Light and Electromagnetic Radiation Notes

Light and Electromagnetic Radiation

Introduction

• Much of our understanding of the atom's structure comes from observing how matter interacts with light.

• These studies initiated quantum mechanics and aided in developing our current atom model.

Interaction of Light with Matter

• When a sample absorbs visible light, the color we perceive is the sum of the remaining colors reflected or transmitted.

  • Opaque objects reflects light, while clear objects transmits light.

  • If an object absorbs all wavelengths of visible light, it appears black because no color reaches our eyes.

• When sunlight (white light) passes through a prism, it separates into a spectrum of colors (ROYGBIV).

Electromagnetic Spectrum

• Electromagnetic radiation involves the emission and transmission of energy through electromagnetic waves.

• It encompasses radiation across a broad range of wavelengths and frequencies.

• Maxwell (1873) proposed that visible light comprises electromagnetic waves.

• Light, or electromagnetic radiation (same thing), is a way energy travels through space.

Concept Check:

• Visible light is a type of radiation we can see, a small part of the electromagnetic spectrum.

• Other types of radiation exist but are not visible to the human eye.

Electromagnetic Spectrum Ranges:

• From shortest to longest wavelength:

  • Gamma rays (shortest)

  • X-rays

  • Ultraviolet

  • Visible light (400-700 nm)

  • Infrared

  • Microwaves

  • Radio waves (FM, Shortwave, AM) (longest)

• Visible light corresponds to wavelengths between 400 and 700 nanometers (nm), covering colors from violet to red.

Visible Light Spectrum (Shortest to Longest Wavelength):

• Violet: 400-420 nm (highest frequency).

• Indigo: 420-440 nm.

• Blue: 440-490 nm.

• Green: 490-570 nm.

• Yellow: 570-585 nm.

• Orange: 585-620 nm.

• Red: 620-780 nm (longest wavelength, lowest frequency).

• All colors blend at high frequencies to produce white light.

Characteristics of Electromagnetic Radiation:

• Low frequency, long wavelength, low quantum energy characterize radio waves. (violet)

• High frequency, short wavelength, high quantum energy characterize gamma rays and X-rays. (red)

Wave Properties

• $λ$ (Lambda) = Wavelength: Distance between identical points on successive waves (m).

• $ν$ (Nu) = Frequency: Number of waves (cycles) per second in s^-1 or Hertz (Hz).

• $c$ = Velocity or speed of light: A constant,

  approximately $3.00 × 10^8 m/s$!!!

Wavelength and Frequency Relationship:

• Wavelength and frequency are inversely related.

• Long wavelength implies low frequency and low energy.

Model 1: Wave and Wavelength

• Wavelength is the distance between two consecutive peaks or troughs of a wave.

Calculations and Conceptual Understanding:

• If a wave travels at 35 cm/sec with a wavelength ($λ$) of 2.5 cm:

  • Time for one wavelength to pass point X: $2.5 cm / (35 cm/sec) ≈ 0.071 sec$.

  • Number of wavelengths passing point X in 1 second: $(35 cm/sec) / (2.5 cm) = 14λ$ .

  • If the wavelength increases, the time to travel one wavelength increases (assuming speed stays constant).

  • If the wavelength increases, the number of wavelengths passing a point in a given time decreases (assuming constant speed).

• Frequency ($ν$) is the number of wavelengths passing a point per second ($ν = cycles/second$ or Hz).

Inverse Proportionality

• Wavelength and frequency are inversely proportional.

• Each color of light has a characteristic wavelength and frequency.

• For a wave traveling at a constant speed, frequency depends inversely on the wavelength.

• Relationship between frequency, wavelength, and speed:

  $c = λν$ or $λ = c/ν$.

  • $λ$ = nm

  • $ν$ = Hertz (Hz) or s^-1

  • $c = 3.00 × 10^8 m/s$ (speed of light)

• True or False: For waves traveling at the same speed, longer wavelength implies greater frequency (FALSE).

Examples

• Microwave ovens generate microwaves with a frequency of $2.45 × 10^9 Hz$. What is the wavelength?
  $λ = c/ν = (3.00 × 10^8 m/s) / (2.45 × 10^9 Hz) = 0.122 m = 1.22 × 10^{-1}m$

• What is the frequency of light with a wavelength of 750 nm?
  $λ = 750 nm = 7.5 × 10^{-7} m$
  $ν = c/λ = (3.00 × 10^8 m/s) / (7.5 × 10^{-7} m) = 4.00 × 10^{14} Hz$

Electromagnetic Radiation and Photons

• Light can be considered an electromagnetic wave with a specific wavelength and frequency.

• Electromagnetic waves consist of oscillating electric and magnetic fields.

  • The electric field oscillates in one plane (e.g., x-y plane) like a cosine wave.

  • The magnetic field oscillates in an orthogonal plane (e.g., x-z plane) like a cosine wave.

• Einstein proposed that electromagnetic radiation could also be viewed as a stream of particles called photons.

• Each photon has a particular amount of energy associated with it.

Photon Energy

• $E_{photon} = hν$, where:

  • $E_{photon}$ is the energy of a photon.

  • $h$ is Planck's constant ($6.63 × 10^{-34} J·s$).

  • $ν$ is the frequency of the radiation.

Relationship between Energy, Frequency, and Wavelength

• Since $ν = c/λ$, the energy of a photon can also be expressed as: $E_{photon} = hc/λ$

Tables of Electromagnetic Radiation

Table 1: Wavelengths, Frequencies, and Energies

Table 2: Regions of the Electromagnetic Spectrum

Units

• 1 nm = $10^{-9}$ m

• 1 m = $10^9$ nm

• The Joule (J) is a unit of energy: 1 J = 1 kg·m²/s² = 1 kPa·L = 1 N·m

Examples

1. A photon with a wavelength of 100 nm is classified in the ultraviolet region.

2. Orange light from a sodium vapor lamp has a wavelength of 589 nm. What is the energy of a single photon?

   • $λ = 589 nm = 5.89 × 10^{-7} m$

   • $ν = c/λ = (3.00 × 10^8 m/s) / (5.89 × 10^{-7} m) = 5.09 × 10^{14} Hz$

   • $E = hν = (6.63 × 10^{-34} J·s) × (5.09 × 10^{14} Hz) = 3.38 × 10^{-19} J$

3. Comparing the energy of red (700 nm) and blue (400 nm) photons:

   • Red photon:

     • $λ = 700 nm = 7 × 10^{-7} m$

     • $ν = c/λ = (3.00 × 10^8 m/s) / (7 × 10^{-7} m) = 4.29 × 10^{14} Hz$

     • $E = hν = (6.63 × 10^{-34} J·s) × (4.29 × 10^{14} Hz) = 2.84 × 10^{-19} J$

   • Blue photon:

     • $λ = 400 nm = 4 × 10^{-7} m$

     • $ν = c/λ = (3.00 × 10^8 m/s) / (4 × 10^{-7} m) = 7.5 × 10^{14} Hz$

     • $E = hν = (6.63 × 10^{-34} J·s) × (7.5 × 10^{14} Hz) = 4.97 × 10^{-19} J$

   • The blue photon is more energetic.

The Photoelectric Effect

• Einstein's analysis of the photoelectric effect led him to his conclusions about light.

• In the photoelectric effect, incident light ejects electrons from a material.

  • The photon must have sufficient energy to eject the electron.

  • Photoelectron spectroscopy determines the energy needed to eject electrons and provides information about the atom's structure.

  • The intensity of the photoelectron is a measure of the number of electrons in that energy level.

• $KE = hν - W$, where:

  • $KE$ is the kinetic energy of the ejected electron.

  • $W$ is the work function, which depends on how strongly electrons are held in the metal.

Dual Nature of Light

• Electromagnetic radiation exhibits both particle and wave properties.

• Light travels through space as a wave and transmits energy as a particle.

De Broglie's Equation

• $λ = h/(mv)$ , where:

  • $λ$ is the wavelength.

  • $h$ is Planck's constant ($6.63 × 10^{-34} J·s$).

  • $m$ is the mass of the particle.

  • $v$ is the velocity of the particle.

• Wavelength can be related to mass.

• Combining $E=mc^2$ and $E = hc/λ$ gives $m = h/(λc)$.

• For a particle not moving at the speed of light: $λ = h/(mv)$

Examples

1. An element absorbs energy at a wavelength of 150 nm, and the total energy emitted is $1.98 × 10^5 J$. Calculate the number of C atoms.

   • Energy of a single photon: $E = (6.63 × 10^{-34} J·s) × (3.00 × 10^8 m/s) / (1.50 × 10^{-7} m) = 1.33 × 10^{-18} J/photon$

   • Number of photons: $(1.98 × 10^5 J) / (1.33 × 10^{-18} J/photon) = 1.49 × 10^{23} photons$

   • Number of C atoms: $1.49 × 10^{23}$ C atoms (assuming each atom emits one photon).

2. What is the wavelength of an electron (mass = $9.11 × 10^{-31} kg$) traveling at $5.31 × 10^6 m/s$?

   • $λ = (6.63 × 10^{-34} J·s) / ((9.11 × 10^{-31} kg) × (5.31 × 10^6 m/s)) = 1.37 × 10^{-10} m = 0.137 nm$

Atomic Emission Spectrum

A. Continuous Spectrum

1. Results when white light is passed through a prism.

2. Contains all wavelengths of visible light, resembling a rainbow.

3. Produced by sunlight or incandescent light.

4. A prism separates light into its constituent colors.

B. Line Spectrum

1. Light from a helium lamp produces discrete lines.

2. All elements emit light when energy is added.

3. Each element has a unique emission spectrum, allowing identification based on emitted colors.

4. Each line corresponds to a discrete wavelength.

5. Elements produce line spectra, not continuous spectra.

6. A line spectrum contains only a few lines, each corresponding to a discrete wavelength.

• The type of spectrum (emission or absorption) depends on the temperature of the gas relative to the background.

C. How Elements Emit Light in the Gas State

• Elements emit colors when heated or vaporized.

• Electrons in atoms can only have certain allowed energies.

• Heating an atom excites its electrons, causing them to jump to higher energy levels (excited state).

• When electrons return to lower energy levels (ground state), they emit energy (radiation) in the form of light.

Bohr Model

• Electrons in a hydrogen atom move around the nucleus in certain allowed circular orbits.

• Bohr's model provided energy levels consistent with the hydrogen emission spectrum.

• Ground state: The lowest possible energy state (n = 1).

• Bright-line spectra confirm that only certain energies exist in the atom; the atom emits photons with definite wavelengths when the electron returns to a lower energy state.

Energy Levels

• Energy levels available to the electron in an atom:

  • $E = -2.178 × 10^{-18} J (Z^2 / n^2)$

  • $E$ = energy in J

  • $Z$ = nuclear charge (number of protons)

  • $n$ = integer related to orbital position (energy level).

Example

• Calculate the energy corresponding to the n=3 electronic state in the Bohr hydrogen atom.

  • For hydrogen, Z = 1.

  • $E = -2.178 × 10^{-18} J (1^2 / 3^2) = -2.42 × 10^{-19} J$

Rydberg Formula

• For a single electron transition in hydrogen from one energy level to another, use the Rydberg formula:

  • $ΔE = R<em>H (1/n</em>{initial}^2 - 1/n_{final}^2)$, where:

    • $ΔE$ is the energy absorbed or released.

    • $R_H$ is the Rydberg constant ($2.178 × 10^{-18} J$).

    • $n_{initial}$ is the initial energy level.

    • $n_{final}$ is the final energy level.

Example

• Calculate the energy change for an electron excited from n=1 to n=3 in the hydrogen atom.

  • $ΔE = 2.178 × 10^{-18} J (1/1^2 - 1/3^2) = 1.94 × 10^{-18} J$

Example

• What wavelength of electromagnetic radiation is associated with this energy change?

  • $E = hc/λ → λ = hc/E$

  • $λ = (6.63 × 10^{-34} J·s) × (3.00 × 10^8 m/s) / (1.94 × 10^{-18} J) = 1.03 × 10^{-7} m = 103 nm$

Shortcomings of the Bohr Model

• The model only works for hydrogen.

• Electrons do not move around the nucleus in circular orbits.

Quantum Mechanical Model

• We do not know the detailed pathway of an electron.

• Heisenberg uncertainty principle: There is a fundamental limitation to how precisely we can know both the position and momentum of a particle at a given time.