Electronic Structure of Atoms

Introduction to Electronic Structure

The electronic structure of atoms refers to the arrangement and energy of electrons within an atom. Understanding this structure is essential for explaining chemical properties and behaviors. The study of electronic structure begins with the nature of electromagnetic radiation and extends to quantum mechanical models of the atom.

The Wave Nature of Light

Electromagnetic Radiation

Electromagnetic radiation consists of oscillating electric and magnetic fields that travel through space at the speed of light. Key properties include:

• Wavelength (λ): The distance between corresponding points on adjacent waves.

• Frequency (ν): The number of waves passing a given point per unit time.

• Speed of Light (c): All electromagnetic radiation travels at the same speed in a vacuum, given by .

• Relationship: Wavelength and frequency are related by .

As wavelength increases, frequency decreases, and vice versa.

Common Wavelength Units

Electromagnetic radiation is classified by wavelength, with units such as meters (m), nanometers (nm), and angstroms (Å). Different regions of the electromagnetic spectrum (radio, microwave, infrared, visible, ultraviolet, X-ray, gamma ray) have characteristic wavelength ranges.

Quantized Energy and Photons

Blackbody Radiation and the Quantum Hypothesis

Classical physics could not explain why heated objects emit light (blackbody radiation) only at certain wavelengths. Max Planck proposed that energy is quantized and comes in discrete packets called quanta (singular: quantum).

• Planck's Equation: where is Planck's constant ().

The Photoelectric Effect

Einstein explained the photoelectric effect by proposing that light consists of particles called photons. Electrons are ejected from metal surfaces only if the light has a frequency above a certain threshold, supporting the quantization of energy.

Atomic Emission Spectra

Atoms emit light at specific wavelengths, producing a line spectrum unique to each element. This observation could not be explained by classical physics and led to the development of quantum models of the atom.

Line Spectra and the Bohr Model

The Hydrogen Spectrum

Johann Balmer and Johannes Rydberg developed mathematical relationships for the emission lines of hydrogen. Niels Bohr explained these lines by proposing that electrons occupy only certain allowed orbits with specific energies.

The Bohr Model

• Electrons move in circular orbits around the nucleus with quantized energies.

• Energy is emitted or absorbed when an electron transitions between orbits: .

• The lowest energy state is the ground state (); higher energy states are excited states ().

A positive means energy is absorbed (photon absorbed); a negative $\Delta E$ means energy is released (photon emitted).

Limitations of the Bohr Model

• Accurately describes only hydrogen (one-electron systems).

• Does not account for electron-electron interactions in multi-electron atoms.

• Electrons exhibit wave-like properties not included in the Bohr model.

The Wave Behavior of Matter

de Broglie Hypothesis

Louis de Broglie proposed that particles, such as electrons, have wave properties. The wavelength of a particle is given by:

where is mass and is velocity.

Heisenberg Uncertainty Principle

Werner Heisenberg showed that it is impossible to know both the exact position and momentum of a particle simultaneously:

This principle is fundamental to quantum mechanics.

Quantum Mechanics and Atomic Orbitals

Schrödinger Equation and Wave Functions

Erwin Schrödinger developed a mathematical model (the Schrödinger equation) that describes electrons as wave functions. The square of the wave function gives the probability density of finding an electron in a particular region of space.

Quantum Numbers

Each atomic orbital is described by a set of quantum numbers:

• Principal Quantum Number (n): Indicates the energy level and size of the orbital ().

• Angular Momentum Quantum Number (l): Defines the shape of the orbital (). For example, (s), (p), (d), (f).

• Magnetic Quantum Number (): Describes the orientation of the orbital in space ().

• Spin Quantum Number (): Describes the spin of the electron; can be or .

Summary of Quantum Numbers

Orbitals with the same value of form an electron shell. Different orbital types within a shell are called subshells.

Representation of Orbitals

s Orbitals

s orbitals () are spherical in shape. The number of radial nodes increases with .

p Orbitals

p orbitals () have two lobes with a node at the nucleus.

d and f Orbitals

d orbitals () have more complex shapes, typically four lobes. f orbitals () are even more complex and have seven orientations.

Hydrogen Atom Orbital Energies

For hydrogen, all orbitals with the same have the same energy (they are degenerate).

Many-Electron Atoms

In atoms with more than one electron, electron-electron repulsions cause orbitals with the same to have different energies. Sublevels (s, p, d, f) split in energy, and energy levels can overlap (e.g., 4s is lower than 3d).

Spin Quantum Number and the Pauli Exclusion Principle

The spin quantum number () describes the intrinsic angular momentum of electrons. The Pauli Exclusion Principle states that no two electrons in the same atom can have the same set of four quantum numbers. This means each orbital can hold a maximum of two electrons with opposite spins.

Electron Configurations

Ground State Electron Configuration

The arrangement of electrons in an atom is called its electron configuration. The most stable (lowest energy) arrangement is the ground state. Electron configurations are written as:

• Number (energy level) + letter (type of orbital) + superscript (number of electrons)

• Example:

Orbital Diagrams and Hund’s Rule

Orbital diagrams use boxes to represent orbitals and arrows for electrons. Hund’s Rule states that electrons fill degenerate orbitals singly first, with parallel spins, before pairing up.

Condensed Electron Configurations

Core electrons (inner shells) are represented by the symbol of the preceding noble gas in brackets. Only valence electrons are written explicitly.

Transition Metals, Lanthanides, and Actinides

Transition metals fill d orbitals after s orbitals. Lanthanides and actinides fill f orbitals. Some elements show anomalies in electron configurations due to the close energy of s, d, and f orbitals (e.g., chromium).

Electron Configurations and the Periodic Table

The periodic table reflects the order in which orbitals are filled. The s and p blocks are main-group elements, d block is transition metals, and f block is inner transition elements. Electron configurations can be determined directly from the periodic table.