IB Physics · Theme E.1 · by Ahmed Gomaa

Structure of the Atom Field Guide

Zoom from an atom down to its nucleus, fire alpha particles at gold, compare the Thomson atom with the nuclear atom, measure a nucleus by how close an alpha particle gets, and turn energy levels into the bright and dark lines of a spectrum. Sections marked HL are for HL only.

Charges, forces, nucleus Results, particles Electrons, graphs Photons, energy

Part 1The nuclear atomSections 01–07

01From the atom to the nucleus: powers of ten

← Back to the notes: The big picture

Zoom in on a gold atom. Its radius is 1.4 × 10⁻¹⁰ m; its nucleus is only 7.0 × 10⁻¹⁵ m, about 20 000 times smaller. At the scale of a 140 m stadium the nucleus is a 7 mm pea, yet it holds more than 99.9 % of the mass. The atom is almost entirely empty space.

zoom (view width 10⁻ⁿ m)

view width—

atom radius—

nucleus radius—

02Build a nucleus: Z, A and isotopes

← Back to the notes: Nuclear notation

Z protons fix the element; A − Z neutrons can vary. Change N and watch the element stay the same: these are isotopes. Change the number of electrons and you make an ion, but the nucleus does not change. Gold-197: Z = 79, N = 118; Au³⁺ has 76 electrons.

protons Z neutrons N electrons

notation—

isotope—

charge—

03The gold foil: how many come off at each angle?

← Back to the notes: The Geiger–Marsden–Rutherford experiment

A detector moves around the foil. Rutherford’s model (electric force only) predicts that the count rate falls very steeply with angle, as 1/sin⁴(θ/2). Compared with 90°, the rate at 5° is about 69 000 times bigger, and at 150° it is about 3.5 times smaller. Most alpha particles go almost straight on; a very few come back.

detector angle θ (°)

angle θ—

count rate ÷ rate at 90°—

meaning—

04Thomson atom or nuclear atom? Follow the force

← Back to the notes: Why the results mean a tiny nucleus

The same +79e charge, spread through the whole atom (Thomson) or packed into a nucleus. The force on an alpha particle grows as it gets closer, but in the Thomson atom it can never be bigger than at the atom’s edge: 1.9 × 10⁻⁶ N. The nucleus gives 364 N at 1.0 × 10⁻¹⁴ m, about 2 × 10⁸ times more: enough to turn the alpha particle back.

distance r from the centre (log)

distance r—

force: Thomson—

force: nucleus—

05HL · Closest approach: kinetic energy becomes potential energy

← Back to the notes: The distance of closest approach

Press play: the alpha particle slows as it climbs the potential energy hill and stops where E_p = E_k. A 6.0 MeV alpha particle stops 3.8 × 10⁻¹⁴ m from the centre of a gold nucleus, about 5 nuclear radii away: it never reaches the nucleus. Double the energy, half the distance.

kinetic energy (MeV)

closest approach d—

d ÷ nuclear radius—

E_k + E_p—

06HL · Nuclear radius and density

← Back to the notes: Nuclear radius and nuclear density

R = R₀A^⅓ is a cube-root curve: gold-197 has 49 times the nucleons of helium-4 but only 3.7 times the radius. Mass and volume are both proportional to A, so the density is the same for every nucleus: 2.3 × 10¹⁷ kg m⁻³.

nucleon number A

radius R—

R ÷ R(He-4)—

density—

07HL · Deviations: where the strong force begins

← Back to the notes: Deviations from Rutherford scattering

At low energy the counts follow Rutherford’s formula (ratio 1). When the closest approach reaches the touching distance, the strong nuclear force acts too and the counts fall below the prediction. For gold this starts near 26 MeV, for aluminium (smaller charge) near 7 MeV. At E₀: R ≈ k(2e)(Ze)/E₀.

alpha energy (MeV)

d for gold—

d for aluminium—

strong force acts for—

Part 2Spectra and energy levelsSections 08–12

08Emission and absorption: the same lines

← Back to the notes: Emission and absorption spectra

Top: the light from a hot gas, a few bright lines on black. Bottom: white light after it passes through the cool gas, a rainbow with dark lines. The lines match exactly, and every element has its own set: they show which elements are present, even in a star.

element—

visible lines—

match?—

09Energy levels and photons

← Back to the notes: Energy levels and photons

A fall from a higher level to a lower one sends out one photon with E = ΔE = hc/λ. Drag the levels: n = 3 → 2 in hydrogen gives 1.89 eV and 658 nm (red); n = 2 → 1 gives 10.2 eV and 122 nm (ultraviolet). The photon is drawn with its wavelength: bigger gap, shorter wave.

from level n to level n

ΔE—

wavelength—

region—

10Reading a level diagram: longest arrow, shortest wavelength

← Back to the notes: Reading an energy-level diagram

Four levels (−12.0, −6.5, −3.6, −1.8 eV) give six lines. The bars show their wavelengths, sorted. The shortest arrow, 4 → 3 (1.8 eV), gives the longest wavelength, 691 nm; the longest arrow, 4 → 1 (10.2 eV), the shortest, 122 nm. Only two of the six are visible.

longest λ—

shortest λ—

visible lines—

11Photon or electron? Excitation and ionisation

← Back to the notes: Excitation, ionisation and the dark lines

A photon is taken whole or not at all: its energy must equal a gap. An electron can give the atom just one gap and keep the rest. Try 11.0 eV: the photon passes through, the electron excites hydrogen to n = 2 and keeps 0.8 eV. Above 13.6 eV, either one ionises the atom.

energy (eV)

photon—

electron—

gaps from n = 1—

12HL · Bohr orbits and the energy ladder

← Back to the notes: The Bohr model of hydrogen

Left: the allowed orbits, r = n²r₁, drawn to scale. Right: their energies, E = −13.6/n² eV (hydrogen only). Higher orbits are much bigger but their energies crowd together near zero. In n = 3: r = 4.8 × 10⁻¹⁰ m, v = 7.3 × 10⁵ m s⁻¹, E = −1.51 eV.

orbit n

radius—

speed—

energy—

13Practice and self-check

← Back to the notes: Extra practice

Answers checked by calculation. Try each one before you open it.

1. How many protons, neutrons and electrons are in the ion ⁵⁶₂₆Fe²⁺?

Protons 26; neutrons 56 − 26 = 30; electrons 26 − 2 = 24.

2. Find the electric force on an alpha particle 2.0 × 10⁻¹⁴ m from the centre of a gold nucleus (Z = 79).

F = k(2e)(79e)/r² = 8.99 × 10⁹ × 158 × (1.60 × 10⁻¹⁹)²/(2.0 × 10⁻¹⁴)² = 91 N.

3. HL · An alpha particle of 7.7 MeV is fired head-on at gold. Find the distance of closest approach.

d = k(2e)(79e)/E_k = 158 × 8.99 × 10⁹ × 1.60 × 10⁻¹⁹/7.7 × 10⁶ = 3.0 × 10⁻¹⁴ m.

4. HL · Find the radius of an iron-56 nucleus.

R = 1.2 × 10⁻¹⁵ × 56^⅓ = 4.6 × 10⁻¹⁵ m.

5. HL · How many times bigger is the volume of a uranium-238 nucleus than that of a helium-4 nucleus?

V ∝ R³ ∝ A: 238/4 = 59.5.

6. Find the wavelength of the photon emitted when hydrogen falls from n = 4 (−0.85 eV) to n = 2 (−3.40 eV).

ΔE = 2.55 eV = 4.08 × 10⁻¹⁹ J; λ = hc/ΔE = 4.9 × 10⁻⁷ m (488 nm, blue-green).

7. A photon has an energy of 2.0 eV. Find its wavelength.

λ = hc/E = 6.63 × 10⁻³⁴ × 3.00 × 10⁸/(2.0 × 1.60 × 10⁻¹⁹) = 6.2 × 10⁻⁷ m (622 nm).

8. An electron of 12.5 eV hits a hydrogen atom in its ground state. Which levels can the atom reach, and what is the least kinetic energy the electron can keep?

Gaps 10.20 eV (n = 2) and 12.09 eV (n = 3) are below 12.5 eV, so n = 2 or n = 3; the electron keeps at least 12.5 − 12.09 = 0.41 eV.

9. Find the minimum frequency of a photon that can ionise a hydrogen atom in the state n = 3 (−1.51 eV).

f = E/h = 1.51 × 1.60 × 10⁻¹⁹/6.63 × 10⁻³⁴ = 3.6 × 10¹⁴ Hz.

10. Explain why the lines in an absorption spectrum are dark.

Photons with energy equal to a gap between energy levels are absorbed by the cool gas, exciting the atoms. The atoms then re-emit photons of the same energy, but in all directions, so very few continue along the beam: those wavelengths are missing.

11. HL · Find the speed of the electron in the n = 2 orbit of hydrogen (r₁ = 5.3 × 10⁻¹¹ m).

v = nh/(2πmr) with r = 4r₁: v = v₁/2 = (6.63 × 10⁻³⁴/(2π × 9.11 × 10⁻³¹ × 5.3 × 10⁻¹¹))/2 = 1.1 × 10⁶ m s⁻¹.

Before the exam, can you…

use ^A_ZX to find protons, neutrons and electrons, for atoms, ions and isotopes?
describe the gold foil experiment and its three results, and say what each result shows?
explain why the Thomson model cannot explain the alpha particles that bounce back?
HL: find the distance of closest approach with E_k = kq₁q₂/d, using 2e and Z?
HL: use R = R₀A^⅓ and show that every nucleus has the same density?
HL: explain the deviations from Rutherford scattering with the strong nuclear force?
explain line spectra with discrete energy levels and E = hf = hc/λ?
read an energy-level diagram: bigger gap, shorter wavelength?
explain how an absorption spectrum forms, and why its lines are dark?
say what a photon and an electron of the same energy do to an atom, and use the ionisation energy?
HL: explain why Bohr introduced mvr = nh/2π and use r ∝ n² and E = −13.6/n² eV (hydrogen only)?