6.1 Electromagnetic Energy

Properties and Behavior of Waves and Light

Properties of waves

Waves transport energy from one location to another without transporting matter. Sound waves and water waves are everyday examples. A traveling wave moves through a medium and is characterized by three key properties:

• Wavelength ($\lambda$): the distance between two consecutive crests (or two consecutive troughs). Measured in meters.
• Frequency ($\nu$): the number of wave cycles passing a fixed point per second. Measured in Hertz (Hz), where 1 Hz = 1 cycle per second.
• Amplitude: the maximum displacement of the wave from its resting (equilibrium) position. Larger amplitude means the wave carries more energy.

Standing waves form when two waves traveling in opposite directions overlap, creating a pattern that appears stationary. Nodes are points where the wave has zero displacement, and antinodes are points of maximum displacement.

Wave-particle duality of light

Light behaves as both a wave and a particle. This is called wave-particle duality, and it's one of the trickiest ideas in this unit.

Wave-like behaviors of light include:

• Diffraction: light bending around obstacles or through narrow slits
• Interference: two light waves overlapping to produce constructive (brighter) or destructive (dimmer) patterns
• Refraction: light bending as it passes from one medium to another (like air into water)

Particle-like behavior shows up when light acts as individual packets of energy called photons. The photoelectric effect is the classic demonstration: light shining on a metal surface ejects electrons, but only if the light's frequency is above a certain threshold. Increasing the brightness (more photons) ejects more electrons, but each photon still needs enough energy on its own. This couldn't be explained by a wave-only model.

Electromagnetic radiation and spectrum

Electromagnetic (EM) radiation is energy that travels through space as waves. Unlike sound, it doesn't need a medium and can travel through a vacuum.

The electromagnetic spectrum organizes all EM radiation by wavelength and frequency:

• Long wavelength / low frequency / low energy: radio waves → microwaves → infrared
• Visible light sits in a narrow band in the middle (roughly 400–700 nm)
• Short wavelength / high frequency / high energy: ultraviolet → X-rays → gamma rays

The key relationship to remember: as wavelength increases, frequency decreases, and vice versa. Energy follows frequency, so shorter wavelengths mean higher energy.

Energy in EM radiation is quantized, meaning it comes in discrete packets (quanta) rather than any arbitrary amount. This idea was revolutionary and laid the groundwork for quantum mechanics.

Quantitative Analysis and Spectra

Light calculations

Two equations do most of the heavy lifting in this section.

Equation 1: Speed of light

$c = \lambda\nu$

• $c$ = speed of light = $3.00 \times 10^8$ m/s
• $\lambda$ = wavelength (in meters)
• $\nu$ = frequency (in Hz, which is $\text{s}^{-1}$)

Since $c$ is constant, this equation tells you that wavelength and frequency are inversely related. If you know one, you can find the other.

Equation 2: Energy of a photon

$E = h\nu$

• $E$ = energy (in Joules)
• $h$ = Planck's constant = $6.626 \times 10^{-34}$ J·s
• $\nu$ = frequency (in Hz)

You can combine both equations to relate energy directly to wavelength: $E = \frac{hc}{\lambda}$. This is useful when a problem gives you wavelength and asks for energy.

Typical problem steps:

1. Identify what you're given (wavelength, frequency, or energy).
2. Convert units if needed (nm to m is common: 1 nm = $10^{-9}$ m).
3. Use $c = \lambda\nu$ to find whichever of wavelength or frequency you're missing.
4. Use $E = h\nu$ to find the photon's energy.

Line vs. continuous emission spectra

When electrons in a substance drop from higher to lower energy levels, they release energy as light. The pattern of that light is called an emission spectrum.

• Line emission spectra show only specific, discrete wavelengths of light, appearing as distinct colored lines against a dark background. Gases at low pressure produce these. Each element has a unique line spectrum, like a fingerprint. Hydrogen's visible lines (the Balmer series) are a classic example.
• Continuous emission spectra contain an unbroken range of wavelengths with no gaps. Hot solids, liquids, and high-pressure gases produce these. An incandescent light bulb is a good example.

The difference matters because line spectra are direct evidence that electrons in atoms occupy discrete energy levels, not a continuous range.

Photons and energy levels in quantum chemistry

Electrons in atoms can only exist at specific, quantized energy levels. When an electron jumps between levels:

• Absorbing a photon moves the electron to a higher energy level.
• Emitting a photon happens when the electron drops to a lower energy level.

The energy of the absorbed or emitted photon exactly equals the energy difference between the two levels. This is why line spectra have specific wavelengths: each line corresponds to a particular electron transition.

Because every element has a unique set of energy levels, every element produces a unique line spectrum. This is how scientists identify elements in distant stars or unknown samples.

Quantum numbers describe the specific energy levels and orbitals electrons occupy:

1. Principal quantum number ($n$): the main energy level (shell). Higher $n$ = higher energy and farther from the nucleus.
2. Angular momentum quantum number ($l$): the shape of the orbital (0 = s, 1 = p, 2 = d, 3 = f).
3. Magnetic quantum number ($m_l$): the orientation of the orbital in space.
4. Spin quantum number ($m_s$): the direction of electron spin ($+\frac{1}{2}$ or $-\frac{1}{2}$).