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.