Atoms: The Quantum World – Electronic Structure and Quantum Theory

Atoms: The Quantum World

Introduction

This topic explores the modern understanding of atomic structure, focusing on the quantum mechanical model of the atom. It covers the interaction of electromagnetic radiation with matter, the quantization of energy, atomic emission spectra, and the development of quantum theory, including the Bohr model and quantum numbers.

The Structure of the Atom

Discovery of the Atomic Nucleus

• Gold Foil Experiment (1911, Ernest Rutherford): Demonstrated that atoms have a small, dense, positively charged nucleus.

• Most alpha particles passed through gold foil, but some were deflected, indicating a concentrated positive charge (the nucleus).

The Modern View of the Atom

• Atoms consist of protons (positively charged) and neutrons (neutral) in the nucleus, and electrons (negatively charged) occupying the space outside the nucleus.

• Atomic radius ≈ 100 pm ($1 \times 10^{-10}$ m); nuclear radius ≈ 5 × $10^{-3}$ pm ($5 \times 10^{-15}$ m).

Particle	Charge (C)	Mass (kg)	Atomic Mass Unit (u)
Proton	+1.6022 × $10^{-19}$	1.6726 × $10^{-27}$	1.0073
Neutron	0	1.6749 × $10^{-27}$	1.0087
Electron	-1.6022 × $10^{-19}$	9.1094 × $10^{-31}$	0.00054858

Electrons in Atoms and Electromagnetic Radiation

Electrons in Chemical Reactions

• In chemical reactions (except nuclear reactions), only electrons participate; nuclei remain unchanged.

Electromagnetic (EM) Radiation

• EM radiation is energy transmitted as oscillating electric and magnetic fields (waves) through space or a medium.

• Wavelength (λ): Distance between identical points on successive waves (meters).

• Frequency (ν): Number of waves passing a point per second (Hz).

• Amplitude (A): Maximum displacement from the wave's midpoint.

• Velocity of light (c): $2.997925 \times 10^8$ m/s (≈ $3.00 \times 10^8$ m/s).

Key Equations:

• $c = \lambda \nu$

• $\lambda = \frac{c}{\nu}$

• $\nu = \frac{c}{\lambda}$

The Electromagnetic Spectrum

• Ranges from gamma rays (shortest wavelength, highest energy) to radio waves (longest wavelength, lowest energy).

• Visible light: 380–760 nm (violet to red).

Wave Properties: Diffraction and Interference

Diffraction and Interference

• Diffraction: Bending of waves around obstacles or through apertures.

• Interference: Combination of two or more waves to form a composite wave.

• Constructive interference: Waves in phase add together (bright bands).

• Destructive interference: Waves out of phase cancel each other (dark bands).

• Young’s Double Slit Experiment: Demonstrates the wave nature of light via alternating bright and dark bands.

Examples of Diffraction

• Dispersion of Light: Passing white light through a prism separates it into its component colors (visible spectrum).

• X-ray Diffraction: X-rays scattered by a crystal produce a pattern that reveals atomic structure.

Prelude to Quantum Theory

Quantization of Energy

• Classical physics could not explain phenomena like the photoelectric effect.

• Max Planck (1900): Energy is quantized; it can only be absorbed or emitted in discrete amounts called quanta.

• Planck’s Equation:

$E = h\nu$

• Where $E$ is energy, $\nu$ is frequency, $h$ is Planck’s constant ($6.62607 \times 10^{-34}$ J·s).

• Higher frequency means higher energy; energy is not continuous but quantized.

Photoelectric Effect and Photons

• Heinrich Hertz (1887): Light striking metal surfaces can eject electrons (photoelectric effect).

• Albert Einstein (1905): Light has particle-like properties; energy is carried in packets called photons.

• A photon must have energy greater than the work function (binding energy) to eject an electron; excess energy appears as kinetic energy.

Atomic Emission Spectra

Continuous and Line Spectra

• Continuous spectrum: Contains all wavelengths (e.g., white light through a prism).

• Line spectrum: Contains only specific wavelengths; characteristic of elements (atomic fingerprint).

Flame Tests and Spectroscopy

• Heating elements excites electrons to higher energy levels; as they return to lower levels, they emit light of characteristic colors.

Ion	Flame Colour
Li+	Red
Na+	Golden yellow
K+	Lilac
Ca2+	Brick red
Sr2+	Crimson
Ba2+	Apple green
Cu2+	Blue/green
Boron (BO33-)	Green

Emission Spectra of Atomic Hydrogen

• The hydrogen spectrum is the most studied atomic spectrum; only certain energies are allowed for the electron.

• Balmer Series: Four visible lines; Balmer’s equation predicts their wavelengths:

$\nu = 3.2881 \times 10^{15} \text{ s}^{-1} \left( \frac{1}{2^2} - \frac{1}{n^2} \right)$, $n \geq 3$

The Bohr Model of the Atom

Postulates of the Bohr Model (1913)

• Electrons move in fixed circular orbits around the nucleus; each orbit has a specific energy.

• Angular momentum is quantized: $mvr = n \frac{h}{2\pi}$, $n = 1, 2, 3, ...$

• Electrons emit or absorb energy as photons when transitioning between orbits.

Bohr Model Equations

• Allowed radii for hydrogen atom:

$r_n = n^2 a_0$

• Energy of allowed orbits:

$E_n = -\frac{R_H}{n^2}$

• Where $R_H$ (Rydberg constant) = $2.17868 \times 10^{-18}$ J.

• Energy is quantized and all allowed values are negative (relative to a free electron).

Key Points

• Energy differences between orbits correspond to photon energies ($\Delta E = h\nu$).

• The Bohr model explains the line spectra of hydrogen and hydrogen-like ions.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Speed of Light	$c = \lambda \nu$	Relates wavelength and frequency
Photon Energy	$E = h\nu$	Energy of a photon
Bohr Energy Levels	$E_n = -\frac{R_H}{n^2}$	Energy of nth orbit in hydrogen
Balmer Equation	$\nu = 3.2881 \times 10^{15} \left( \frac{1}{2^2} - \frac{1}{n^2} \right)$	Frequency of hydrogen spectral lines

Example Applications

• Calculating Frequency: For light with $\lambda = 589$ nm, $\nu = \frac{c}{\lambda}$.

• Photon Energy: For $\lambda = 242.4$ nm, $E = \frac{hc}{\lambda}$.

• Hydrogen Energy Levels: Is there an energy level for $E = -1.00 \times 10^{-19}$ J? Use $E_n = -\frac{R_H}{n^2}$ to solve for $n$.

Additional info: These notes cover the foundational quantum concepts necessary for understanding atomic structure, electron configurations, and periodic trends, as outlined in a typical General Chemistry I syllabus.