42.3 Content
42.3.1 Limitations of Rutherford’s Model
Classical Problem: Accelerating charges radiate energy (Maxwell’s equations)
An orbiting electron is constantly accelerating (centripetal). According to classical physics:
- Electron radiates electromagnetic energy
- Loses energy → spirals inward
- Continuous spectrum emitted (not discrete lines)
- Atom collapses in ~10⁻¹¹ seconds
Reality: Atoms are stable and emit discrete spectral lines!
Classical physics predicted atoms would collapse almost instantly. The fact that matter is stable proves classical physics is incomplete at atomic scales.
42.3.2 Bohr’s Model (1913)
Bohr’s postulates saved the atom:
- Stationary states: Electrons can only orbit at specific radii without radiating
- Quantised angular momentum: \(L = mvr = n\hbar\) where \(\hbar = h/(2\pi)\)
- Photon emission/absorption: Transitions between states emit/absorb photons with \(E = hf\)
42.3.3 Interactive: Bohr Energy Levels
42.3.4 Energy Levels of Hydrogen
The energy of the nth level:
\[E_n = \frac{-13.6}{n^2}\ \text{eV}\]
| Level | n | Energy (eV) |
|---|---|---|
| Ground state | 1 | -13.6 |
| First excited | 2 | -3.4 |
| Second excited | 3 | -1.51 |
| Third excited | 4 | -0.85 |
| Ionised | ∞ | 0 |
Negative energy means the electron is bound. Zero energy means just free. The 13.6 eV ionisation energy is needed to remove the electron completely.
42.3.5 The Balmer Series
Visible hydrogen lines from transitions to n = 2:
\[\frac{1}{\lambda} = R_H \left[\frac{1}{n_f^2} - \frac{1}{n_i^2}\right]\]
where \(R_H = 1.097 \times 10^7\ \text{m}^{-1}\) (Rydberg constant)
| Transition | Wavelength | Colour |
|---|---|---|
| 3 → 2 (Hα) | 656 nm | Red |
| 4 → 2 (Hβ) | 486 nm | Blue-green |
| 5 → 2 (Hγ) | 434 nm | Violet |
| 6 → 2 (Hδ) | 410 nm | Deep violet |
42.3.6 Interactive: Hydrogen Emission Spectrum
42.3.7 Other Spectral Series
| Series | Final Level | Wavelength Region |
|---|---|---|
| Lyman | n = 1 | Ultraviolet |
| Balmer | n = 2 | Visible |
| Paschen | n = 3 | Infrared |
| Brackett | n = 4 | Far infrared |
42.3.8 Photon Energy and Wavelength
When an electron transitions between levels:
\[E_{photon} = E_i - E_f = hf = \frac{hc}{\lambda}\]
where: - \(h = 6.63 \times 10^{-34}\) J·s (Planck’s constant) - \(f\) = frequency (Hz) - \(\lambda\) = wavelength (m) - \(c = 3.0 \times 10^8\) m/s
42.3.9 Interactive: Energy Level Transitions
42.3.10 Limitations of Bohr’s Model
While successful for hydrogen, Bohr’s model fails for:
| Limitation | Description |
|---|---|
| Multi-electron atoms | Can’t predict spectra of helium or heavier elements |
| Fine structure | Can’t explain splitting of spectral lines in magnetic fields |
| Chemical bonding | No explanation for molecular formation |
| Intensity | Can’t predict relative brightness of spectral lines |
| Ad hoc | Quantisation postulated without deeper explanation |
42.3.11 de Broglie’s Matter Waves (1924)
Louis de Broglie proposed that particles have wave properties:
\[\lambda = \frac{h}{mv} = \frac{h}{p}\]
where: - \(\lambda\) = de Broglie wavelength (m) - \(h = 6.63 \times 10^{-34}\) J·s - \(m\) = mass (kg) - \(v\) = velocity (m/s) - \(p\) = momentum (kg·m/s)
All matter exhibits wave properties, but the wavelength is only significant for very small masses. An electron at 10⁶ m/s has λ ≈ 7 × 10⁻¹⁰ m (atomic scale). A cricket ball has λ ≈ 10⁻³⁴ m (undetectable).
42.3.12 de Broglie Explains Bohr’s Quantisation
The allowed orbits correspond to standing waves around the nucleus:
\[2\pi r = n\lambda\]
Only integer numbers of wavelengths fit around the orbit, explaining why only certain radii are allowed.
42.3.13 Interactive: de Broglie Standing Waves
42.3.14 Schrödinger’s Wave Mechanics (1926)
Erwin Schrödinger developed a wave equation describing electron behaviour:
Key Concepts:
- Wave function (\(\psi\)): Mathematical description of quantum state
- Probability density (\(|\psi|^2\)): Probability of finding electron at a location
- Orbitals: 3D probability distributions replace definite orbits
- Quantum numbers: n, l, m, s describe the state completely
Electrons don’t follow definite paths (orbits). Instead, they exist as probability clouds (orbitals). The “1s orbital” is a spherical cloud where you’re most likely to find the electron.
42.3.15 Interactive: Probability Orbitals
42.3.16 Heisenberg’s Uncertainty Principle
You cannot simultaneously know both position and momentum precisely:
\[\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\]
This is a fundamental limit of nature, not a limitation of measurement technology.