understand that there are discrete electron energy levels in isolated atoms (e.g. atomic hydrogen)
Energy Levels in Atoms and Line Spectra
Discrete Electron Energy Levels
In an isolated atom, electrons can only occupy certain energy levels. These levels are quantised, meaning the electron’s energy can take only specific values. When an electron jumps from a higher level to a lower one, it emits a photon; when it absorbs a photon, it jumps to a higher level.
Bohr Model of Hydrogen
For hydrogen (single proton + electron), Niels Bohr showed that the allowed energies are given by $$E_n = -\frac{13.6\ \text{eV}}{n^2}$$ where $n = 1,2,3,\dots$ is the principal quantum number. The negative sign indicates that the electron is bound; $E_1 = -13.6\ \text{eV}$ is the ground state.
Energy Level Table (Hydrogen)
| $n$ (level) | Energy $E_n$ (eV) |
|---|---|
| 1 | $-13.6$ |
| 2 | $-3.4$ |
| 3 | $-1.51$ |
| 4 | $-0.85$ |
Emission of Photons and Line Spectra
When an electron drops from level $n_i$ to a lower level $n_f$, the emitted photon has energy $$\Delta E = E_{n_i} - E_{n_f} = 13.6\ \text{eV}\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right).$$ Using $E = hf = \frac{hc}{\lambda}$, the wavelength of the emitted light is $$\frac{1}{\lambda} = R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right),$$ where $R_H = 1.097\times10^7\ \text{m}^{-1}$ is the Rydberg constant for hydrogen.
Spectral Series of Hydrogen
| Series | Final level $n_f$ | Typical wavelength range |
|---|---|---|
| Lyman | 1 | UV ($<122\ \text{nm}$) |
| Balmer | 2 | Visible ($365-656\ \text{nm}$) |
| Paschen | 3 | IR ($>820\ \text{nm}$) |
Key Points to Remember
- Electrons in isolated atoms occupy only discrete energy levels.
- Energy differences between levels give rise to photons with specific wavelengths → line spectra.
- For hydrogen, $E_n = -13.6\ \text{eV}/n^2$ and the wavelengths follow the Rydberg formula.
- Different series (Lyman, Balmer, Paschen, …) correspond to electrons falling to different final levels.
🌟 Understanding quantised energy levels helps explain the colours we see in gas‑discharge tubes and stars! 🌟
Revision
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