Bohr's theory of the hydrogen atom revolutionized our understanding of atomic structure. It explained the discrete nature of atomic spectra and introduced the concept of quantized energy levels, paving the way for modern quantum mechanics.

The model proposed electrons orbiting the nucleus in specific energy states, with transitions between states causing photon emission or absorption. While it had limitations, Bohr's theory successfully predicted hydrogen's spectral lines and laid crucial groundwork for future atomic models.

Bohr's Theory of the Hydrogen Atom

Significance of atomic spectra

• Unique patterns of light emitted or absorbed by atoms
  • Hydrogen's emission spectrum consists of distinct wavelengths (Lyman series in ultraviolet, Balmer series in visible, Paschen series in infrared)
  • Absorption spectrum is the inverse of the emission spectrum
• Classical physics could not explain the discrete nature of atomic spectra
  • Expected a continuous spectrum based on classical electromagnetic theory
• Bohr's theory aimed to provide a theoretical explanation for the observed atomic spectra

Key principles of Bohr's model

• Electrons orbit the nucleus in circular paths called stationary states or energy levels
  • Each state has a specific radius and energy
  • Electron energies are quantized, meaning they can only have certain discrete values
• Electrons transition between states by absorbing or emitting photons
  • Absorbing a photon causes an electron to jump to a higher energy level
  • Emitting a photon causes an electron to drop to a lower energy level
• Energy of the absorbed or emitted photon equals the difference in energy between the two states
  • $\Delta E = hf$, where $h$ is Planck's constant and $f$ is the photon frequency
• Angular momentum of an electron in a stationary state is quantized
  • $mvr = n\frac{h}{2\pi}$, where $n$ is an integer, $m$ is electron mass, $v$ is velocity, and $r$ is orbit radius
• The force of attraction between the electron and nucleus is described by Coulomb's law

Energy-level diagrams for hydrogen

• Show the relative energies of an atom's stationary states
  • Each horizontal line represents a specific energy level
  • Lowest energy level (ground state) is usually denoted as $n=1$
  • Higher energy levels have increasing $n$ values ($n=2, 3, 4, ...$)
• Arrows between levels represent transitions caused by photon absorption or emission
  • Upward arrows indicate absorption, downward arrows indicate emission
• Wavelength of the absorbed or emitted photon is related to the energy difference between levels
  • $\frac{1}{\lambda} = R(\frac{1}{n_1^2} - \frac{1}{n_2^2})$, where $R$ is the Rydberg constant, and $n_1$ and $n_2$ are the initial and final levels
• The energy required to remove an electron from the atom completely is called the ionization energy

Bohr's model vs earlier theories

• Improved upon the Rutherford model
  • Rutherford proposed a "solar system" model with electrons orbiting a positively charged nucleus
  • Bohr introduced the concept of quantized energy levels to explain atomic spectra
• Differed from Thomson's "plum pudding" model
  • Thomson suggested electrons were embedded in a positively charged "pudding"
  • Bohr's model had electrons orbiting a dense, positively charged nucleus

Successes and limitations of Bohr's theory

• Successes:
  1. Explained the discrete nature of atomic spectra for hydrogen
  2. Accurately predicted the wavelengths of light in hydrogen's emission spectrum
  3. Introduced the concept of quantized energy levels in atoms
• Limitations:
  1. Could not accurately explain the spectra of atoms with more than one electron
  2. Did not account for the wave-like properties of electrons
  3. Unable to explain the fine structure and hyperfine structure of spectral lines
• Despite limitations, laid the groundwork for the development of quantum mechanics

Quantum mechanical developments

• de Broglie wavelength: Introduced the concept of wave-particle duality for electrons
• Schrödinger equation: Provided a more accurate description of electron behavior in atoms
• Atomic number: Determines the number of protons in the nucleus and electrons in a neutral atom