M

Quantum Theory and Electronic Structure of Atoms

Section 2: Quantum Theory and Electronic Structure of Atoms

2.1 From Classical Physics to Quantum Theory

Limitations of Classical Physics in the Late 19th Century

  • Matter: Viewed as particles with precisely knowable location and momentum.
  • Molecules: Classical physics could not explain the forces holding molecules together.
  • Energy Emission/Absorption: Assumed atoms/molecules could emit or absorb any type and extent of energy continuously.

Max Planck's Revolutionary Proposals (German physicist)

  • Quantization of Energy: Molecules emit energy only in discrete quantities, called quanta.
  • Heisenberg Uncertainty Principle (implied): One cannot know both the exact position and momentum of a particle simultaneously.
  • Consequence: When matter is heated, it emits one or more quanta of energy.

Properties of Waves

  • A wave is a vibrating disturbance that transmits energy (not matter).
  • Amplitude: Height of the wave, related to the intensity or brightness of radiation.
  • Frequency (\nu): Number of cycles per second, measured in s^{-1} or Hertz (Hz).
  • Wavelength (\lambda): Distance a wave travels in one complete cycle.
  • Speed of Wave (u): For light, denoted as c, a constant value of 3.00 \times 10^8 \text{ m/s}.
  • Relationship: The speed of light is the product of its wavelength and frequency: c = \lambda \times \nu

Electromagnetic Radiation

  • Definition: All forms of radiation described by wave-like electric and magnetic fields.
  • Examples: Visible light, X-rays, microwaves, cell phone signals.
  • Common Properties: All electromagnetic (EM) radiation shares fundamental wave properties.

The Electromagnetic Spectrum

  • Characterization: EM waves are characterized by their frequency or wavelength.
  • Visible Spectrum: A very narrow band of wavelengths our eyes can detect.
    • Ranges from approximately 380 \text{ nm} (purple) to 750 \text{ nm} (red).
  • Black Body Radiation: Hot objects emit electromagnetic radiation; the spectrum depends on temperature.

Planck's Quantum Theory

  • Core Idea: The energy of EM waves is quantized rather than continuous.
  • Quantum: A discrete bundle or packet of energy.
  • Photon: A quantum of electromagnetic energy.
  • Planck's Equation: The energy (E) of a single photon is directly proportional to its frequency (\nu): E = h\nu
    • Planck's Constant (h): A fundamental physical constant with a value of 6.626 \times 10^{-34} \text{ J\cdot s}.
  • Example Calculation: Energy of one photon from a red stop light (\lambda = 652 \text{ nm}).
    • First, calculate frequency: \nu = c/\lambda = (3.00 \times 10^8 \text{ m/s}) / (652 \times 10^{-9} \text{ m}) \approx 4.60 \times 10^{14} \text{ Hz}.
    • Then, calculate energy: E = h\nu = (6.626 \times 10^{-34} \text{ J\cdot s}) \times (4.60 \times 10^{14} \text{ Hz}) = 3.05 \times 10^{-19} \text{ J}.
  • Bond Breaking Application: Determining the maximum wavelength to break chemical bonds.
    • To break an O-O single bond (strength = 142 \text{ kJ/mol}): requires converting bond strength from per mole to per photon.
      • Energy per bond: (142 \times 10^3 \text{ J/mol}) / (6.022 \times 10^{23} \text{ bonds/mol}) \approx 2.36 \times 10^{-19} \text{ J/bond}.
      • Maximum wavelength: \lambda = hc/E = (6.626 \times 10^{-34} \text{ J\cdot s}) \times (3.00 \times 10^8 \text{ m/s}) / (2.36 \times 10^{-19} \text{ J}) \approx 8.43 \times 10^{-7} \text{ m} = 843 \text{ nm}. This falls in the infrared region.
    • To break an O=O double bond (strength = 498 \text{ kJ/mol}):
      • Energy per bond: (498 \times 10^3 \text{ J/mol}) / (6.022 \times 10^{23} \text{ bonds/mol}) \approx 8.27 \times 10^{-19} \text{ J/bond}.
      • Maximum wavelength: \lambda = hc/E = (6.626 \times 10^{-34} \text{ J\cdot s}) \times (3.00 \times 10^8 \text{ m/s}) / (8.27 \times 10^{-19} \text{ J}) \approx 2.40 \times 10^{-7} \text{ m} = 240 \text{ nm}. This falls in the ultraviolet region.

2.2 Atomic Spectroscopy and The Bohr Model

Emission Spectra

  • Continuous Spectrum: Produced by light sources like light bulbs or stars, containing many different wavelengths. When passed through a prism, it separates into a rainbow of colors.
  • Line Spectrum (Atomic Spectrum): Produced when atoms absorb energy (e.g., when heated) and then emit energy as specific electromagnetic radiation. The pattern of emitted radiation consists of discrete lines (specific wavelengths/frequencies).
    • Hydrogen Atom: Emits a characteristic pink light, which, when passed through a prism, shows a distinct line spectrum.
    • Each element has a unique line spectrum, acting as a