The quantum mechanical model of the atom revolutionized our understanding of atomic structure. It explains how electrons behave in atoms, using wave functions and probability distributions. This model is crucial for grasping atomic properties and chemical behavior.

This section dives into wave-particle duality, the Schrödinger equation, and quantum numbers. We'll explore atomic orbitals, electron configurations, and the principles that govern how electrons are arranged in atoms. These concepts form the foundation for understanding periodic trends.

Wave-Particle Duality and Quantum Mechanics

Fundamental Concepts of Wave-Particle Duality

• Wave-particle duality describes the dual nature of matter and energy behaving as both waves and particles
• Light exhibits properties of both waves (diffraction, interference) and particles (photoelectric effect)
• Electrons and other subatomic particles also demonstrate wave-like behavior (electron diffraction)
• De Broglie wavelength relates particle momentum to wavelength using the equation $λ = h/p$
• Double-slit experiment demonstrates wave-particle duality for both light and matter

Mathematical Foundations of Quantum Mechanics

• Schrödinger equation serves as the fundamental equation of quantum mechanics
• Time-independent Schrödinger equation describes stationary states of quantum systems $Hψ = Eψ$
• Wave function (ψ) represents the quantum state of a particle
• Probability density given by |ψ|² determines the likelihood of finding a particle in a specific location
• Heisenberg uncertainty principle states the inherent limit in simultaneously measuring position and momentum
• Uncertainty principle expressed mathematically as $Δx · Δp ≥ h/4π$

Quantization of Energy and Atomic Structure

• Energy levels in atoms are quantized, existing only in discrete values
• Bohr model introduced the concept of quantized energy levels in hydrogen atoms
• Quantum numbers describe the energy states of electrons in atoms
• Principal quantum number (n) determines the main energy level
• Emission and absorption spectra result from electrons transitioning between energy levels
• Energy of photons emitted or absorbed calculated using $E = hν$

Atomic Orbitals and Electron Distribution

Characteristics and Types of Atomic Orbitals

• Atomic orbitals represent the quantum mechanical description of electron behavior in atoms
• Shape of orbitals determined by the angular momentum quantum number (l)
• s orbitals have spherical shape, p orbitals have dumbbell shape, d orbitals have more complex geometries
• Orbitals increase in size and energy as the principal quantum number (n) increases
• Orbital diagrams visually represent the distribution of electrons in atomic orbitals

Electron Probability and Spatial Distribution

• Electron probability distribution describes the likelihood of finding an electron in a particular region
• Radial probability distribution shows the probability of finding an electron at a specific distance from the nucleus
• Angular probability distribution indicates the directional probability of electron location
• Electron density maps provide a visual representation of electron probability distributions
• Nodes represent regions of zero electron probability within atomic orbitals

Quantum Numbers and Orbital Characteristics

• Four quantum numbers fully describe an electron's state in an atom
• Principal quantum number (n) determines the main energy level and orbital size
• Angular momentum quantum number (l) defines the subshell and orbital shape
• Magnetic quantum number (ml) specifies the orbital orientation in space
• Spin quantum number (ms) indicates the intrinsic angular momentum of an electron
• Quantum numbers are interconnected and follow specific rules and restrictions

Electron Configuration

Principles Governing Electron Arrangement

• Pauli exclusion principle states no two electrons in an atom can have the same set of quantum numbers
• Electrons in the same orbital must have opposite spins (↑↓)
• Aufbau principle dictates electrons fill orbitals from lowest to highest energy
• Energy order of orbitals follows the pattern 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p
• Hund's rule states electrons in degenerate orbitals occupy separate orbitals with parallel spins before pairing

Electron Configuration Notation and Representations

• Electron configuration notation uses spectroscopic notation (1s² 2s² 2p⁶)
• Noble gas configuration provides a shorthand for inner shell electrons ([Ar] 4s² 3d¹⁰)
• Orbital box diagrams visually represent electron distribution in orbitals
• Valence electrons are the outermost electrons involved in chemical bonding
• Core electrons are inner shell electrons not typically involved in bonding

Exceptions and Special Cases in Electron Configuration

• Chromium and copper exhibit electron configuration exceptions due to half-filled and fully-filled subshell stability
• Lanthanides and actinides have complex electron configurations involving f orbitals
• Excited state configurations occur when electrons are promoted to higher energy levels
• Ionization changes electron configuration by removing or adding electrons
• Electron configurations of transition metals often involve d orbital filling