Table of Contents
- Introduction
- Historical Context and the Failure of Classical Models
- Basic Postulates of Bohr’s Model
- Quantization of Angular Momentum
- Energy Levels of the Hydrogen Atom
- Spectral Lines and the Rydberg Formula
- Radius of Bohr Orbits
- Velocity and Time Period of Electron
- Limitations of the Bohr Model
- Successes of the Bohr Model
- Bohr–Sommerfeld Extension
- Comparison with Quantum Mechanics
- Experimental Verification
- Legacy and Influence on Modern Theory
- Conclusion
1. Introduction
The Bohr model of hydrogen was a revolutionary step in atomic theory. Introduced by Niels Bohr in 1913, it merged classical mechanics with new quantum ideas to explain the discrete energy levels and spectral lines observed in hydrogen. Though ultimately superseded by quantum mechanics, the Bohr model remains an essential pedagogical tool.
2. Historical Context and the Failure of Classical Models
Classical electromagnetism predicted that electrons orbiting a nucleus would radiate energy continuously and spiral into the nucleus. However, atoms were observed to be stable, and hydrogen emission spectra showed discrete lines — not a continuous range.
These contradictions called for a new model.
3. Basic Postulates of Bohr’s Model
- Electrons orbit the nucleus in fixed circular paths without radiating energy.
- Only orbits with quantized angular momentum are allowed.
- Radiation is emitted or absorbed when an electron jumps between energy levels.
4. Quantization of Angular Momentum
Bohr postulated that angular momentum is quantized:
\[
L = n\hbar = n\frac{h}{2\pi}, \quad n = 1, 2, 3, \dots
\]
Where:
- \( h \) is Planck’s constant
- \( n \) is the principal quantum number
This condition selects discrete orbits.
5. Energy Levels of the Hydrogen Atom
From Coulomb’s law and centripetal force:
\[
\frac{ke^2}{r^2} = \frac{mv^2}{r}
\]
Combine with quantized angular momentum:
\[
E_n = -\frac{13.6\ \text{eV}}{n^2}
\]
Where:
- \( E_n \) is energy of the \( n^\text{th} \) level
- \( 13.6 \ \text{eV} \) is the ground state energy
6. Spectral Lines and the Rydberg Formula
Transition between levels emits a photon:
\[
h\nu = E_{n_i} – E_{n_f}
\]
The corresponding wavelength is given by the Rydberg formula:
\[
\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} – \frac{1}{n_i^2} \right)
\]
Where:
- \( R_H \approx 1.097 \times 10^7 \ \text{m}^{-1} \) is the Rydberg constant
Explains Lyman, Balmer, and Paschen series.
7. Radius of Bohr Orbits
From angular momentum and centripetal force:
\[
r_n = \frac{n^2 \hbar^2}{k e^2 m_e} = n^2 a_0
\]
Where \( a_0 = 0.529 \ \text{Ã…} \) is the Bohr radius (radius of ground state orbit).
8. Velocity and Time Period of Electron
Velocity in the \( n^\text{th} \) orbit:
\[
v_n = \frac{ke^2}{\hbar} \cdot \frac{1}{n}
\]
Time period of revolution:
\[
T_n = \frac{2\pi r_n}{v_n}
\]
Reinforces quantization of motion.
9. Limitations of the Bohr Model
- Applies only to hydrogen-like atoms (one electron)
- Cannot explain fine structure, Zeeman effect, or electron spin
- Circular orbit assumption conflicts with quantum principles
- Violates Heisenberg’s uncertainty principle
10. Successes of the Bohr Model
- Explained hydrogen spectrum accurately
- Predicted quantized energy levels
- Introduced key concepts of quantum theory
- Inspired future development of wave mechanics
11. Bohr–Sommerfeld Extension
- Introduced elliptical orbits
- Included quantization of radial and angular motions
- Provided improved spectral accuracy
Still semi-classical and later replaced by Schrödinger’s theory.
12. Comparison with Quantum Mechanics
- Bohr: electrons in fixed orbits
- Schrödinger: electrons as wavefunctions in orbitals
- Bohr’s energy levels match the quantum mechanical hydrogen atom, but wave mechanics provides deeper and more general framework
13. Experimental Verification
- Hydrogen spectral lines matched precisely
- Balmer series: visible spectrum lines
- Lyman and Paschen series: UV and IR, respectively
- Rydberg constant confirmed via spectroscopic measurements
14. Legacy and Influence on Modern Theory
Bohr’s model:
- Bridged classical and quantum physics
- Influenced the Copenhagen interpretation
- Demonstrated necessity of quantization
- Provided stepping stone toward full quantum theory
15. Conclusion
The Bohr model of hydrogen was a revolutionary yet transitional model in atomic physics. While it fell short of explaining complex atoms, it captured the essence of quantization and paved the way for modern quantum mechanics. Its elegance and predictive power continue to make it a cornerstone of physics education.
