Chapter: Atoms

• Introduction & Early Models

  • Dalton's Atomic Theory
    • Indivisible particles
  • Thomson's Plum Pudding Model
    • Sphere of positive charge with embedded electrons
    • Disproved by alpha-particle scattering
  • Rutherford's Nuclear Model
    • Central dense positive nucleus
    • Electrons revolve around the nucleus
    • Limitations: Unable to explain stability and line spectra

• Alpha-Particle Scattering Experiment

  • Setup
    • Radioactive source ($\text{Bi}^{214}$ or $\text{Ra}$)
    • Gold foil (very thin, ~1000 nm)
    • Zinc sulphide ($\text{ZnS}$) screen and microscope
  • Observations
    • Most particles passed undeflected (atom is mostly empty space)
    • Few particles deflected at small angles
    • Very few ($\sim 1$ in $8000$) deflected by $180^\circ$ (presence of massive positive nucleus)
  • Distance of Closest Approach ($r_0$)
    • Kinetic energy converted to electrostatic potential energy
    • $r_0 = \frac{1}{4\pi\epsilon_0} \frac{2Ze^2}{K}$
  • Impact Parameter ($b$)
    • Perpendicular distance of the velocity vector from the central line of the nucleus

• Bohr's Model of Hydrogen Atom

  • Postulates
    • Stationary Orbits: Electrons revolve without radiating energy
    • Quantization of Angular Momentum: $L = mvr = \frac{nnh}{2\pi}$ where $n = 1, 2, 3...$
    • Frequency Condition: Energy emitted/absorbed during transition ($h\nu = E_2 - E_1$)
  • Derivations for Hydrogen-like Atoms
    • Radius of Orbit ($r_n$): $r_n \propto \frac{n^2}{Z}$ ($r_1 \approx 0.53\text{ \AA}$)
    • Velocity of Electron ($v_n$): $v_n \propto \frac{Z}{n}$
    • Total Energy ($E_n$): $E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$
      • Kinetic Energy: $K = -E$
      • Potential Energy: $U = 2E$
  • Limitations
    • Applicable only to single-electron systems (Hydrogen, $\text{He}^+$, $\text{Li}^{2+}$)
    • Could not explain relative intensities of spectral lines
    • Fails to explain Zeeman and Stark effects

• Hydrogen Line Spectra

  • Spectral Series
    • Lyman Series: Ultraviolet region ($n_1 = 1, n_2 = 2, 3, 4...$)
    • Balmer Series: Visible region ($n_1 = 2, n_2 = 3, 4, 5...$)
    • Paschen Series: Infrared region ($n_1 = 3, n_2 = 4, 5, 6...$)
    • Brackett Series: Infrared region ($n_1 = 4, n_2 = 5, 6, 7...$)
    • Pfund Series: Far Infrared region ($n_1 = 5, n_2 = 6, 7, 8...$)
  • Rydberg Formula
    • $\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$
    • Rydberg Constant ($R$): $\approx 1.097 \times 10^7 \text{ m}^{-1}$

• De Broglie's Explanation of Bohr's Postulate

  • Wave Nature of Electron
    • Electron orbit as a stationary wave
    • Circumference equals integral multiple of wavelength ($2\pi r = n\lambda$)
    • Substituting $\lambda = \frac{h}{mv}$ yields Bohr's quantization condition ($mvr = \frac{nh}{2\pi}$)