Atomic Structure_01.10.2024

Hydrogen Spectrum

Understanding Hydrogen Spectrum
- High voltage discharge in hydrogen gas produces a line spectrum
- Diffraction grating creates distinct atomic spectral series
- Principal quantum numbers denoted as n:
  - Lyman series (n = 1)
  - Balmer series (n = 2)
  - Paschen series (n = 3)
- Various wavelengths associated with transitions:
  - Wavelength values expressed as ( \lambda ) in nm
  - Example: 820 nm (n=4), 364.56 nm (n=6)

Characteristics and Measurements
- Electric discharge in hydrogen yields 4 intense visible lines and 5 ultraviolet lines
- Measured by Huggins with observed wavelengths:
  - 656.28 nm, 486.13 nm, 434.05 nm, 410.17 nm, 364.56 nm
- Includes a converging limit in the ultraviolet region
- Balmer empirical relation describes the wavelength of spectral lines

Foundational Concepts
- Proposed by Swedish physicist Johannes Rydberg
- Generalizes Balmer series relations for other elements
- Expressed in terms of series terms and quantum numbers

Details of Each Spectrum Series
- Lyman:
  - Series Term: 1, n (2,3,4,5…)
  - 1st line: 122 nm, Converging line: 91 nm
- Balmer:
  - Series Term: 2, n (3,4,5,6…)
  - 1st line: 656 nm, Converging line: 365 nm
- Paschen:
  - Series Term: 3, n (4,5,6,7…)
  - 1st line: 1875 nm, Converging line: 826 nm
- Brackett:
  - Series Term: 4, n (5,6,7…)
  - 1st line: 4050 nm, Converging line: 1458 nm
- Pfund:
  - Series Term: 5, n (6,7,8…)
  - 1st line: 7400 nm, Converging line: 2280 nm

Calculations and Discrepancies
- Rydberg constant (R): ( R = 1.0973731 \times 10^7 , m^{-1} )
- Experimental value: ( 1.0967758 \times 10^7 , m^{-1} )
- Discrepancy of ~60 cm^-1 explained by considering electron and nucleus motion about their center of mass

Foundations of the Model
- Addressed failures of Rutherford model leading to instability of atoms
- Based on quantum conditions for the electron's movement in defined orbits
- Stability provided by stationary (non-radiating) orbits which do not emit radiation
- Angular momentum quantization given by ( mvr = n \frac{h}{2\pi} )

Key Postulates
- Nucleus at the center with positive charge
- Electrons occupy discrete energy levels (not continuous)
- Motion controlled by Newtonian principles, leading to stationary orbits
- Electrons can emit or absorb radiation during transitions

Kinetic and Potential Energy
- Total energy of the electron system is given via kinetic and potential energy relations
- Energy levels quantized and derived for hydrogen as function of principal quantum numbers

Link to Electron Energies
- Rydberg equation reveals quantized energies for hydrogen’s electron
- Energy transitions relate directly to emitted photon frequencies, establishing quantization of electron energies

Advantages of Bohr’s Model
- Explains stability of atoms and origins of spectral lines
- Predicts resonant and ionization potentials which align with experimental data
- Incorporates isotopic shifts demonstrating robustness in calculations
- Aids in understanding electron-proton mass ratio

Observational Facts
- Empirical relations confirmed through experimental data
- Discovery of heavy hydrogen (deuterium) supported predictions of spectral lines
- Quantized energy levels show consistency across various hydrogen lines

Limitations Encountered
- Fine structure of spectral lines not explained
- No information on electron distribution in multi-electron systems
- Dual reliance on classical and quantum theories leads to conceptual inconsistencies

Quantum Nature of Atomic Energy Levels
- Experiment involved electric interactions in evacuated and gas-filled tubes
- Observed current variations as function of accelerating voltage (Va)
- Findings confirmed quantized energy states through specific excitation potentials

Historical Context
- The transition from classical to wave-particle duality concepts in physics
- Key contributions from Planck, Einstein, de Broglie, Schrödinger leading to quantum mechanics

Understanding Particles as Waves
- Defined through wave functions representing probabilities in space
- Represents electron behavior as a quantifiable disturbance in defined regions
- Heisenberg uncertainty principle expressed through wave packets ( ext{ΔxΔp} ext{≥} h/2 )

Distinction of Waves
- Group velocity relates to a moving particle, while phase velocity pertains to individual wave properties
- Wave packets can exhibit varying velocities based on constituent wave frequencies

Synthesis of Quantum Mechanics
- Integration of particle and wave theories establishes a robust framework for understanding atomic structures and behavior
- Ongoing experimentation and theoretical development continues to validate and expand upon these foundational principles.