Learning Objectives
- Understand Rutherford's atomic model and alpha scattering
- Study Bohr's model of the hydrogen atom
- Calculate energy levels, radii, and spectral series
- Understand hydrogen spectrum and spectral lines
- Compare Bohr's model with quantum mechanical model
Key Concepts
Rutherford's Alpha Scattering Experiment
Alpha particles were scattered by a thin gold foil. Most passed straight, some deflected at small angles, very few bounced back.
Conclusions: Most of atom is empty space. Positive charge and mass are concentrated in a tiny nucleus (10⁻¹⁵ m). Electrons orbit the nucleus.
Distance of closest approach: d = 2kZe²/(½mv²) = kZe²/KE
Impact parameter: b = (kZe²/KE) cot(θ/2)
Limitation: Orbiting electrons should radiate energy (classical EM theory), spiral inward, and atom should collapse. This does not happen.
Bohr's Model of Hydrogen Atom
Postulates:
- Electrons revolve in fixed circular orbits (stationary orbits) without radiating energy.
- Angular momentum is quantized: L = mvr = nh/(2π) = nℏ (n = 1, 2, 3, ...)
- Electrons emit/absorb photons when transitioning between orbits: hν = E₂ - E₁.
Bohr's Formulas for Hydrogen-like Atoms
Radius of nth orbit: rₙ = n²a₀/Z, where a₀ = 0.529 Å (Bohr radius).
Velocity: vₙ = Zv₀/n, where v₀ = 2.18 × 10⁶ m/s.
Energy: Eₙ = -13.6 Z²/n² eV
Time period: Tₙ ∝ n³/Z². Current: Iₙ ∝ Z²/n³.
Hydrogen Spectrum
Rydberg formula: 1/λ = R(1/n₁² - 1/n₂²), R = 1.097 × 10⁷ m⁻¹.
Spectral Series:
- Lyman (n₁=1): UV region. n₂ = 2, 3, 4, ... First line: n₂=2.
- Balmer (n₁=2): Visible region. n₂ = 3, 4, 5, ... (Hα, Hβ, Hγ, ...)
- Paschen (n₁=3): Infrared. n₂ = 4, 5, 6, ...
- Brackett (n₁=4): Infrared. n₂ = 5, 6, 7, ...
- Pfund (n₁=5): Far infrared. n₂ = 6, 7, 8, ...
Maximum number of spectral lines from level n: n(n-1)/2.
Series limit: shortest wavelength in each series (n₂ → ∞).
Summary
Rutherford's experiment revealed the nuclear model of the atom. Bohr's model quantized electron orbits and angular momentum, successfully explaining the hydrogen spectrum. Energy levels are given by Eₙ = -13.6/n² eV. Transitions between levels produce specific spectral lines grouped into series (Lyman, Balmer, Paschen, etc.).
Important Terms
- Bohr Radius: a₀ = 0.529 Å, radius of ground state of hydrogen
- Ground State: Lowest energy state (n = 1), E = -13.6 eV
- Excited State: Any state with n > 1
- Ionisation Energy: Energy to remove electron from ground state (13.6 eV for H)
- Rydberg Constant: R = 1.097 × 10⁷ m⁻¹
- Spectral Series: Group of lines from transitions to the same lower level
Quick Revision
- rₙ = 0.529 n²/Z Å; Eₙ = -13.6 Z²/n² eV
- L = nℏ; vₙ ∝ Z/n; rₙ ∝ n²/Z
- 1/λ = R(1/n₁² - 1/n₂²)
- Lyman (UV), Balmer (visible), Paschen (IR)
- Number of lines from n: n(n-1)/2
- Ionisation energy of H = 13.6 eV