Overview
This lecture explains the Bohr model for the hydrogen atom, including quantized energy levels, electron transitions, and the use of related equations to calculate energies and wavelengths.
The Bohr Model of the Hydrogen Atom
- The planetary model depicted electrons orbiting the nucleus like planets, but predicted unstable atoms.
- Bohr proposed electrons exist in quantized orbits and do not emit radiation unless moving between orbits.
- Energy is emitted or absorbed as photons when electrons transition between energy levels.
- The energy of a photon from a transition is |Ef – Ei| = hν = hc/λ, where h is Planck’s constant.
Quantized Energy Levels and the Rydberg Equation
- Electron energy levels in hydrogen are given by En = k/n², where n = 1, 2, 3,...
- k is a constant (2.179 × 10⁻¹⁸ J for hydrogen).
- Substituting orbital energies into the energy formula leads to the Rydberg equation.
- The Rydberg constant, R, calculated from Bohr’s model closely matches experimental values.
Electron States and Transitions
- The lowest energy state (n = 1) is called the ground state; higher n values are excited states.
- When electrons return to lower energy states, photons are emitted; when excited, they absorb photons.
- The energy required to move an electron between levels equals the energy released when it returns.
Hydrogen-like Atoms and Orbit Sizes
- Bohr’s model applies to all one-electron (hydrogen-like) ions, with Z as the nuclear charge.
- For hydrogen-like ions: En = kZ²/n².
- Orbit radius is r = n²/Z × a₀, with a₀ (the Bohr radius) = 5.292 × 10⁻¹¹ m.
- As n increases, electrons are farther from the nucleus and energy levels approach zero (the ionization limit).
Example Calculations
- For hydrogen, n = 3: En = –2.42 × 10⁻¹⁹ J.
- For hydrogen, n = 6: En = –6.05 × 10⁻²⁰ J.
- Moving an electron from n = 4 to n = 6 in hydrogen absorbs 7.57 × 10⁻²⁰ J (infrared radiation, λ = 2.63 × 10⁻⁶ m).
- Transition from n = 5 to n = 3 in He⁺: 6.20 × 10⁻¹⁹ J, λ = 3.21 × 10⁻⁷ m.
Limitations and Legacy of Bohr’s Model
- Bohr’s model fails for atoms with more than one electron due to electron–electron interactions.
- The model introduced the concept of quantized energy levels, foundational for quantum mechanics.
Key Terms & Definitions
- Planetary Model — Atom model with electrons orbiting the nucleus.
- Bohr Model — Model stating electrons occupy quantized orbits with fixed energies.
- Quantum Number (n) — Integer describing an electron’s energy level.
- Ground State — Lowest possible energy state of an atom.
- Excited State — Higher energy state when electrons are in higher orbits.
- Hydrogen-like Atom — Atom/ion with only one electron.
- Rydberg Equation — Formula predicting hydrogen’s spectral lines using quantum numbers.
- Bohr Radius (a₀) — The smallest allowed orbit radius for hydrogen (5.292 × 10⁻¹¹ m).
Action Items / Next Steps
- Practice calculating energies and wavelengths for electron transitions using the provided formulas.
- Review diagrams of energy levels and electron transitions for hydrogen.
- Prepare for further study of quantum mechanics beyond the Bohr model.