Rutherford and Bohr atom models

Rutherford's experiment (1911). Alpha particles pass through a thin gold foil: most are deflected only slightly, but a few scatter through angles greater than $90^\circ$, which cannot be explained by a uniform distribution of charge throughout the atom.

The conclusion: the positive charge and nearly all the mass are concentrated in a very small region — the nucleus, with a radius of order $10^{-15}$ m, while the whole atom has a size of order $10^{-10}$ m.

However, classical electrodynamics makes Rutherford's model unstable: an electron in orbit would radiate continuously and fall into the nucleus within about $10^{-10}$ s, and the emitted spectrum would be continuous — yet a line spectrum is observed.

Bohr's postulates (1913). There exist stationary orbits on which the electron does not radiate, and the angular momentum is quantized: $m v r = n\hbar$, where $n = 1, 2, 3, \ldots$ Radiation is emitted or absorbed only when the electron jumps between orbits, and the photon energy equals the energy difference: $h\nu = E_m - E_n$.

For the hydrogen atom this gives $E_n = -13.6/n^2$ eV and $r_n = n^2 a_0$, where $a_0 \approx 0.053$ nm is the Bohr radius; the model explains the Lyman, Balmer, and Paschen series and the Rydberg formula.

For a hydrogen atom, find the energy (in eV) and the orbital radius of the electron in the state $n = 3$. Use $E_n = -13.6/n^2$ eV and $r_n = n^2 a_0$, where $a_0 = 0.053$ nm.

An electron in a hydrogen atom makes a transition from $n = 3$ to $n = 2$ (the H$\alpha$ line of the Balmer series). Find the energy of the emitted photon in eV and its wavelength in nm. Use $E_n = -13.6/n^2$ eV and $hc = 1240$ eV·nm.

A hydrogen atom is excited to the level $n = 4$. List all possible emission lines, find the longest and the shortest wavelengths, and assign each line to the Lyman, Balmer, or Paschen series.

Safe self-study plan in a simulation. Do not work with real discharge tubes or high voltage — use a computer model.

1. Open the PhET simulation “Models of the Hydrogen Atom” (Physics → Quantum Phenomena) or a similar interactive model of hydrogen energy levels.
2. Switch to “Experiment” mode and fire white light at the hydrogen atom. Record the wavelengths at which absorption occurs.
3. Compare the “Rutherford” and “Bohr” models: the first should give a continuous spectrum, the second a line spectrum. Record this in a two-column table.
4. Switch to the Bohr model and follow the transitions $n = 4 \to 2$, $n = 3 \to 2$, $n = 2 \to 1$. Before each transition, predict the color or spectral region from $\lambda = 1240/\Delta E$ nm, then check your prediction.
5. Sum up: in two sentences, explain why a line spectrum is direct evidence for discrete energy levels.

Show together the two key facts of the topic: (1) how alpha-particle scattering reveals a small, dense nucleus, and (2) how discrete energy levels explain the hydrogen line spectrum.

A two-panel diagram. Left: a beam of alpha particles strikes a thin gold foil; almost all particles pass straight through, while only a few scatter through large angles and backwards — those trajectories are marked with arrows, labelled “nucleus, $r \sim 10^{-15}$ m” and “atom, $r \sim 10^{-10}$ m”. Right: an energy-level diagram of the hydrogen atom with horizontal levels $n=1,2,3,4$ and transition arrows; below it, a wavelength scale marked with the lines 122 nm, 486 nm, and 656 nm, labelled Lyman and Balmer series.

Inside an ionisation smoke detector, behind a metal foil, sits an americium-241 source with an activity of about $33\ \text{kBq}$. Each decay fires an $\alpha$-particle with kinetic energy $5.49\ \text{MeV}$, that is $8.8\times 10^{-13}\ \text{J}$. In air one ion-electron pair costs about $34\ \text{eV}$, so a single particle creates roughly $1.6\times 10^{5}$ ion pairs and is stopped after a path of only $4\ \text{cm}$. The ions carry a weak current between the electrodes; smoke absorbs the $\alpha$-particles, the current drops — and the siren sounds.

What is at work here is Rutherford's nuclear model: nearly all the mass and all the positive charge sit in a nucleus of radius $\approx 7\times 10^{-15}\ \text{m}$, while the atom itself has a radius of $\approx 1.4\times 10^{-10}\ \text{m}$. So the $\alpha$-particle flies through emptiness almost all the time and loses energy only in frequent weak collisions with electrons (hence exactly $4\ \text{cm}$ of range and $1.6\times 10^{5}$ ion pairs), while a large deflection is possible only on a close approach to a nucleus — this is exactly what explains the rare back-scattering in the gold-foil experiment.

In a school hydrogen discharge tube and in the Sun's chromosphere during a total eclipse, a red line of wavelength $656.3\ \text{nm}$ is visible — the H-$\alpha$ line. Each of its photons carries the energy $E=hc/\lambda$; with $hc=1240\ \text{eV}\cdot\text{nm}$ this gives $1240\ \text{eV}\cdot\text{nm}/656.3\ \text{nm}=1.89\ \text{eV}$.

This is a transition in Bohr's model of the hydrogen atom: the electron drops from level $n=3$ to level $n=2$, and the photon energy equals the difference of the level energies $E_3-E_2 = 13.6\ \text{eV}\left(\frac{1}{2^2}-\frac{1}{3^2}\right)=1.89\ \text{eV}.$ The discreteness of the orbits explains why the line is always the same and has an exact wavelength instead of being smeared across the whole spectrum.

In a dental X-ray machine electrons are accelerated by a voltage of $60\ \text{kV}$, so each electron reaches the tungsten target with an energy of $60\ \text{keV}$. The continuous spectrum cuts off at $\lambda_{min}=hc/(eU)=1240\ \text{eV}\cdot\text{nm}/60000\ \text{eV}=0.021\ \text{nm}$, and on top of it sits a sharp tungsten line at $0.0589\ \text{nm}$ with a photon energy of $1240\ \text{eV}\cdot\text{nm}/0.0589\ \text{nm}=21.1\ \text{keV}$.

The sharp line is radiation emitted when an electron of a tungsten atom falls into a vacancy in an inner Bohr shell: the level energy is fixed, so the wavelength is fixed too. The continuous background (from $0.021\ \text{nm}$ upwards) comes from a fast electron being braked in the Coulomb field of a nucleus — a direct consequence of the fact that the charge and mass of the atom are concentrated in the nucleus.