlm_l
    • msm_s
  • Quantum numbers provide a complete description of an electron's energy, angular momentum, orientation, and spin within an atom
    • Essential for understanding the electronic structure and properties of atoms (atomic radius, ionization energy, electron affinity)
    • Used to predict the behavior of electrons in chemical bonding and spectroscopy (absorption, emission, and fluorescence spectra)

Physical meaning of quantum numbers

  • Principal quantum number nn
    • Represents the main or of an electron
    • Takes positive integer values: 1, 2, 3, ...
    • Higher values of nn correspond to higher energy levels and larger atomic radii
      • n=1n=1: (lowest energy)
      • n=2n=2:
      • n=3n=3:
      • n=4n=4: (higher energy)
  • Angular momentum quantum number ll
    • Describes the or suborbital of an electron within a main energy level
    • Takes integer values from 0 to n−1n-1
    • Determines the shape of the (s, p, d, f)
      • l=0l=0: (spherical)
      • l=1l=1: (dumbbell)
      • l=2l=2: (cloverleaf)
      • l=3l=3: (complex)
    • Related to the magnitude of an electron's angular momentum (rotational motion around the nucleus)
  • Magnetic quantum number mlm_l
    • Describes the orientation of an orbital in space relative to an external magnetic field
    • Takes integer values from −l-l to +l+l, including 0
    • Determines the number of orbitals within a
      • s subshell: 1 orbital (ml=0m_l=0)
      • p subshell: 3 orbitals (ml=−1,0,+1m_l=-1, 0, +1)
      • d subshell: 5 orbitals (ml=−2,−1,0,+1,+2m_l=-2, -1, 0, +1, +2)
      • f subshell: 7 orbitals (ml=−3,−2,−1,0,+1,+2,+3m_l=-3, -2, -1, 0, +1, +2, +3)
    • Orbitals with different mlm_l values have different orientations in space (x, y, z axes)
  • msm_s
    • Describes the intrinsic angular momentum (spin) of an electron
    • Takes values of +12+\frac{1}{2} () or −12-\frac{1}{2} ()
    • Each orbital can accommodate a maximum of two electrons with opposite spins
    • Electron spin is a fundamental property with no classical analog (intrinsic angular momentum)
    • Responsible for the magnetic properties of atoms and materials (paramagnetic, ferromagnetic, antiferromagnetic)

Electron Configurations and Quantum Principles

Electron configurations and quantum principles

  • Pauli exclusion principle
    • No two electrons in an atom can have the same set of four quantum numbers
    • Each orbital can accommodate a maximum of two electrons with opposite spins
    • Crucial for understanding the electronic structure and stability of atoms and molecules
    • Explains the periodic table's structure and the chemical properties of elements
  • Hund's rule
    • Electrons occupy orbitals of the same energy (degenerate orbitals) singly before pairing up
    • Electrons in singly occupied orbitals have the same spin (parallel spins)
    • Minimizes electron-electron repulsion and maximizes the total spin of the atom
    • Responsible for the magnetic properties of atoms with unpaired electrons (paramagnetic)
  • notation
    • Represents the distribution of electrons in an atom's orbitals
    • Written as: 1s22s22p63s23p6...1s^2 2s^2 2p^6 3s^2 3p^6 ...
      • The number represents the principal quantum number nn
      • The letter represents the subshell (s, p, d, f)
      • The superscript represents the number of electrons in the subshell
    • Examples:
      • Helium: 1s21s^2
      • Carbon: 1s22s22p21s^2 2s^2 2p^2
      • Neon: 1s22s22p61s^2 2s^2 2p^6
      • Sodium: 1s22s22p63s11s^2 2s^2 2p^6 3s^1
    • Electrons fill orbitals in order of increasing energy
    • Energy order: 1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f<5d<6p<7s<5f<6d<7p1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p
    • Exceptions occur due to electron-electron repulsion and the stability of half-filled and fully-filled subshells
      • Chromium: 1s22s22p63s23p64s13d51s^2 2s^2 2p^6 3s^2 3p^6 4s^1 3d^5 instead of 1s22s22p63s23p64s23d41s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^4
      • Copper: 1s22s22p63s23p64s13d101s^2 2s^2 2p^6 3s^2 3p^6 4s^1 3d^{10} instead of 1s22s22p63s23p64s23d91s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^9
  • The combination of the Pauli exclusion principle, Hund's rule, electron configuration notation, and the Aufbau principle provides a comprehensive framework for understanding the electronic structure of atoms and predicting their chemical properties

Quantum Mechanical Principles

  • : A mathematical description of the quantum state of an electron in an atom
  • : The fundamental equation of quantum mechanics that describes the behavior of wave functions
  • : The square of the wave function, representing the likelihood of finding an electron in a particular region of space
  • : The principle that a quantum system can exist in multiple states simultaneously until measured
  • : The fundamental limit on the precision with which certain pairs of physical properties can be determined simultaneously
  • : The occurrence of multiple quantum states with the same energy level in an atom

Key Terms to Review (42)

AC current: AC current (Alternating Current) is an electric current that reverses its direction periodically. It is the form of electrical energy commonly delivered to businesses and residences.
Angular Momentum Quantum Number: The angular momentum quantum number, often represented by the symbol 'l', is a quantum number that determines the shape of an electron's orbital and its angular momentum in an atom. It plays a crucial role in defining the energy levels and sublevels that electrons can occupy, as well as influencing the overall behavior of electrons in various energy states.
Antielectron: An antielectron, also known as a positron, is the antimatter counterpart of an electron. It has the same mass as an electron but carries a positive charge.
Aufbau Principle: The aufbau principle is a fundamental concept in quantum mechanics that describes the way electrons occupy the available energy levels in an atom. It is a set of rules that determine the order in which electrons fill the atomic orbitals as the atomic number increases.
D orbital: The d orbital is a type of atomic orbital that can hold a maximum of 10 electrons and is characterized by its distinct shape and energy level. These orbitals play a crucial role in the electron configuration of transition metals, influencing their chemical properties and behavior. The presence of d orbitals allows for various oxidation states and complex ion formation, impacting the overall chemistry of an element.
Degeneracy: Degeneracy refers to the phenomenon where multiple quantum states or configurations can have the same energy level. This concept is crucial in understanding the behavior of quantum systems, particularly in the context of quantum numbers and the rules governing them.
Electron: An electron is a fundamental subatomic particle that carries a negative electric charge and is found in all atoms, playing a crucial role in various physical and chemical phenomena. Electrons are responsible for the flow of electric current, the formation of chemical bonds, and the behavior of matter at the atomic and molecular levels. The concept of the electron is central to understanding topics such as static electricity, electric fields, magnetic fields, the photoelectric effect, quantum mechanics, and the structure of atoms. Electrons are the building blocks of matter and are essential for understanding the fundamental nature of the universe.
Electron Configuration: Electron configuration is the arrangement of electrons in an atom, which is determined by the quantum mechanical rules governing the distribution of electrons in an atom's orbitals. This concept is fundamental to understanding the behavior and properties of atoms and their interactions.
Energy level: An energy level is a specific quantized state that an electron can occupy in an atom, defined by a set of quantum numbers. Each energy level corresponds to a certain amount of energy, and electrons can transition between these levels by absorbing or emitting energy in the form of photons. The arrangement and energies of these levels play a crucial role in determining the chemical properties and behaviors of elements.
F orbital: The f orbital is a type of atomic orbital that can hold a maximum of 14 electrons, characterized by complex shapes and angular momentum quantum numbers of l = 3. These orbitals are important in the arrangement of electrons in elements, especially those in the lanthanide and actinide series, where their unique properties influence chemical behavior and bonding.
Hund's Rule: Hund's Rule states that electrons will fill degenerate orbitals singly before pairing up in the same orbital. This principle helps minimize electron-electron repulsion and leads to lower energy configurations. The rule is crucial when determining the electron configurations of atoms and is closely related to quantum numbers and the Pauli Exclusion Principle, as it impacts how we understand the arrangement of electrons in an atom.
K Shell: The K shell is the innermost electron shell or energy level in an atom. It is the shell closest to the nucleus and has the lowest energy level of all the electron shells.
L: L is a fundamental quantity used in the description of oscillations and quantum mechanical systems. It represents the angular momentum or the orbital angular momentum of a particle or system, and it is a crucial concept in both the study of periodic motion and the understanding of the behavior of atoms and subatomic particles.
L shell: The L shell refers to the second electron shell of an atom, which can hold up to 8 electrons. It is characterized by its principal quantum number n=2 and contains the 2s and 2p subshells. The arrangement of electrons within this shell is crucial for determining the chemical properties and reactivity of an element.
M shell: The M shell is the third principal energy level or shell in an atom, which can accommodate a maximum of 18 electrons. It is the second outermost shell after the K and L shells, and it plays a crucial role in understanding the quantum numbers and the Pauli exclusion principle in atomic structure.
Magnetic Quantum Number: The magnetic quantum number is a quantum number that describes the orientation of an electron's angular momentum, or orbital angular momentum, within an atom. It is one of the four quantum numbers used to fully describe the state of an electron in an atom.
Magnitude of the intrinsic (internal) spin angular momentum: Magnitude of the intrinsic (internal) spin angular momentum is a fundamental property of particles, such as electrons, indicating the magnitude of their internal spin. It quantifies the amount of angular momentum arising from the particle's intrinsic spin characteristics.
Ml: In quantum mechanics, 'ml' is one of the quantum numbers that describes the orientation of an electron's orbital within an atom. It is specifically known as the magnetic quantum number and can take integer values ranging from -l to +l, where 'l' is the azimuthal quantum number. This value determines the number of distinct orientations an orbital can have in a magnetic field, helping to define the shape and spatial distribution of electron clouds around the nucleus.
Ms: In quantum mechanics, 'ms' refers to the magnetic spin quantum number, which describes the intrinsic angular momentum or 'spin' of an electron. It can take on one of two values: +1/2 or -1/2, representing the two possible orientations of an electron's spin in a magnetic field. This concept is crucial for understanding how electrons fill atomic orbitals and how they interact with magnetic fields, playing a significant role in determining the properties of atoms and molecules.
N: N is a variable or constant that represents a specific quantity or value, and it is commonly used in various scientific and mathematical contexts. This term is particularly relevant in the topics of Friction, Variation of Pressure with Depth in a Fluid, Hooke's Law: Stress and Strain Revisited, and Quantum Numbers and Rules, where it serves different purposes and carries distinct meanings.
N Shell: The N shell is the fourth principal energy level or shell of an atom, which can accommodate up to 32 electrons. It is an important concept in understanding the quantum numbers and the Pauli exclusion principle, which govern the arrangement and behavior of electrons within an atom.
Orbital: An orbital is a mathematical function that describes the wave-like behavior of electrons in an atom, representing the probability distribution of an electron's position around the nucleus. These orbitals can have different shapes and orientations, which are essential for understanding how atoms bond and interact with each other.
P orbital: A p orbital is a type of atomic orbital that can hold a maximum of six electrons and has a distinct dumbbell shape, oriented along specific axes in three-dimensional space. These orbitals are important for understanding the arrangement of electrons in atoms, especially in relation to the quantum numbers that describe their state and behavior, as well as the Pauli Exclusion Principle which dictates how electrons can occupy these orbitals without violating specific rules.
Pauli exclusion principle: The Pauli Exclusion Principle states that no two fermions, such as electrons, can occupy the same quantum state simultaneously within a quantum system. This principle is fundamental in explaining the structure of atoms and the behavior of electrons in atoms.
Pauli Exclusion Principle: The Pauli exclusion principle is a fundamental principle in quantum mechanics that states that two identical fermions (particles with half-integer spin, such as electrons, protons, and neutrons) cannot occupy the same quantum state simultaneously. This principle is essential in understanding the structure of atoms, molecules, and the behavior of matter at the quantum level.
Principal Quantum Number: The principal quantum number, denoted as 'n', is an integer that specifies the energy level of an electron in an atom. It helps define the electron's distance from the nucleus and plays a crucial role in determining the electron's energy and the overall structure of the atom.
Probability Density: Probability density is a fundamental concept in quantum mechanics that describes the likelihood of finding a particle in a specific region of space. It is a mathematical function that represents the probability distribution of a particle's position or other quantum mechanical properties.
Quantum numbers: Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. They provide information about the electron’s energy level, orbital shape, orbital orientation, and spin direction.
Quantum Numbers: Quantum numbers are a set of numerical values that describe the unique quantum state of an electron in an atom, providing essential information about its energy level, orbital shape, orientation, and spin. They connect the quantization of energy to electron configurations, patterns in spectra, and the fundamental principles governing atomic structure and behavior.
Quantum Superposition: Quantum superposition is a fundamental principle of quantum mechanics that states that a quantum system can exist in multiple states or configurations simultaneously until it is observed or measured. This principle is central to understanding the particle-wave duality, quantum numbers, and quantum tunneling in the context of introductory college physics.
S Orbital: The s orbital is one of the fundamental atomic orbitals in quantum mechanics, representing the spherical electron distribution around the nucleus of an atom. The s orbital is the simplest and most symmetric of the atomic orbitals, and it plays a crucial role in understanding the behavior of electrons in atoms, particularly in the context of quantum numbers and the Pauli exclusion principle.
Schrödinger Equation: The Schrödinger equation is a fundamental equation in quantum mechanics that describes the wave function of a particle and how it evolves over time. It is a central concept that connects the particle-wave duality and the quantization of energy, and is essential for understanding the behavior of quantum systems, including the structure of atoms and the tunneling phenomenon.
Shell: In the context of quantum mechanics and atomic structure, a shell refers to the discrete energy levels or orbitals that electrons occupy around the nucleus of an atom. These shells are defined by the principal quantum number, which determines the overall energy of the electron.
Spin down: Spin down refers to one of the two possible orientations of an intrinsic property called spin in quantum mechanics. It is commonly associated with electrons and other fermions, where 'spin down' represents a state of lower energy compared to its counterpart, 'spin up'. This concept plays a critical role in understanding how particles behave in magnetic fields and is integral to the rules governing quantum numbers.
Spin projection quantum number: The spin projection quantum number ($m_s$) specifies the orientation of the intrinsic angular momentum (spin) of an electron within an atom. It can take on values of $+\frac{1}{2}$ or $-\frac{1}{2}$, corresponding to 'spin up' and 'spin down' states.
Spin quantum number: The spin quantum number ($m_s$) describes the intrinsic angular momentum of an electron within an atom. It can take on values of $+\frac{1}{2}$ or $-\frac{1}{2}$, corresponding to the two possible spin states.
Spin Quantum Number: The spin quantum number is a fundamental property of subatomic particles that describes the intrinsic angular momentum or 'spin' of the particle. It is one of the key quantum numbers used to fully characterize the state of an electron within an atom.
Spin up: Spin up refers to a specific orientation of a quantum particle's intrinsic angular momentum, or spin, where the particle is aligned in a particular direction, typically denoted as 'up' in the context of quantum mechanics. This term connects to the principles of quantum numbers and the rules that govern the arrangement of electrons in atoms, helping to define how particles interact with magnetic fields and how they are characterized in terms of their quantum states.
Subshell: A subshell is a division of electron shells separated by electron orbitals within an atom. Subshells are designated by the letters s, p, d, and f, and they describe the shape and energy of the orbitals.
Subshell: A subshell is a subdivision of an electron shell within an atom, representing a specific set of quantum numbers that describe the energy levels and spatial distribution of electrons. Subshells are crucial in understanding the Pauli Exclusion Principle and the arrangement of electrons within an atom's structure.
Uncertainty Principle: The uncertainty principle is a fundamental concept in quantum mechanics that states it is impossible to simultaneously know both the exact position and exact momentum of a particle. This principle highlights the limitations of measurement in the quantum realm and reveals how energy is quantized, wave-particle duality manifests, and how particles interact under quantum rules.
Wave function: A wave function is a mathematical description of the quantum state of a system, representing the probabilities of finding a particle in various positions and states. It connects deeply with the behavior of particles at the quantum level, demonstrating the dual nature of matter as both particles and waves, and illustrating how energy levels are quantized.
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30.9 The Pauli Exclusion Principle