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

Wavelength (nm)

Frequency (Hz)

Energy (J)

333.1

9.000 × 10^{14}

5.963 × 10^{-19}

499.7

6.000 × 10^{14}

3.976 × 10^{-19}

999.3

3.000 × 10^{14}

1.988 × 10^{-19}

Table 2: Regions of the Electromagnetic Spectrum

Region

Wavelength Range

Radio wave

3 km - 30 cm

Microwave

30 cm - 1 mm

Infrared (IR)

1 mm - 800 nm

Visible (VIS)

780 nm - 400 nm

Ultraviolet (UV)

400 nm - 10 nm

X-Ray

10 nm - 0.10 nm

Gamma Ray

< 0.1 nm

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 = RH (1/n{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.