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IB DP Physics · Theme E Nuclear and quantum physics · E.1 · ~6 min read

E.1 Structure of the atom

The Geiger–Marsden–Rutherford experiment, nuclear notation, emission and absorption spectra, discrete energy levels and photons, and (HL) nuclear radius and density, distance of closest approach and the Bohr model of hydrogen.

Rutherford scatteringNuclear notationEmission & absorption spectraEnergy levelsPhotons E = hfNuclear radius (HL)Bohr model (HL)
RSby Riyaz Sir
mentorriyaz.in
No tag = SL and HLHL ONLY HL students onlyDATA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Rutherford scattering showed that an atom is mostly empty space with a tiny, dense, positive nucleus.
  • Nuclear notation AZX: A = nucleon (mass) number, Z = proton number; N = A − Z neutrons. Isotopes have the same Z, different N.
  • Electrons in atoms have discrete energy levels. A transition between levels emits or absorbs a photon with E = hf = ΔE, so each element has its own line spectrum.
  • HL: nuclear radius R = R0A1/3 (so all nuclei have about the same density), closest-approach calculations, and the Bohr model with E = −13.6/n2 eV.

By the end you can

  • Describe the Geiger–Marsden–Rutherford experiment and what it showed.
  • Use nuclear notation and identify isotopes.
  • Explain emission and absorption spectra with energy levels and photons.
  • Calculate photon energies, frequencies and wavelengths from energy-level transitions.
  • Use R = R0A1/3 and explain why nuclear density is constant. HL
  • Calculate the distance of closest approach and explain deviations from Rutherford scattering. HL
  • Use the Bohr model of hydrogen. HL

1The nuclear atom

gold nucleusmost α pass straight through (blue)a few deflected (orange); about 1 in 8000 bounce back (red): a tiny, dense, positive nucleus
Alpha particles fired at thin gold foil. The rare large deflections need a tiny, concentrated positive charge.
ObservationConclusion
most α particles pass straight throughthe atom is mostly empty space
a few are deflected through large anglesthere is a concentrated positive charge
about 1 in 8000 bounce backthe nucleus is tiny and contains almost all the mass
  • Before Rutherford the "plum pudding" model spread positive charge through the whole atom; it could not produce large deflections.

2Nuclear notation

AZX
A = nucleon number (protons + neutrons), Z = proton number (atomic number)
  • Example: 23892U has 92 protons and 238 − 92 = 146 neutrons. 23592U is an isotope of it with 143 neutrons.
  • Units: atomic masses are given in u (unified atomic mass units); 1 u = 1.66 × 10−27 kg ≈ 931.5 MeV c−2.

3Energy levels and spectra

emissionabsorptionSame element, same wavelengths: bright lines from a hot gas,dark lines when white light passes through a cooler gas.
Bright lines from a glowing gas match the dark lines when light shines through the same cool gas.
n = 1-13.60 eVn = 2-3.40 eVn = 3-1.51 eVn = 4-0.85 eVn = 5-0.54 eV0 (ionised)10.2 eV: UV1.89 eV: red, 656 nmEmission: a falling electron emits one photon with E = hf = ΔE. (Not to scale.)
Energy levels of hydrogen. Each downward arrow is a possible photon.
E = hf = hcλ
photon energy; h = 6.63 × 10−34 J s
  • Emission: an excited electron drops to a lower level and emits a photon of energy equal to the gap. Absorption: a photon is absorbed only if its energy exactly matches a gap; the electron jumps up.
  • Electronvolt: 1 eV = 1.60 × 10−19 J, the energy an electron gains through 1 V.
Worked example

Find the wavelength of the photon emitted when hydrogen's electron falls from n = 3 (−1.51 eV) to n = 2 (−3.40 eV).

  1. ΔE = 3.40 − 1.51 = 1.89 eV = 3.02 × 10−19 J M1
  2. λ = hc/ΔE = 6.63 × 10−34 × 3.00 × 108/3.02 × 10−19 M1
  3. = 6.6 × 10−7 m (656 nm, red) A1
Common mistakeUsing the energy of one level instead of the gap between two. The photon carries the difference, ΔE.
Exam tipSpectral lines are discrete because energy levels are discrete. That sentence earns the explanation mark in most "why lines?" questions.

4Size and density of the nucleus HL

R = R0A1/3 DATA BOOKLET
R0 = 1.20 × 10−15 m (1.20 fm)
Worked example

Find the radius of a lead-208 nucleus and the density of nuclear matter.

  1. R = 1.20 × 10−15 × 2081/3 = 7.1 × 10−15 m A1
  2. density = A × 1.66 × 10−27/(⁴⁄₃πR03A) — the A cancels M1
  3. = 2.3 × 1017 kg m−3, the same for every nucleus A1

Distance of closest approachHL ONLY

  • A head-on α particle stops where its kinetic energy has all become electric potential energy: Ek = k(2e)(Ze)/d.
Worked example

A 5.0 MeV α particle is fired head-on at a gold nucleus (Z = 79). Find the distance of closest approach.

  1. Ek = 5.0 × 106 × 1.6 × 10−19 = 8.0 × 10−13 J M1
  2. d = k(2e)(79e)/Ek = 8.99 × 109 × 158 × (1.6 × 10−19)2/8.0 × 10−13 M1
  3. = 4.5 × 10−14 m: an upper limit for the nuclear radius A1
  • Deviations: at higher energies the α particles get close enough to feel the strong nuclear force, and scattering no longer follows the Coulomb-only prediction. This gives an estimate of the nuclear radius.

5The Bohr model of hydrogen HL

E = −13.6n2 eV DATA BOOKLET
energy of level n in hydrogen
mvr = nh2π DATA BOOKLET
the electron's angular momentum is quantised
  • Bohr's idea: the electron orbits only in allowed orbits where its angular momentum is a whole number of h/2π. It works for hydrogen but fails for atoms with more electrons; quantum mechanics (E.2) replaces it.
Worked example

Find the ionisation energy of hydrogen from its ground state, and the photon energy for n = 4 → n = 2.

  1. ionisation: 0 − (−13.6) = 13.6 eV A1
  2. E4 = −0.85 eV, E2 = −3.40 eV: photon 2.55 eV (486 nm, blue-green) M1A1
Linking questionThe energy-level idea explains greenhouse absorption (B.2), the photoelectric effect (E.2) and how astronomers read stellar composition and redshift (C.5, E.5).
Idea for your IA / explorationUse a diffraction grating and a hydrogen or helium discharge tube to measure line wavelengths, and compare them with predicted energy-level gaps.

Test yourself — cover the answers first

Quick check

  1. What did the large-angle α scattering show?
  2. How many neutrons are in 146C?
  3. What is an isotope?
  4. Why do atoms give line spectra rather than continuous ones?
  5. Convert 2.0 eV to joules.
  6. HL What happens to nuclear radius if A is multiplied by 8?

Exam-style

  1. A photon of wavelength 122 nm is absorbed by hydrogen in its ground state. Which level does the electron reach?
  2. HL Find the ratio of the radii of 216Po and 27Al nuclei.
Show answers
  1. a tiny, dense, positive nucleus
  2. 8
  3. same number of protons, different number of neutrons
  4. energy levels are discrete, so only certain photon energies occur
  5. 3.2 × 10−19 J
  6. it doubles
  7. E = hc/λ = 1.63 × 10−18 J = 10.2 eV
    −13.6 + 10.2 = −3.4 eV
    n = 2
  8. R ∝ A1/3
    (216/27)1/3 = 81/3
    2
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.