Spectral Lines of Hydrogen – Comprehensive Study Notes

Scope of Chapter 5

• Spectral Lines of Hydrogen (Modern Physics, PHY 310)
• Sub-topics covered
  • Electronic transitions in the hydrogen atom
  • Energy emitted or absorbed during a transition
  • Spectral series: Lyman, Balmer, Paschen (brief mention of Brackett & Pfund in diagram)

Electronic Energy in the Hydrogen Atom

• Origin of electronic energy
  • Motion of the single electron around the proton
  • Electrostatic interactions: e⁻–proton & (in multi-electron atoms) e⁻–e⁻
• Quantization principle
  • Only discrete electronic energies are allowed → “stationary states”
  • A change of level requires absorption/emission of a definite quantum $\Delta E = h\nu$
• Definitions
  • Ground state: lowest energy level ( $n = 1$ ); energy $E_1 = -13.6\,\text{eV}$
  • Excited state: any n > 1 ( $E<em>n > E</em>1$, numerically less negative)
  • Ionization limit: $n \to \infty$, $E = 0\,\text{eV}$
• Observed spectral feature
  • Line spacing decreases systematically as $n$ increases → lines crowd toward series limits

Bohr Energy-Level Scheme (recap)

• Allowed energies: $E_n = -\frac{13.6\,\text{eV}}{n^2}$ (hydrogen only)
• Energy of photon during transition $n<em>2 \to n</em>1$:
  $\Delta E = E<em>{n</em>1} - E<em>{n</em>2} = h\nu = \frac{hc}{\lambda}$
• Alternative wavelength form (Rydberg): $\frac{1}{\lambda} = R<em>H \left( \frac{1}{n</em>1^{2}} - \frac{1}{n_2^{2}} \right)$ where
  • $R_H = 1.0974 \times 10^{7}\,\text{m}^{-1}$
  • $n<em>2 > n</em>1$ (upper → lower)

Physical Interpretation of Spectra

• Emission: excited e⁻ drops to lower level → photon out (bright lines on dark background)
• Absorption: incoming continuum light; only photons with precise $h\nu$ move e⁻ upward → missing (dark) lines exactly at the emission frequencies
• Rutherford-model failures (why Bohr was needed)
  1. Predicted a continuous, not line, spectrum
  2. Predicted atomic collapse (radiating e⁻ should spiral into nucleus) → contradicted matter’s stability

Temperature & Excitation

• Room T: almost all H atoms in ground state
• High T or electrical discharge → frequent e⁻–atom collisions excite population to n > 1, enabling emission series

Energy Difference vs. Wavelength

• Inverse proportionality: $\lambda \propto 1/\Delta E$
  • Large $\Delta E$ (e.g.
    $n=2 \to 1$) → short $\lambda$ (< 400 nm, ultraviolet)
  • Small $\Delta E$ (e.g.
    $n=4 \to 3$) → long $\lambda$ (> 700 nm, infrared)

Main Hydrogen Series

• Lyman series (UV)
  • Lower level $n_1 = 1$
  • Formula: $\frac{1}{\lambda} = R<em>H \left( 1 - \frac{1}{n</em>2^{2}} \right)$
  • Wavelength band: 91 nm – 122 nm
  • First line ( $n_2 = 2$ )
  • $\Delta E = 10.2\,\text{eV}$, $\lambda = 1.22 \times 10^{-7}\,\text{m}$ (UV)
• Balmer series (visible)
  • Lower level $n_1 = 2$
  • Historical Balmer formula (1885) fits 4 visible lines
  • Representative wavelengths
  • $n_2=3$ → $656.3\,\text{nm}$ (red, H-α)
  • $n_2=4$ → $486.1\,\text{nm}$ (blue-green, H-β)
  • $n_2=5$ → $434.1\,\text{nm}$ (violet, H-γ)
  • $n_2=6$ → $410.2\,\text{nm}$ (violet, H-δ)
  • Example calc (text): $n=6 \to 2$ gives $\lambda \approx 4.10 \times 10^{-7}\,\text{m}$ (violet)
• Paschen series (infrared)
  • Lower level $n_1 = 3$
  • Formula: $\frac{1}{\lambda} = R<em>H \left( \frac{1}{3^{2}} - \frac{1}{n</em>2^{2}} \right)$
  • IR region wavelengths (longer than 700 nm)
• (Diagrams also mention Brackett $n<em>1=4$ and Pfund $n</em>1=5$ series)

Representative Energy-Level Values (eV)

• $n=1$: $E_1 = -13.6$
• $n=2$: $E_2 = -3.4$
• $n=3$: $E_3 = -1.5$
• $n=4$: $E_4 = -0.85$
• Ionization limit $n=\infty$: $E=0$ (by convention)
• Ionization energy from ground: $|E_1| = 13.6\,\text{eV}$

Worked Numerical Examples (from slides)

1. n = 2 → n = 1 (rest H atom)
   • Use Rydberg formula → $\lambda = 1.22 \times 10^{-7}\,\text{m}$ (UV Lyman-α)
2. n = 6 → n = 2
   • Balmer-series photon → $\lambda \approx 4.10 \times 10^{-7}\,\text{m}$ (violet)

Practical / Real-World Connections

• Spectral fingerprints allow identification of H in astrophysical objects (sun, nebulae)
• Basis for quantum mechanics development; validates Planck/Bohr postulates
• Modern spectroscopy (e.g.
  plasma diagnostics, astronomy red-shift studies)

Philosophical & Historical Notes

• Quantization overcame classical catastrophe predicted by Rutherford’s orbit model
• Einstein quote (slide end): “If A is success in life, then A = x + y + z; Work = x; y = play; z = keeping your mouth shut.”
  • Highlights balance of effort, leisure, and prudence—added for inspirational closure