Chapter 2 Notes: The Quantum-Mechanical Model of the Atom
Waves and Basic Properties
Waves have peaks and troughs, repeating over time. Wavelength (\lambda) is the distance between peaks, and frequency (\nu) is waves per unit time.
Velocity (v), wavelength, and frequency are related by: General: \nu = \frac{v}{\lambda} and for Light in vacuum: \nu = \frac{c}{\lambda} where c = 3.00\times 10^{8} \,\mathrm{m\,s^{-1}}. Faster waves or shorter wavelengths mean higher frequencies.
Electromagnetic Radiation and the Spectrum
Visible light is a small part of the electromagnetic spectrum, lying between infrared (IR) (longer wavelength, lower frequency) and ultraviolet (UV) (shorter wavelength, higher frequency).
Red light (≈ 750 nm) is near IR; violet light (≈ 400 nm) is near UV. X-rays have higher frequency than visible light; radio waves have lower frequencies.
The Light-Quantum Connection and Planck’s Constant
Planck hypothesized light is quantized into photons, with energy (E) proportional to frequency (\nu).
Planck's constant: h = 6.626\times 10^{-34} \ \mathrm{J\,s}.
Photon energy: E = h\nu or E = \frac{hc}{\lambda}.
This explained phenomena like black-body radiation where classical theory failed.
Photoelectric Effect and Work Function
Electrons are ejected from a surface if incident light photon energy (h\nu) exceeds a threshold binding energy (work function, \phi), regardless of intensity.
Kinetic energy of ejected electron: K.E. = h\nu - \phi.
Bohr Model of the Atom: Successes and Limitations
Electrons occupy discrete orbits; transitions between orbits involve absorption or emission of photons with specific energies determined by the Rydberg formula.
Bohr's energy levels for hydrogen-like atoms: E_n = -2.18\times 10^{-18}\ \mathrm{J} \cdot \frac{1}{n^{2}}.
The model explained hydrogen's line spectra but failed for multi-electron atoms and chemical bonding, and couldn't justify electron stability in orbits.
de Broglie’s Matter-Wave Hypothesis
Louis de Broglie proposed wave-particle duality for all matter; particles exhibit wave-like characteristics.
de Broglie wavelength: \lambda = \frac{h}{p} = \frac{h}{mv}.
Wave behavior is significant only for microscopic particles (e.g., electrons); for macroscopic objects, the wavelength is imperceptibly small.
Schrödinger’s Wave Equation and Orbital Concepts
Schrödinger's wave equation models electron behavior as standing waves in atoms, leading to orbitals.
The wavefunction (\psi) itself is not physical; its square, \psi^{2} (Born interpretation), gives the probability density of finding an electron in a given region. Nodes are regions of zero electron probability.
Quantum Numbers and Orbital Structure
Orbitals are characterized by three quantum numbers:
Principal quantum number (n): Integer (n \ge 1); determines orbital size and energy. Higher n means larger, higher energy orbital.
Angular momentum quantum number (l): Integer from 0 to (n-1); determines orbital shape ($l=0 for s, l=1 for p, l=2 for d, etc.).
Magnetic quantum number (m_l): Integer from (-l) to (+l); determines orbital orientation in space.
Radial nodes for an orbital = (n - l - 1).
Uncertainty Principle and Its Consequences
Heisenberg's uncertainty principle states that precise knowledge of a particle's position ($\Delta x) reduces certainty of its momentum ($\Delta p), and vice versa: \Delta x \; \Delta p \ge \frac{h}{4\pi}$$.
This implies electrons cannot be precisely located within an atom, challenging the simplistic Bohr orbits.
Key Conceptual Takeaways
Light exhibits both wave (interference) and particle (photon) properties depending on the measurement.
Quantization of energy explains discrete spectral lines and the photoelectric effect threshold.
Schrödinger's equation provides a probabilistic framework for electron distribution (orbitals) characterized by quantum numbers.
The Bohr model is a useful but limited stepping stone to quantum-mechanical orbital theory.
The Uncertainty Principle fundamentally limits simultaneous knowledge of position and momentum for microscopic systems.