Failures of Rutherford's Model

Rutherford's nuclear atom solved the alpha scattering puzzle but immediately raised two new ones. Why doesn't the orbiting electron spiral into the nucleus, given that classical electromagnetism requires accelerating charges to radiate? And why are atomic spectra not continuous but a set of sharp, discrete lines? These failures forced the leap from classical to quantum atomic theory.

Concept

Rutherford pictured the atom as a miniature Coulomb solar system: electrons in circular orbits around the central positive nucleus, balanced by the electrostatic attraction.

$\frac{m_e v^2}{r} = \frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r^2}$

This Newtonian picture allows any orbital radius $r$ — a continuum of allowed energies $E(r)$. Two failures emerge:

1. Radiative instability (Larmor's formula). A charged particle in centripetal acceleration radiates EM waves with power $P_{rad} = \dfrac{e^2 a^2}{6\pi\varepsilon_0 c^3}$. This drains energy from the orbit, so $r$ shrinks continuously. A classical hydrogen atom collapses in about $10^{-11}$ s.
2. Continuous spectrum. As $r$ shrinks the orbital frequency rises continuously, so the emitted EM frequency would also vary continuously — a smear, not lines. Real atoms emit only specific frequencies.

Derivation

Collapse time estimate. For a hydrogen electron in circular orbit at radius $r$, the centripetal acceleration is

$a = \frac{v^2}{r} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_e r^2}$

Using Larmor:

$P_{rad} = \frac{e^2 a^2}{6\pi\varepsilon_0 c^3} = \frac{e^2}{6\pi\varepsilon_0 c^3}\left(\frac{e^2}{4\pi\varepsilon_0 m_e r^2}\right)^2$

The total mechanical energy of the orbit is $E = -\dfrac{e^2}{8\pi\varepsilon_0 r}$, so $dE/dr = \dfrac{e^2}{8\pi\varepsilon_0 r^2}$. Equating energy loss rate to radiated power and integrating from $r = a_0 \approx 5.3\times 10^{-11}$ m down to $r=0$,

$\tau \approx \frac{a_0^3 m_e^2 c^3 (4\pi\varepsilon_0)^2}{4 e^4} \approx 1.6\times 10^{-11}\,\text{s}$

Matter would not exist on macroscopic timescales — yet hydrogen is perfectly stable.

Spectral problem. The classical orbital frequency is

$f_{orb} = \frac{v}{2\pi r} \quad\text{with}\quad v = \sqrt{\frac{e^2}{4\pi\varepsilon_0 m_e r}}$

As $r$ varies continuously, $f_{orb}$ varies continuously, predicting a continuous emission spectrum. Experiment, however, shows discrete lines (the Balmer series in hydrogen, for instance) — which classical mechanics cannot reproduce.

Worked Example

A classical hydrogen electron starts at $r = a_0$ with orbital energy $E_0 = -13.6$ eV. Estimate the time over which it would lose half this energy by Larmor radiation.

Using the spiralling solution $r(t)^3 = a_0^3 - 4 t \, (e^2/(4\pi\varepsilon_0))^2 / (m_e^2 c^3)$, half-energy corresponds to $r = 2 a_0$? No — half-energy means $E = -27.2$ eV, hence $r = a_0/2$. Plugging $r = a_0/2$ in the cubic relation:

$t_{half} = \frac{a_0^3 - (a_0/2)^3}{4} \cdot \frac{m_e^2 c^3 (4\pi\varepsilon_0)^2}{e^4} = \frac{7}{32} a_0^3 \cdot \frac{m_e^2 c^3 (4\pi\varepsilon_0)^2}{e^4}$

Plugging numbers gives $t_{half}\sim 3.5\times 10^{-12}$ s. Far shorter than the timescale of any chemistry — the atom would be gone in picoseconds.

Common Confusions

• Rutherford's model is right about the nucleus but wrong about the orbiting electron. The nucleus did survive; what failed is the picture of the electron as a tiny classical planet.
• Larmor radiation is real — for instance, a synchrotron does radiate. The failure is that there is no mechanism in classical theory to stop the orbital decay.
• Discrete spectra are not just a calculational nuisance. They reveal a deep new principle: quantisation.
• The fix (Bohr) is to postulate stable, non-radiating "stationary" orbits — a clear break from classical EM.

Key Takeaways

• Classical orbiting electron is centripetally accelerated and must radiate (Larmor).
• Predicted collapse time of a hydrogen atom is about $10^{-11}$ s. Real atoms are stable.
• Classical orbital frequency varies continuously, predicting a continuous spectrum. Real spectra are discrete.
• Both failures point to a new postulate: only certain orbits are allowed, and transitions between them produce discrete photons.
• This sets the stage for Bohr's quantisation of angular momentum and the photon-emission rule $h\nu = E_i - E_f$.