Coaching Centre
From the Universe to the Atom: review
Module 8 review: a summary of the origins of the elements, the structure of the atom, the quantum mechanical atom, the nucleus and the Standard Model, all the formulas in one place, how the ideas connect, key words, and a mixed HSC-style test.
Module 8 runs from the largest scale there is to the smallest. Yet the same few ideas keep returning: $E = mc^2$ turns radiation into matter in the early universe and mass into starlight in stars; quantised energy levels explain both stellar spectra and the hydrogen atom; and every step inward, from atom to nucleus to quark, was taken by firing something at a target and watching what bounced off. Use this page to pull it all together, then test yourself.
The module at a glance
| Topic | Big ideas | Key evidence and people |
|---|---|---|
| Origins of the elements (8A–8E) | Radiation became matter as the universe cooled ($E = mc^2$, pair production); H and He from Big Bang nucleosynthesis; expansion $v = H_0d$; stellar spectra and classes OBAFGKM; H–R diagram and stellar evolution; p–p chain, CNO cycle, triple-alpha, neutron capture | CMB (Penzias and Wilson); Leavitt's Cepheids, Slipher's redshifts, Hubble and Lemaître; Cannon and Payne; Bethe; neutrinos from the Sun |
| Structure of the atom (8F–8G) | Cathode rays are electrons; $q/m$ from crossed fields; charge is quantised; the nuclear atom; the neutron | Thomson's q/m; Millikan's oil drops; Geiger–Marsden scattering and Rutherford's model; Chadwick's beryllium radiation |
| Quantum mechanical atom (8H–8I) | Bohr's stationary states and $E_n = -13.6/n^2$ eV; Rydberg equation; matter waves $\lambda = h/mv$; Schrödinger's orbitals | Balmer series; Davisson–Germer and G. P. Thomson diffraction |
| Properties of the nucleus (8J–8K) | α, β⁻, β⁺, γ decay; band of stability; $N_t = N_0e^{-\lambda t}$; mass defect and binding energy; fission and fusion | Becquerel, the Curies, Rutherford's transmutation, Meitner and Frisch, Fermi's first reactor; OPAL; ITER and NIF |
| Deep inside the atom (8L) | Quarks, leptons and bosons; baryons and mesons; conservation of charge, baryon and lepton number; accelerators | Particle zoo; Ω⁻ prediction; SLAC deep inelastic scattering; W, Z, top quark, Higgs boson |
Formulas
| Formula | Use |
|---|---|
| $E = mc^2$ | mass–energy equivalence; 1 u = 931.5 MeV/c² |
| $\lambda_{max} = \dfrac{b}{T}$ | surface temperature of a star from its peak; b = 2.898 × 10−3 m K |
| $v = H_0d$ | Hubble's law; age of universe ≈ 1/H0 (with H0 in s−1) |
| $z = \dfrac{\Delta\lambda}{\lambda}$, $v \approx zc$ | redshift and recession velocity |
| $v = \dfrac{E}{B}$, $E = \dfrac{V}{d}$ | velocity selector in Thomson's experiment |
| $\dfrac{q}{m} = \dfrac{v}{Br} = \dfrac{E}{B^2r}$ | charge-to-mass ratio from magnetic deflection |
| $q = \dfrac{mgd}{V}$ | balanced oil drop |
| $E = hf = \dfrac{hc}{\lambda}$ | photon energy in a transition |
| $\dfrac{1}{\lambda} = R\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)$ | hydrogen lines; R = 1.097 × 107 m−1 |
| $E_n = -\dfrac{13.6}{n^2}$ eV | energy levels of hydrogen |
| $\lambda = \dfrac{h}{mv}$ | de Broglie wavelength |
| $N_t = N_0e^{-\lambda t}$, $\lambda = \dfrac{\ln 2}{t_{1/2}}$ | radioactive decay (also activity A) |
| $\Delta m = Zm_p + Nm_n - m_{\text{nucleus}}$, $E_B = \Delta mc^2$ | mass defect and binding energy |
| $E = (m_{\text{before}} - m_{\text{after}})c^2$ | energy released in any nuclear reaction |
Putting it together
- $E = mc^2$ everywhere. It explains pair production and annihilation after the Big Bang (8A), the Sun's power and mass loss (8E), binding energy, fission and fusion (8K), and the creation of new particles in accelerators (8L).
- Energy levels link stars and atoms. Stellar absorption lines (8C) are the same transitions as hydrogen's emission lines (8H). Temperature decides which levels are occupied, and so the spectral class.
- Scattering experiments reveal structure. Geiger–Marsden found the nucleus (8G); Chadwick's recoils found the neutron (8G); Davisson–Germer showed electron waves (8I); SLAC found quarks (8L). Each probe needed a wavelength smaller than the thing being studied.
- The binding energy curve connects stars and reactors. Stars fuse up the left side until iron (8E), reactors split nuclei down from the right (8K), and elements beyond iron need neutron capture.
- Models change with evidence. Plum pudding → Rutherford → Bohr → Schrödinger; three particles → particle zoo → Standard Model. HSC "evaluate" and "assess" questions want the evidence, what each model explained, what it could not, and what replaced it.
In the Sun's core, the p–p chain turns four protons into helium-4 and releases 26.7 MeV. (a) What fraction of the mass is converted? (b) Light from a hydrogen gas cloud near a distant galaxy shows the Hα line (656.3 nm) at 689.1 nm. Find the galaxy's distance using $H_0 = 70$ km s⁻¹ Mpc⁻¹. (c) Which transition in hydrogen produces the line, and what is its photon energy?
Four H atoms have mass $4 \times 1.007825 = 4.0313$ u, equivalent to $4.0313 \times 931.5$ $= 3755$ MeV. Fraction converted $= 26.7/3755 = 0.71\%$.
$z = \dfrac{689.1 - 656.3}{656.3} = 0.0500$, so $v = zc = 0.0500 \times 3.00 \times 10^{5}$ $= 1.50 \times 10^{4}$ km s⁻¹, and $d = \dfrac{v}{H_0}$ $= \dfrac{1.50 \times 10^{4}}{70}$ $= 214$ Mpc.
Hα is $n = 3 \rightarrow n = 2$. $E = E_3 - E_2$ $= -1.51 - (-3.40)$ $= 1.89$ eV $= 3.03 \times 10^{-19}$ J, and $\lambda = hc/E = 656$ nm as emitted.
Key words
| Term | Meaning |
|---|---|
| Pair production | a photon converting into a particle and its antiparticle |
| Annihilation | a particle and its antiparticle converting into photons |
| Cosmic microwave background | radiation released at recombination, now a 2.7 K black-body spectrum |
| Standard candle | an object whose luminosity can be found independently of its distance |
| Hubble constant | the gradient of recession velocity against distance for galaxies |
| Spectral class | OBAFGKM, a temperature sequence based on absorption lines |
| Main sequence | the band on the H–R diagram where stars fuse hydrogen in their cores |
| CNO cycle | hydrogen fusion using C, N and O nuclei as catalysts |
| Cathode rays | streams of electrons in an evacuated tube |
| Stationary state | an allowed electron energy level in which it doesn't radiate |
| Wave function | the quantity describing a particle's wave; its square gives probability |
| Decay constant | the probability per unit time that a nucleus decays |
| Mass defect | the difference between the mass of the separate nucleons and the nucleus |
| Binding energy | the energy needed to separate a nucleus into its nucleons |
| Critical | a chain reaction with k = 1, running at steady power |
| Hadron | a particle made of quarks: a baryon (qqq) or a meson (q q̄) |
| Lepton | a fundamental particle that doesn't feel the strong force, such as the electron or a neutrino |
| Gauge boson | a force-carrying particle: gluon, photon, W or Z |
Test yourself
Which observation is the best evidence that the light elements formed in a hot, dense early universe?
Big Bang nucleosynthesis predicts about 25% helium before any stars formed, matching what is observed.
A main-sequence star has a surface temperature of 25 000 K. Its spectral class and main source of energy are
25 000 K is class B, a massive star with a hot core where the CNO cycle dominates.
In Thomson's experiment the electric field is switched off and the magnetic field doubled. The radius of the beam's path
$r = mv/qB$, with v unchanged.
Which transition in hydrogen emits a photon of 10.2 eV?
$-3.40 - (-13.6) = 10.2$ eV.
A neutron and an alpha particle have the same speed. The ratio of the neutron's de Broglie wavelength to the alpha's is about
λ ∝ 1/m at the same speed; the alpha is about 4 times as massive.
After 3 half-lives, the activity of a sample is 50 Bq. Its original activity was
50 × 2³ = 400 Bq.
Fission of a heavy nucleus releases energy because
The fragments are more tightly bound, so total mass decreases and energy is released.
The Λ⁰ particle has quark composition uds. It is a
Three quarks make a baryon; charge $\tfrac23 - \tfrac13 - \tfrac13 = 0$.
Label the chart of nuclides.
Which part of the module does each piece of evidence belong to?
Order these models and discoveries in time.
Cooler gas in a star's atmosphere produces an spectrum. The Sun gets most of its energy from the chain. Hydrogen lines ending on n = 2 form the series. Joining light nuclei to form a heavier nucleus is called . Protons and neutrons are , made of three quarks.
Scientific models of the atom have changed as new experimental evidence became available. Evaluate this statement with reference to at least three models of the atom from 1897 to the present.
The statement is strongly supported: each major change in the model of the atom followed a specific experimental result the previous model could not explain.
Thomson's discovery of the electron (1897), through the identical $q/m$ of cathode rays from every material, showed atoms contain light negative particles. His plum pudding model placed electrons in a sphere of spread-out positive charge. In 1909, Geiger and Marsden found that about 1 in 8000 alpha particles bounced back from gold foil, which spread-out charge could not do. Rutherford therefore proposed a tiny, dense, positive nucleus with orbiting electrons, confirmed by his quantitative scattering predictions.
Rutherford's model could not explain atomic stability (accelerating electrons should radiate and spiral in) or line spectra. Bohr (1913) added stationary states and quantised transitions, which reproduced hydrogen's spectrum and the Rydberg constant. But it failed for multi-electron atoms and line intensities and had no justification for its rules. De Broglie's matter waves, confirmed by Davisson and Germer's electron diffraction (1927), explained Bohr's orbits as standing waves, and Schrödinger's wave mechanics replaced orbits with orbitals, explaining all atoms. Meanwhile Chadwick's 1932 experiment added the neutron to the nucleus, and SLAC's deep inelastic scattering (from 1967) showed that protons and neutrons are made of quarks.
Overall, the history shows models being kept as long as they fitted the evidence and modified or replaced when an experiment contradicted them, often with parts of the old model retained (the nucleus, energy levels). The current model is still being tested, for example at the LHC.
Thorium-232 (atomic mass 232.038054 u, half-life $1.40 \times 10^{10}$ years) undergoes alpha decay to radium-228 (228.031069 u). (a) Write the nuclear equation. (b) Calculate the energy released, in MeV. (He-4 = 4.002603 u) (c) What fraction of the thorium-232 present when the Earth formed, 4.54 billion years ago, remains today?
(a) $^{232}_{\ 90}\text{Th} \rightarrow {}^{228}_{\ 88}\text{Ra} + {}^4_2\text{He}$
(b) $\Delta m = 232.038054 - (228.031069 + 4.002603) = 0.004382$ u; $E = 0.004382 \times 931.5 = 4.08$ MeV.
(c) $\lambda = \dfrac{\ln 2}{1.40 \times 10^{10}} = 4.95 \times 10^{-11}$ y⁻¹. $\dfrac{N}{N_0} = e^{-\lambda t} = e^{-4.95 \times 10^{-11} \times 4.54 \times 10^{9}} = e^{-0.225} = 0.80$. About 80% remains, which is why thorium is still common in the Earth's crust.