Lecture Notes: The Bohr Model

Introduction

• Topic: The Bohr Model
• Learning Objective: Understanding quantized energy states in the Bohr Model, particularly for the hydrogen atom.

Background

• Before Niels Bohr, no application of quantized energy states existed for atomic models.
• Bohr proposed quantized energy states for electrons orbiting a nucleus, inspired by:
  • Photoelectric effect
  • Black body radiation

Bohr's Contribution

• Derived an equation similar to Rydberg independently:
  • Equation: ( \frac{1}{\lambda} = \frac{k}{h \cdot c} )
  • Constants:
    • ( k = 2.179 \times 10^{-18} ) Joules
    • ( h ) (Planck's constant)
    • ( c ) (speed of light)
• Calculated the Rydberg constant using fundamental constants, which was a major achievement.

Structure of the Bohr Model

• Nucleus: Positively charged
• Electron Orbits:
  • Integer values assigned: ( n = 1, 2, 3, \ldots )
  • Ground State: ( n = 1 ) (lowest energy)
  • Excited States: ( n \geq 2 )
• Explains line spectra of hydrogen by linking integer values to quantized energy states.

Electron Transitions

• Electrons can absorb light and move to higher energy states (excited states).
• When electrons drop to lower energy states, they emit light.
• Energy Calculation:
  • Formula: ( \Delta E = k \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) )
  • Energy emission leads to line spectra observed in hydrogen.

Example Problem

• Problem: Find energy and wavelength for electron transition from ( n = 4 ) to ( n = 6 ).
• Energy Calculation:
  • ( \Delta E = 2.179 \times 10^{-18} J \left( \frac{1}{16} - \frac{1}{36} \right) )
  • Result: ( 7.566 \times 10^{-20} J )
• Wavelength Calculation:
  • Formula: ( \lambda = \frac{hc}{E} )
  • Result: ( 2.626 \times 10^{-6} m )
  • Spectrum: Infrared region of the electromagnetic spectrum.

Conclusion

• Bohr Model provides a quantized approach to atomic structure, explaining the discrete energy levels and spectral lines observed in hydrogen and other elements.