Understanding Spectral Lines and Equations

Aug 14, 2024

Lecture Notes: Spectral Lines and the Balmer-Rydberg Equation

Introduction to Spectral Lines

  • Famous experiment by Isaac Newton: light through a prism.
  • Continuous Spectrum: Result of white light separation, blending colors.
  • Line Spectrum: Discrete lines of color, unique to each element.

Hydrogen Emission Spectrum

  • Experiment: Pass current through hydrogen gas.
  • Electrons absorb energy and jump to higher energy levels.
  • When electrons fall, they emit light (emission).
  • Hydrogen's spectrum shows distinct lines, not continuous.
    • Red line: Wavelength of 656 nm.
    • Blue-green line: Wavelength of 486 nm.
    • Blue line: Wavelength of 434 nm.
    • Violet line: Wavelength of 410 nm.
  • Each element has a unique emission spectrum.

Balmer-Rydberg Equation

  • Purpose: Calculate emitted light's wavelength.
  • Formula: [ \frac{1}{\lambda} = R \left( \frac{1}{i^2} - \frac{1}{j^2} \right) ]
    • ( \lambda ): Wavelength.
    • ( R ): Rydberg constant, ( 1.097 \times 10^7 \ m^{-1} ).
    • ( i ): Lower energy level.
    • ( j ): Higher energy level.

Example Calculation

  • Transition: Electron from n=3 to n=2.
  • Calculation steps:
    • ( \frac{1}{\lambda} = 1.097 \times 10^7 (\frac{1}{2^2} - \frac{1}{3^2}) )
    • ( \frac{1}{4} - \frac{1}{9} = 0.138 )
    • Calculate ( \lambda = 656 ) nm (matches red line).

Explaining Other Lines

  • Other transitions (n=4 to n=2, etc.) correspond to blue-green, blue, violet lines.
  • Diagram: Visualize electron transitions.

Beyond the Visible Spectrum

  • Transition example from n=2 to n=1.
    • Calculation gives 122 nm, falls in UV spectrum (not visible).
  • Different series of lines exist beyond visible (Balmer series).

Conclusion

  • Bohr Model: Although not entirely accurate, helps explain energy levels.
  • Energy Quantization: Spectral lines demonstrate that energy is quantized.
  • Balmer Series: Specific to hydrogen, explains visible spectrum transitions.

The Balmer-Rydberg equation and Bohr model provide fundamental insights into atomic structure and light emission, demonstrating quantized energy levels.