3.2 Many-electron atoms and electron configurations

Many-electron atoms are complex systems that require sophisticated models to understand. The quantum mechanical approach becomes increasingly challenging as the number of electrons grows, necessitating approximation methods to solve the Schrödinger equation.

Electron configurations describe how electrons are arranged in atoms. The Pauli exclusion principle, aufbau principle, and Hund's rule govern these arrangements. Understanding these principles is crucial for predicting atomic properties and chemical behavior.

Multi-electron atoms

Quantum mechanical model for multi-electron atoms

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• The Schrödinger equation can be applied to atoms with multiple electrons, but it becomes increasingly complex and cannot be solved exactly for atoms with more than one electron
• The wave function for a multi-electron atom depends on the coordinates of all the electrons, making it a function of many variables
• Approximation methods, such as the Hartree-Fock method and density functional theory, are used to solve the Schrödinger equation for multi-electron atoms

Principles governing electron arrangement in multi-electron atoms

• The Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers (n, l, ml, ms), which determines the arrangement of electrons in an atom
• The aufbau principle describes the order in which electrons fill orbitals in a multi-electron atom, starting from the lowest energy orbital and progressing to higher energy orbitals
  • Electrons fill orbitals in the order of increasing energy: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p
• Hund's rule states that when there are degenerate orbitals (same n and l quantum numbers), electrons will occupy separate orbitals with parallel spins before pairing up in the same orbital with opposite spins
  • This minimizes electron-electron repulsion and results in a lower energy state for the atom

Electron configuration notation

Standard notation for electron configurations

• Electron configurations are written using the standard notation, which includes the principal quantum number (n), the orbital type (s, p, d, f), and the number of electrons in each orbital as a superscript
• For example, the electron configuration of carbon (Z=6) is 1s²2s²2p², and the electron configuration of iron (Z=26) is 1s²2s²2p⁶3s²3p⁶4s²3d⁶
• The superscript indicates the number of electrons in each orbital, and the order of the orbitals follows the aufbau principle

Noble gas notation for electron configurations

• The electron configuration can also be abbreviated using the noble gas notation, where the electron configuration of the preceding noble gas is represented by its chemical symbol in square brackets
• For example, the electron configuration of iron can be written as [Ar] 4s²3d⁶, where [Ar] represents the electron configuration of argon (1s²2s²2p⁶3s²3p⁶)
• This notation simplifies the representation of electron configurations for elements with many electrons

Ground state electron configuration

Determining the ground state electron configuration

• The ground state electron configuration of an atom can be determined by following the aufbau principle, Pauli exclusion principle, and Hund's rule
• Electrons are added to the lowest available energy orbitals first, with no more than two electrons per orbital (Pauli exclusion principle) and with parallel spins in degenerate orbitals (Hund's rule)
• For example, the ground state electron configuration of phosphorus (Z=15) is 1s²2s²2p⁶3s²3p³, which follows the aufbau principle and Hund's rule

Exceptions to the aufbau principle

• Exceptions to the aufbau principle occur when the energy difference between two orbitals is small, and half-filled or completely filled subshells are more stable
• For example, chromium (Z=24) has the electron configuration [Ar] 4s¹3d⁵ instead of [Ar] 4s²3d⁴, because the half-filled 3d subshell is more stable
• Similarly, copper (Z=29) has the electron configuration [Ar] 4s¹3d¹⁰ instead of [Ar] 4s²3d⁹, because the completely filled 3d subshell is more stable

Valence electrons and chemical properties

Definition and significance of valence electrons

• Valence electrons are the electrons in the outermost shell (highest principal quantum number) of an atom, which participate in chemical bonding and reactions
• The number of valence electrons determines an element's chemical properties, such as its reactivity, oxidation state, and the types of bonds it can form
• For example, alkali metals (group 1) have one valence electron and are highly reactive, while noble gases (group 18) have a complete octet of valence electrons and are chemically inert

Relationship between valence electrons and periodic trends

• Elements in the same group of the periodic table have similar chemical properties because they have the same number of valence electrons
• As you move down a group, the valence electrons are in higher principal quantum number shells, resulting in a larger atomic radius and lower ionization energy
• As you move from left to right across a period, the number of valence electrons increases, resulting in a smaller atomic radius, higher ionization energy, and higher electronegativity

Octet rule and chemical reactivity

• Atoms tend to gain, lose, or share electrons to achieve a stable octet configuration (8 valence electrons) in their outermost shell, which is similar to the electron configuration of the nearest noble gas
• The reactivity of an element is related to its electron configuration and the stability of its valence shell
  • Elements with one or two valence electrons (alkali and alkaline earth metals) tend to lose electrons and form cations
  • Elements with six or seven valence electrons (halogens and chalcogens) tend to gain electrons and form anions
• For example, sodium (Na) has one valence electron and readily loses it to form Na⁺, while chlorine (Cl) has seven valence electrons and readily gains one to form Cl⁻, resulting in the formation of the ionic compound NaCl