Electromagnetic Radiation, Atomic Structure, and Quantum Theory: Study Notes

Electromagnetic Radiation

Nature and Properties of Electromagnetic Radiation

Electromagnetic radiation consists of oscillating electric and magnetic fields that propagate through space as waves. The properties of these waves are characterized by their wavelength, frequency, and energy.

• Wavelength (λ): The distance between successive crests of a wave, measured in meters (m).

• Frequency (ν): The number of wave cycles that pass a given point per second, measured in hertz (Hz).

• Speed of Light (c): All electromagnetic waves travel at the speed of light in a vacuum, $c = 3.00 \times 10^8$ m/s.

Relationship between wavelength, frequency, and speed:

$c = \lambda \nu$

As wavelength increases, frequency decreases, and vice versa.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, from radio waves to gamma rays. Different regions of the spectrum are characterized by their wavelength and frequency.

• Radio waves: Longest wavelength, lowest frequency, lowest energy per photon.

• Microwaves

• Infrared radiation

• Visible light: The portion of the spectrum visible to the human eye (approximately 400–700 nm).

• Ultraviolet radiation

• X-rays

• Gamma rays: Shortest wavelength, highest frequency, highest energy per photon.

Order of increasing frequency (and decreasing wavelength): radio waves < microwaves < infrared < visible < ultraviolet < X-rays < gamma rays

Order of increasing energy per photon: Same as increasing frequency.

Energy of Electromagnetic Radiation

The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength.

$E = h \nu$

$E = \frac{hc}{\lambda}$

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

Example: Calculate the energy of a photon with a wavelength of 532 nm.

$E = \frac{(6.626 \times 10^{-34} \text{ J·s})(3.00 \times 10^8 \text{ m/s})}{532 \times 10^{-9} \text{ m}}$

The Wave Nature of Matter and the Uncertainty Principle

Wave-Particle Duality

Electrons and other particles exhibit both wave-like and particle-like properties. This duality is demonstrated by interference patterns observed when electrons pass through closely spaced slits, similar to light waves.

• Interference patterns: Alternating bright and dark bands resulting from constructive and destructive interference of waves.

De Broglie Wavelength

Louis de Broglie proposed that all matter has a wavelength associated with its momentum.

$\lambda = \frac{h}{mv}$

• m: mass of the particle (kg)

• v: velocity of the particle (m/s)

For macroscopic objects, the de Broglie wavelength is extremely small and not observable, but for electrons and other subatomic particles, it is significant.

Heisenberg Uncertainty Principle

The uncertainty principle states that it is impossible to simultaneously know both the exact position and exact momentum of a particle.

$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$

• Δx: uncertainty in position

• Δp: uncertainty in momentum

This principle is fundamental to quantum mechanics and explains why electrons do not have precise orbits.

Bohr’s Atomic Model

Structure and Postulates

Niels Bohr proposed a model of the atom in which electrons orbit the nucleus in fixed energy levels without radiating energy. Transitions between these levels involve absorption or emission of photons.

• Electrons occupy specific orbits with quantized energies.

• Energy is absorbed or emitted only when an electron changes orbits.

• The energy difference between orbits determines the wavelength of emitted or absorbed light.

Energy Levels and Spectra

The energy of an electron in a hydrogen atom is given by:

$E_n = -\frac{2.178 \times 10^{-18} \text{ J}}{n^2}$

• n: principal quantum number (n = 1, 2, 3, ...)

When an electron transitions from a higher energy level ($n_i$) to a lower one ($n_f$), a photon is emitted with energy:

$\Delta E = E_{n_f} - E_{n_i}$

The wavelength of the emitted photon is:

$\lambda = \frac{hc}{\Delta E}$

Rydberg Equation for Hydrogen Spectrum

The Rydberg equation predicts the wavelengths of spectral lines in hydrogen:

$\frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$

• R: Rydberg constant ($1.097 \times 10^7$ m-1)

• ni: initial energy level (ni > nf)

• nf: final energy level

Quantum Numbers and Atomic Orbitals

Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the electrons within them:

• Principal quantum number (n): Indicates the main energy level (shell).

• Angular momentum quantum number (l): Indicates the shape of the orbital (0 = s, 1 = p, 2 = d, 3 = f).

• Magnetic quantum number (ml): Indicates the orientation of the orbital.

• Spin quantum number (ms): Indicates the spin of the electron (+1/2 or -1/2).

n	l	ml	ms	Number of Orbitals
1	0	0	+1/2, -1/2	1 (s)
2	0, 1	-1, 0, +1	+1/2, -1/2	1 (s), 3 (p)
3	0, 1, 2	-2, -1, 0, +1, +2	+1/2, -1/2	1 (s), 3 (p), 5 (d)

Atomic Orbitals and Electron Configuration

Atomic orbitals are regions in space where there is a high probability of finding an electron. The arrangement of electrons in an atom is described by its electron configuration, following the Aufbau principle, Pauli exclusion principle, and Hund's rule.

• Aufbau principle: Electrons fill the lowest energy orbitals first.

• Pauli exclusion principle: No two electrons in an atom can have the same set of four quantum numbers.

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

Example: Electron configuration of carbon (Z = 6): 1s2 2s2 2p2

Atomic Spectroscopy

Emission and Absorption Spectra

When atoms absorb energy, electrons are promoted to higher energy levels. When they return to lower levels, they emit photons of specific energies, producing an emission spectrum. Each element has a unique line spectrum.

• Emission spectrum: Bright lines on a dark background, corresponding to specific wavelengths emitted by excited atoms.

• Absorption spectrum: Dark lines on a continuous spectrum, corresponding to wavelengths absorbed by atoms.

Applications

• Identification of elements in stars and other samples.

• Understanding atomic structure and energy levels.

Summary Table: Key Equations

Equation	Description
$c = \lambda \nu$	Relationship between speed, wavelength, and frequency
$E = h \nu$	Energy of a photon
$E = \frac{hc}{\lambda}$	Energy of a photon in terms of wavelength
$\lambda = \frac{h}{mv}$	De Broglie wavelength
$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$	Heisenberg uncertainty principle
$\frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$	Rydberg equation for hydrogen
$E_n = -\frac{2.178 \times 10^{-18} \text{ J}}{n^2}$	Energy of electron in hydrogen atom

Additional info: These notes integrate worksheet questions, diagrams, and textbook-style explanations to provide a comprehensive review of electromagnetic radiation, quantum theory, and atomic structure for General Chemistry students.