5.1 Multi-electron atoms and the Pauli exclusion principle

Multi-electron atoms are complex systems where electrons interact with each other and the nucleus. The Pauli exclusion principle, which states that no two electrons can have identical quantum states, shapes the electronic structure of these atoms.

This principle leads to the filling of electron orbitals in a specific order, forming shells and subshells. It explains periodic trends and is responsible for the stability of matter, preventing all electrons from collapsing into the lowest energy state.

Pauli Exclusion Principle Implications

Fundamental Concept and Atomic Structure

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• Pauli exclusion principle states no two electrons in an atom can have the same set of quantum numbers
• Shapes electronic structure of multi-electron atoms
• Leads to filling of electron orbitals in specific order (Aufbau principle and Hund's rule)
• Results in formation of electron shells and subshells
  • Determines chemical and physical properties of elements
• Explains periodic trends across the periodic table
  • Atomic size
  • Ionization energy
  • Electron affinity

Stability and Consequences

• Responsible for stability of matter
• Prevents all electrons from collapsing into lowest energy state
• Violation would lead to dramatic changes in behavior of matter and universe
  • Collapse of atomic structures (atoms would shrink dramatically)
  • Changes in chemical bonding (molecules would become unstable)
  • Alteration of stellar evolution (stars would behave differently)

Electron Spin in Atomic Structure

Fundamental Properties

• Electron spin intrinsic angular momentum of electron
• Characterized by spin quantum number (s) with values of +1/2 or -1/2
• Quantum mechanical property with no classical analogue
  • Often visualized as electron rotating about its axis (not physically accurate)
• Spin magnetic moment interacts with external magnetic fields
  • Leads to phenomena like Zeeman effect (splitting of spectral lines in magnetic field)

Experimental Evidence and Applications

• Stern-Gerlach experiment provided evidence for quantization of electron spin
  • Beam of silver atoms split into two distinct beams in non-uniform magnetic field
• Crucial role in determining magnetic properties of atoms and materials
  • Ferromagnetism (iron, nickel, cobalt)
  • Antiferromagnetism (manganese oxide, chromium)
• Spin-orbit coupling interaction between electron's spin and orbital angular momentum
  • Leads to fine structure in atomic spectra (splitting of spectral lines)

Quantum Mechanical Considerations

• Combination of spin and spatial wavefunctions must be antisymmetric for fermions like electrons
  • Required by Pauli exclusion principle
• Spin states can be represented as |↑⟩ and |↓⟩ in quantum mechanics
• Spin angular momentum magnitude given by $\sqrt{s(s+1)}\hbar$ where s = 1/2 for electrons

Electron Configurations for Multi-electron Atoms

Notation and Orbital Filling

• Electron configurations written using spectroscopic notation
  • Orbitals denoted by principal quantum number (n) and angular momentum quantum number (l)
  • Example: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ (for zinc)
• Maximum number of electrons in each subshell determined by 2(2l+1)
  • s orbitals (l=0): 2 electrons
  • p orbitals (l=1): 6 electrons
  • d orbitals (l=2): 10 electrons
  • f orbitals (l=3): 14 electrons
• Hund's rule states electrons in degenerate orbitals occupy them singly with parallel spins before pairing
  • Minimizes electron-electron repulsion
  • Example: Carbon ground state 1s² 2s² 2p² (two unpaired electrons in 2p orbitals)

Aufbau Principle and Exceptions

• Aufbau principle dictates order of orbital filling: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p
• Exceptions occur due to increased stability of half-filled or completely filled subshells
  • Chromium: [Ar] 4s¹ 3d⁵ instead of [Ar] 4s² 3d⁴
  • Copper: [Ar] 4s¹ 3d¹⁰ instead of [Ar] 4s² 3d⁹
• Lanthanides and actinides show complex filling patterns due to similar energies of f, d, and s orbitals

Valence Electrons and Excited States

• Valence electrons in outermost shell largely determine chemical properties
  • Can be predicted using periodic table (group number for main group elements)
• Excited state configurations involve promotion of electrons to higher energy levels
  • Subject to selection rules based on changes in quantum numbers (Δl = ±1, Δm = 0, ±1)
  • Example: Sodium excited state 1s² 2s² 2p⁶ 3p¹ (instead of ground state 3s¹)

Electron Shielding and Atomic Properties

Shielding Effect and Effective Nuclear Charge

• Electron shielding (screening) reduces effective nuclear charge experienced by outer electrons
• Shielding effect increases with principal quantum number (n) and decreases with angular momentum quantum number (l)
• Penetration affects degree of shielding
  • Probability of electron being found close to nucleus
  • Order of penetration: s > p > d > f for orbitals of same principal quantum number
• Effective nuclear charge (Zeff) calculated as actual nuclear charge minus shielding effect
  • Determines strength of electron-nucleus attraction
  • Example: Lithium Zeff for 2s electron ≈ 1.3 (actual Z = 3)

Periodic Trends and Anomalies

• Shielding explains trend of decreasing ionization energy and increasing atomic radius across a period
  • Outer electrons less tightly bound due to increased shielding
• Responsible for irregular trends in first ionization energies
  • Beryllium anomaly (higher than expected ionization energy due to filled 2s subshell)
  • Nitrogen anomaly (higher than oxygen due to half-filled 2p subshell)
• Affects electron affinity trends
  • Noble gases have very low electron affinities due to complete outer shells

Multi-electron Atom Behavior

• Shielding effect leads to departure from hydrogen-like behavior in multi-electron atoms
• Results in more complex electron energy levels and spectral patterns
  • Splitting of energy levels due to electron-electron interactions
  • Multiple emission lines in spectra (compared to simple hydrogen spectrum)
• Influences chemical bonding and molecular properties
  • Affects electronegativity and atomic size, impacting bond strengths and molecular geometries