Quantum-Mechanical Model of the Atom and the Nature of Light

Science Connections

Context and Relevance

Understanding chemistry requires connecting scientific concepts to real-world applications and current research. This course integrates podcasts and readings from leading scientists to provide context and insight into the development and impact of chemical knowledge.

• Podcasts and readings are assigned to supplement course material.

• Post-reading/listening questions are testable and may appear on quizzes or tests.

Quantum-Mechanical Model of the Atom

Overview

The quantum-mechanical model describes the structure and behavior of atoms using principles of quantum theory. This model evolved from classical models as new evidence from spectroscopy and the study of light emerged.

• Key topics include the nature of light, atomic spectroscopy, the Bohr model, wave-particle duality, quantum mechanics, atomic orbitals, and electron configurations.

Atomic Theories from the Last Two Centuries

Historical Development

Atomic theory has evolved through contributions from several scientists:

• John Dalton (1804): Proposed that matter is composed of indivisible atoms.

• J.J. Thomson (1903): Discovered the electron; proposed the "plum pudding" model.

• Ernest Rutherford (1911): Demonstrated the existence of a dense, positively charged nucleus.

• Niels Bohr (1913): Introduced quantized electron orbits to explain atomic spectra.

• Erwin Schrödinger (1926): Developed the quantum-mechanical model using wave equations.

Laws, Theories, and Models in Science

Definitions and Differences

• Law: A statement based on repeated experimental observations that describes some aspect of the world (e.g., Law of Conservation of Mass).

• Theory: A well-substantiated explanation of some aspect of the natural world that can incorporate laws, hypotheses, and facts (e.g., Atomic Theory).

• Model: A simplified representation used to explain and predict phenomena (e.g., Bohr Model of the atom).

Nature of Light

Electromagnetic Radiation

Light is a form of electromagnetic radiation, which exhibits both wave-like and particle-like properties.

• Electromagnetic (EM) Spectrum: The range of all types of EM radiation, from gamma rays to radio waves.

• Speed of Light: In a vacuum, $c = 3.00 \times 10^8$ m/s.

Wave-Particle Duality

• Light can behave as both a wave and a particle (photon).

• Key relationships:

  • Wavelength ($\lambda$) and frequency ($\nu$): $c = \lambda \nu$

  • Energy of a photon: $E = h\nu = \dfrac{hc}{\lambda}$

  • Where $h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s).

Atomic Spectroscopy and the Bohr Model

Interaction of Light and Matter

• Spectroscopy: The study of the interaction between light and matter, often visualized as spectra (intensity vs. wavelength).

• Absorption: Atoms absorb energy and electrons move to higher energy levels (excited state).

• Emission: Electrons return to lower energy levels, emitting photons of specific energies.

Bohr Model

• Electrons occupy quantized orbits with specific energies.

• Energy difference between orbits corresponds to photon energy: $\Delta E = E_{final} - E_{initial}$

• Explains line spectra of hydrogen and other elements.

The Wave Nature of Matter

de Broglie Wavelength

• All matter exhibits wave-like properties.

• de Broglie equation: $\lambda = \dfrac{h}{mv}$

• Where $m$ is mass and $v$ is velocity.

Heisenberg Uncertainty Principle

• It is impossible to know both the exact position and momentum of a particle simultaneously.

• Mathematically: $\Delta x \cdot m \Delta v \geq \dfrac{h}{4\pi}$

Quantum Mechanics and Electrons in Atoms

Schrödinger Equation and Atomic Orbitals

• Electrons are described by wave functions ($\psi$), which give the probability of finding an electron in a region of space.

• Orbitals are regions where electrons are most likely to be found.

• Each orbital can hold a maximum of 2 electrons.

Shape of Atomic Orbitals

Types of Orbitals

• s orbitals: Spherical shape, 0 nodal planes.

• p orbitals: Dumbbell shape, 1 nodal plane, 3 orientations (px, py, pz).

• d orbitals: Cloverleaf shape, 2 nodal planes, 5 orientations.

• f orbitals: Complex shapes, 3 nodal planes, 7 orientations.

Quantum Numbers

Describing Electrons in Atoms

• Each electron in an atom is described by a unique set of four quantum numbers:

Quantum Number	Symbol	Description	Allowed Values
Principal	n	Shell (energy level)	1, 2, 3, ...
Angular Momentum	l	Subshell (shape)	0 to n-1
Magnetic	m_l	Orbital orientation	-l to +l
Spin	m_s	Electron spin	+1/2 or -1/2

Electron Configurations

Filling Order and Rules

• Electrons fill orbitals in order of increasing energy (Aufbau principle).

• Hund's Rule: Electrons occupy degenerate orbitals singly before pairing.

• Pauli Exclusion Principle: No two electrons in the same atom can have the same set of four quantum numbers.

Methods of Representation

• Energy Level Diagram: Shows relative energies of shells and subshells.

• Orbital Diagram: Uses arrows to represent electron spins in orbitals.

• Expanded Electron Configuration: Lists all occupied subshells (e.g., 1s2 2s2 2p6).

• Condensed Electron Configuration: Uses noble gas core notation (e.g., [Ne] 3s2 3p3).

Exceptions

• Some transition metals (e.g., Cr, Cu) have electron configurations that deviate from the expected order due to stability associated with half-filled or fully filled subshells.

Summary Table: Quantum Numbers and Orbitals

Orbital Type	l	Number of Orientations (ml)	Shape
s	0	1 (0)	Sphere
p	1	3 (-1, 0, +1)	Dumbbell
d	2	5 (-2 to +2)	Cloverleaf
f	3	7 (-3 to +3)	Complex

Key Equations

• Speed of light: $c = \lambda \nu$

• Photon energy: $E = h\nu = \dfrac{hc}{\lambda}$

• de Broglie wavelength: $\lambda = \dfrac{h}{mv}$

• Heisenberg uncertainty: $\Delta x \cdot m \Delta v \geq \dfrac{h}{4\pi}$

Example: Electron Configuration of Phosphorus (Z = 15)

• Expanded: 1s2 2s2 2p6 3s2 3p3

• Condensed: [Ne] 3s2 3p3

• Number of unpaired electrons: 3

• Number of valence electrons: 5