Atomic Models

Lessons

Senior 5 · Physics

Atomic Models

4 lessons

Lesson 1 of 4

Thomson's and Rutherford's Atomic Models

25 min

Seeing inside something you cannot see

An atom is about $10^{-10}\,\mathrm{m}$ across, a thousand times smaller than the wavelength of visible light, so no microscope can show its inside. How, then, do we know that an atom has a tiny heavy centre? In 1909 Hans Geiger and Ernest Marsden, working in Ernest Rutherford's laboratory in Manchester, fired alpha particles at a gold foil thinner than a page and counted where they went. Most went straight through, as expected. But now and then one bounced back. Rutherford later said it was almost as if you fired a shell at a sheet of tissue paper and it came back and hit you. This lesson follows how that surprise replaced one model of the atom with another.

Thomson's plum pudding model

In 1897 J. J. Thomson showed that cathode rays are made of tiny negatively charged particles, much lighter than any atom (Lesson 4): electrons. Since atoms are neutral, they must also contain positive charge. In 1904 Thomson proposed that an atom is a sphere of evenly spread positive charge, about $10^{-10}\,\mathrm{m}$ across, with electrons embedded in it like the fruit in a pudding, enough of them to make the whole atom neutral.

The model explained why atoms are neutral and how electrons can be knocked out of them, leaving positive ions. But it predicted that alpha particles, which are fast and heavy, would pass through a thin foil deflected by at most a fraction of a degree, because the positive charge is spread too thinly to push hard on anything.

The alpha-scattering experiment

Alpha particles are helium nuclei, with charge $+ 2 e$ and mass about 7300 times that of an electron. Geiger and Marsden counted the tiny flashes they made on a zinc sulfide screen at different angles $\theta$ . They found that:

most alpha particles passed straight through the foil, or were deflected by only a few degrees;
some were deflected through large angles;
a very small fraction, roughly one in several thousand, bounced back through more than 90°.

Rutherford's nuclear model (1911)

Rutherford worked out that the results could only be explained if:

almost all of the atom's mass and all of its positive charge are concentrated in a tiny central nucleus, so that the strong electric repulsion close to it can turn a fast alpha particle right round;
the nucleus is very small compared with the atom, so most alpha particles never come close to one and pass straight through: the atom is mostly empty space;
the electrons, which are far too light to deflect an alpha particle, move in the space around the nucleus.

His calculation of the number of particles scattered at each angle, assuming an inverse-square electric force, matched the counts exactly. From the results, a nucleus is roughly $10^{-15}$ to $10^{-14}\,\mathrm{m}$ across, about 10 000 to 100 000 times smaller than the atom.

Definition

The nucleus is the very small, dense, positively charged centre of an atom, containing nearly all of its mass. In Rutherford's nuclear model, electrons move around the nucleus in the mostly empty space of the atom. Example: a gold atom is about $3 \times 10^{-10}\,\mathrm{m}$ across, but its nucleus is only about $1.4 \times 10^{-14}\,\mathrm{m}$ across.

Distance of closest approach

An alpha particle fired straight at a nucleus slows down as the repulsion grows, stops for an instant, and is pushed back. At that instant all its kinetic energy has become electric potential energy. For a nucleus of charge $Z e$ and an alpha particle of charge $2 e$ , the distance $d$ of closest approach satisfies

E_k = \frac{1}{4\pi\varepsilon_0}\frac{(2e)(Ze)}{d} \quad\Longrightarrow\quad d = \frac{2Ze^2}{4\pi\varepsilon_0E_k}

The nucleus can be no larger than $d$ (otherwise the alpha particle would touch it), so $d$ gives an upper limit to the size of the nucleus.

Worked example 1: how small is a gold nucleus?

An alpha particle with kinetic energy 5.0 MeV is fired head-on at a gold nucleus ( $Z = 79$ ). Find the distance of closest approach. ( $\frac{1}{4\pi\varepsilon_0} = 8.99 \times 10^9\,\mathrm{N\,m^2\,C^{-2}}$ , $e = 1.60 \times 10^{-19}\,\mathrm{C}$ .)

Solution

$E_k = 5.0 \times 10^6 \times 1.60 \times 10^{-19} = 8.0 \times 10^{-13}\,\mathrm{J}$ .

$d = \frac{8.99 \times 10^9 \times 2 \times 79 \times (1.60 \times 10^{-19})^2}{8.0 \times 10^{-13}} = 4.5 \times 10^{-14}\,\mathrm{m}$ .

The gold nucleus must be smaller than this, while the gold atom is about $3 \times 10^{-10}\,\mathrm{m}$ across: the atom is at least several thousand times wider than its nucleus.

Worked example 2: a scale model

If a hydrogen atom (radius about $5 \times 10^{-11}\,\mathrm{m}$ ) were enlarged until its nucleus (radius about $1 \times 10^{-15}\,\mathrm{m}$ ) was the size of a 1 cm marble on the centre spot of Amahoro Stadium's pitch, how far away would the electron be?

Solution

Scale factor: $\frac{0.5 \times 10^{-2}\,\mathrm{m}}{1 \times 10^{-15}\,\mathrm{m}} = 5 \times 10^{12}$ . Atom radius: $5 \times 10^{-11} \times 5 \times 10^{12} = 250\,\mathrm{m}$ . The electron would be about 250 m away, well outside the stadium, with nothing but empty space in between.

The problem with Rutherford's model

If the electrons orbit the nucleus like planets round the Sun, they are always accelerating towards the centre. Classical electromagnetism says that an accelerating charge radiates electromagnetic waves and loses energy. An orbiting electron should therefore spiral into the nucleus in about $10^{-11}\,\mathrm{s}$ , giving out a continuous range of frequencies on the way. But atoms are stable, and a heated gas gives out only certain sharp frequencies (a line spectrum, Lesson 3). Rutherford's model got the nucleus right but could not explain how the electrons behave. Niels Bohr's answer, in 1913, is the subject of the next lesson.

Warning

Safety. Alpha sources used in schools are sealed and weak, but they must still be handled only by the teacher (or by students under direct supervision, where local rules allow), with tongs or a holder, held away from the body and eyes, and returned to their locked store straight after use. Never touch the source face. Alpha particles are stopped by a sheet of paper or a few centimetres of air, but a source swallowed or breathed in would be very harmful.

Warning

Common mistakes. (1) Saying the alpha particles bounced off electrons: electrons are about 7300 times lighter and cannot turn an alpha particle back. (2) Saying "most alpha particles were deflected": most went straight through. (3) Forgetting that the alpha particle has charge $2 e$ , not $e$ , in the closest-approach formula. (4) Thinking Rutherford's model explained line spectra: it could not.

Tip

Link each observation to one conclusion: most go through → mostly empty space; some deflected a lot → concentrated positive charge; a few bounce back → the charge and mass are in a tiny, massive nucleus.

Rutherford Gold Foil Experiment (Backstage Science). Watch a modern version of the experiment, and notice how rarely the detector records a particle at large angles.

✦

Practice

Describe Thomson's plum pudding model and state one thing it explained.
Sketch the apparatus of the alpha-scattering experiment, and explain why it is evacuated.
State the three main observations of the experiment and the conclusion drawn from each.
Find the distance of closest approach of a 7.7 MeV alpha particle to a gold nucleus ( $Z = 79$ ).
Why were alpha particles, and not electrons, used to probe the atom?
Explain why a thin foil was needed, rather than a thick gold sheet.
State the main weakness of Rutherford's model.
A student says: "The experiment proved that atoms have no electrons, because the alpha particles went straight through." Evaluate this.

Show answersHide answers

A sphere of uniformly spread positive charge with electrons embedded in it; it explained that atoms are neutral and can lose electrons to form positive ions.
Source in a lead block, narrow beam, thin gold foil, zinc sulfide screen and microscope rotating round the foil. The chamber is evacuated because alpha particles travel only a few centimetres in air.
Most pass straight through → the atom is mostly empty space; some are deflected through large angles → positive charge is concentrated; a very few rebound → the charge and most of the mass are in a tiny nucleus.
$d = \frac{8.99 \times 10^9 \times 2 \times 79 \times (1.60 \times 10^{-19})^2}{7.7 \times 10^6 \times 1.60 \times 10^{-19}} = 3.0 \times 10^{-14}\,\mathrm{m}$ .
Alpha particles are heavy and fast, so they are not deflected by the light electrons; only a large concentration of positive charge can deflect them strongly. They are also positively charged, so they are repelled by the nucleus.
So that each alpha particle is scattered by at most one nucleus; in a thick sheet, many small deflections would add up and the particles would also be stopped.
An orbiting (accelerating) electron should radiate energy continuously and spiral into the nucleus, so atoms would be unstable and would give continuous, not line, spectra.
Wrong. The particles passed through because the atom is mostly empty space and electrons are too light to deflect them, not because electrons are absent. Electrons were already known from Thomson's work, and the atom's neutrality requires them.

Key points

Thomson (1904): positive sphere with embedded electrons; it could not explain large-angle scattering.
Geiger and Marsden (1909): most alpha particles pass through a gold foil, some are strongly deflected, a very few rebound.
Rutherford (1911): a tiny, dense, positive nucleus; the atom is mostly empty space.
Closest approach: $d = \frac{2Ze^2}{4\pi\varepsilon_0E_k}$ ; nuclei are about $10^{-15}$ – $10^{-14}\,\mathrm{m}$ across.

Sources and credits

6.4 Bohr's Model of the Hydrogen Atom (introduction to Rutherford's model), from University Physics Volume 3 by Samuel J. Ling, Jeff Sanny and William Moebs, OpenStax, licensed under CC BY 4.0; and 30.1 Discovery of the Atom and 30.2 Discovery of the Parts of the Atom, from College Physics 2e by Paul Peter Urone and Roger Hinrichs, OpenStax, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Rutherford Gold Foil Experiment - Backstage Science, BackstageScience, embedded from YouTube.

The three diagrams, the worked examples and the practice exercises are written by Ganzaa.

1Seeing inside something you cannot see

2Thomson's plum pudding model

3The alpha-scattering experiment

4Rutherford's nuclear model (1911)

5Distance of closest approach

6The problem with Rutherford's model