The Pauli Exclusion Principle is a key concept in atomic physics. It states that no two electrons in an atom can have the same set of quantum numbers, shaping how electrons fill orbitals and determining atomic structure.

This principle, along with electron configurations, forms the foundation for understanding atomic behavior. It explains why elements have unique properties and react differently, setting the stage for exploring chemical bonding and periodic trends.

The Pauli Exclusion Principle

Fundamental Principle and Quantum Numbers

• The Pauli Exclusion Principle is a fundamental principle in quantum mechanics that governs the behavior of electrons in atoms
• It states that no two electrons in an atom can have the same set of four quantum numbers: principal quantum number ($n$), angular momentum quantum number ($l$), magnetic quantum number ($m_l$), and spin quantum number ($m_s$)
• Each electron in an atom must have a unique combination of these four quantum numbers, which describe its energy, angular momentum, magnetic moment, and spin
• The principal quantum number ($n$) represents the main energy level or shell of the electron (e.g., 1, 2, 3, etc.)
• The angular momentum quantum number ($l$) represents the subshell or orbital type of the electron (e.g., s, p, d, f)
• The magnetic quantum number ($m_l$) represents the specific orbital within a subshell (e.g., $p_x$, $p_y$, $p_z$)
• The spin quantum number ($m_s$) represents the intrinsic angular momentum or spin of the electron, which can be either +1/2 or -1/2

Implications for Electron Configurations

• The Pauli Exclusion Principle limits the number of electrons that can occupy each atomic orbital to a maximum of two, with opposite spins ($m_s$ = +1/2 and -1/2)
• This means that each orbital can hold up to two electrons, and these electrons must have opposite spins
• As a result of the Pauli Exclusion Principle, electrons in an atom fill orbitals in a specific order, leading to the development of the Aufbau principle and Hund's rule
• The Aufbau principle states that electrons fill orbitals in order of increasing energy, starting from the lowest energy orbital (1s) and progressing to higher energy orbitals (2s, 2p, 3s, etc.)
• Hund's rule states that when there are degenerate orbitals (orbitals with the same energy), electrons will occupy these orbitals singly with parallel spins before pairing up with opposite spins
• The Pauli Exclusion Principle plays a crucial role in determining the electron configurations of atoms and, consequently, their chemical properties and behavior

Electron Configurations of Atoms

Ground State Electron Configurations

• The ground state electron configuration of an atom represents the lowest energy arrangement of its electrons in atomic orbitals
• To determine the ground state electron configuration of an atom, follow these steps:
  1. Determine the total number of electrons in the atom based on its atomic number
  2. Fill the orbitals according to the Aufbau principle, placing electrons in the lowest energy orbitals first (1s, 2s, 2p, 3s, 3p, 4s, 3d, etc.)
  3. Apply Hund's rule when filling degenerate orbitals, ensuring that electrons occupy orbitals singly with parallel spins before pairing up
  4. Ensure that the Pauli Exclusion Principle is followed, with no two electrons having the same set of four quantum numbers
• Electron configurations can be written using the standard notation (e.g., 1s²2s²2p⁶) or the noble gas notation (e.g., [Ne]3s²3p³) for elements beyond neon
• Examples of ground state electron configurations:
  • Carbon (6 electrons): 1s²2s²2p²
  • Aluminum (13 electrons): [Ne]3s²3p¹
  • Iron (26 electrons): [Ar]4s²3d⁶

Exceptions to the Aufbau Principle

• While the Aufbau principle generally predicts the correct electron configurations for most atoms, there are some exceptions due to the relative energies of certain orbitals
• In some cases, electrons may occupy higher energy orbitals before completely filling lower energy orbitals, resulting in slightly different electron configurations
• Examples of exceptions to the Aufbau principle:
  • Chromium (24 electrons): [Ar]4s¹3d⁵ instead of [Ar]4s²3d⁴
  • Copper (29 electrons): [Ar]4s¹3d¹⁰ instead of [Ar]4s²3d⁹
• These exceptions occur due to the slightly lower energy of the half-filled (3d⁵) or completely filled (3d¹⁰) d-subshells, which provides greater stability

Atomic Stability and Reactivity

Stable Electron Configurations

• Atoms with completely filled valence shells (i.e., noble gas configurations) are generally more stable and less reactive than atoms with partially filled valence shells
• Noble gases (e.g., He, Ne, Ar, Kr, Xe) have completely filled valence shells with 2 electrons in the outermost s-subshell (except for helium, which has 2 electrons in the 1s orbital)
• Atoms with electron configurations similar to noble gases (e.g., ions with noble gas configurations) also tend to be stable and less reactive
• Examples of stable electron configurations:
  • Neon (10 electrons): 1s²2s²2p⁶
  • Potassium ion, K⁺ (18 electrons): [Ar]4s⁰, isoelectronic with argon
  • Sulfide ion, S²⁻ (18 electrons): [Ne]3s²3p⁶, isoelectronic with argon

Reactive Electron Configurations

• Atoms with one or two electrons in their outermost s-subshell (e.g., alkali and alkaline earth metals) are highly reactive due to their tendency to lose electrons and form cations
• Atoms with nearly filled valence shells (e.g., halogens) are also highly reactive due to their tendency to gain electrons and form anions
• Atoms with half-filled or completely filled d-subshells (e.g., Cr, Cu, Ag, Au) often exhibit unique properties and reactivities due to their electron configurations
• Electron configurations can help predict the formation of chemical bonds, as atoms tend to gain, lose, or share electrons to achieve stable noble gas configurations
• Examples of reactive electron configurations:
  • Sodium (11 electrons): [Ne]3s¹, tends to lose one electron to form Na⁺ ion
  • Chlorine (17 electrons): [Ne]3s²3p⁵, tends to gain one electron to form Cl⁻ ion
  • Oxygen (8 electrons): 1s²2s²2p⁴, tends to gain two electrons to form O²⁻ ion or share electrons in covalent bonds

Core vs Valence Electrons

Core Electrons

• Core electrons are the electrons occupying the inner, lower-energy shells of an atom (i.e., all shells except the outermost shell)
• Core electrons are generally not involved in chemical bonding and reactions, as they are more tightly bound to the nucleus and shielded by the valence electrons
• The number of core electrons in an atom can be determined by the electron configuration of the previous noble gas
• For example, in the electron configuration of sodium ([Ne]3s¹), the electrons in the [Ne] core (1s²2s²2p⁶) are the core electrons
• Core electrons contribute to the overall stability of the atom and are responsible for the screening effect, which reduces the effective nuclear charge experienced by the valence electrons

Valence Electrons

• Valence electrons are the electrons occupying the outermost shell (highest principal quantum number, $n$) of an atom
• The number of valence electrons determines an atom's chemical properties, including its ability to form chemical bonds and participate in reactions
• In the periodic table, elements within the same group (column) have the same number of valence electrons and exhibit similar chemical properties
• For example, all alkali metals (Group 1) have one valence electron in their outermost s-subshell, while all halogens (Group 17) have seven valence electrons in their outermost s- and p-subshells
• Valence electrons are involved in chemical bonding, as they can be lost, gained, or shared to form ionic, covalent, or metallic bonds
• The distinction between core and valence electrons is essential for understanding periodic trends, chemical bonding, and the reactivity of elements