Overview
This lecture focuses on the observation of discrete light emission from excited electrons in atoms, introduces the Rydberg formula, and outlines how to calculate wavelengths and energies of emitted photons in hydrogen-like atoms.
Observations of Electron Emission
- Electrons in excited states fall to lower states, emitting energy as light.
- Emitted light consists of specific discrete colors, not a continuous spectrum (no rainbow).
- Each element emits a unique set of wavelengths, indicating different allowed electron energies for each atom.
Interpretation of Observations
- Electrons can only occupy certain discrete energy levels within atoms.
- The emission of specific colors means electrons transition only between set energy states, not any random energy.
Measuring Wavelengths and Energy Differences
- Wavelengths of emitted light are easily measured in the lab with inexpensive diffraction gratings.
- Wavelength is linked to frequency and energy (via E = hν and c = λν).
- Measuring the wavelength of emitted light reveals the energy difference between two electron energy levels.
The Rydberg Formula for Hydrogen
- The inverse wavelength of emitted light is given by:
1/λ = RH × (1/n_final² - 1/n_initial²), where RH is the Rydberg constant for hydrogen.
- n_final and n_initial are positive integers representing the energy levels.
- This formula only applies directly to hydrogen or hydrogen-like atoms.
Example Calculation: Wavelength for Electron Transition
- For an electron falling from n=5 to n=4 in hydrogen:
Plug values into the Rydberg formula to get 1/λ, then invert to find λ.
- Remember: After calculating the inverse wavelength, take its reciprocal to find λ.
- The constant RH is in inverse nanometers, so resulting wavelength is in nanometers.
Calculating Energy of Emitted Photons
- Energy of emitted photon: E = hν = hc/λ.
- If wavelength is in nanometers, convert to meters before using in the energy equation.
- For emission (electron falling), the energy is negative, indicating energy release.
Modified Rydberg Formula for Energy
- Direct formula for energy difference:
ΔE = -RH’ × (1/n_final² - 1/n_initial²), where RH’ is a constant in joules for hydrogen.
- This shortcut eliminates the need to calculate wavelength first.
Key Terms & Definitions
- Excited State — an electron energy level higher than the ground state.
- Photon — a particle of light emitted when electrons transition to lower energy levels.
- Rydberg Formula — equation relating wavelengths of emitted light to electron transitions in hydrogen.
- Rydberg Constant (RH) — a numerical constant specific to hydrogen atoms.
- Plank’s Constant (h) — a fundamental constant that relates energy and frequency of a photon.
Action Items / Next Steps
- Practice calculating wavelengths and energies for electron transitions using both Rydberg formulas.
- Remember to convert nanometers to meters when calculating energy.
- Note the sign convention: negative for emitted energy, positive for absorbed.