Physics • Year 12 • Module 8 • Lesson 7

Nucleosynthesis and the Origin of Elements

Build HSC Band 5–6 extended-response technique: quantitative mass-defect reasoning, multi-criteria nucleosynthesis comparison, and experimental design evaluating cosmic evidence.

Master · Extended Response

1. Multi-step calculation — mass defect and energy of helium fusion (Band 5–6)

8 marks Band 5–6

Given data. Atomic mass unit: 1 u = 931.5 MeV/c² = 1.661 × 10⁻²⁷ kg. Speed of light: c = 3.00 × 10⁸ m s⁻¹. Relevant masses: proton m_p = 1.007276 u; neutron m_n = 1.008665 u; helium-4 nucleus m(⁴He) = 4.001506 u; carbon-12 nucleus m(¹²C) = 11.996709 u.

Q1. A star on the main sequence fuses hydrogen into helium-4 via the proton–proton chain. The net reaction can be summarised as:

4 protons → ⁴He nucleus + 2 positrons + 2 neutrinos

(Ignore positron masses and neutrino masses for this calculation.)

(a) Calculate the mass defect Δm for the fusion of four protons into one helium-4 nucleus. Give your answer in atomic mass units. (2 marks)
(b) Convert this mass defect to energy in MeV using $E = \Delta m c^2$ and the conversion 1 u = 931.5 MeV/c². (2 marks)
(c) The Sun’s luminosity is 3.85 × 10²⁶ W. Using your answer to (b), calculate the number of proton–proton fusion reactions occurring per second in the Sun. (2 marks)
(d) Explain what your answer to (b) tells us about why hydrogen fusion dominates stellar energy production for most of a star’s lifetime, and relate this to the binding energy curve. (2 marks)

Stuck? (a) Δm = 4 × m_p − m(⁴He); (b) multiply by 931.5; (c) convert MeV to J (1 MeV = 1.602 × 10⁻¹³ J), divide luminosity by energy per reaction; (d) H is most abundant fuel and H fusion is energetically efficient because ⁴He is well up the binding energy curve from ¹H.

2. Data + scenario: primordial helium abundance as evidence for the Big Bang (Band 5–6)

8 marks Band 5–6

Scenario. Astronomers measure the helium-4 abundance in very old, metal-poor regions of the universe (such as dwarf irregular galaxies), which have been only minimally processed by stars. The table below summarises observations of three low-metallicity galaxies.

Galaxy	Metallicity (Z / Z⊙)	Observed He-4 mass fraction (Y)	Best estimate when Z → 0
I Zwicky 18	0.02	0.245 ± 0.006	Y_p = 0.245 ± 0.004
SBS 0335−052	0.035	0.248 ± 0.007
DDO 68	0.018	0.243 ± 0.005

Illustrative data adapted from Peimbert & Peimbert (2010) and Izotov & Thuan (2014). Z⊙ = solar metallicity.

Q2. Analyse and evaluate the data and scenario above to assess how the primordial helium abundance provides evidence for Big Bang nucleosynthesis, and evaluate the limitations of this measurement. In your response you must:

Explain why astronomers measure helium abundance in low-metallicity, metal-poor galaxies rather than in the Sun or the Milky Way.
Use the data to state the estimated primordial helium-4 mass fraction Y_p and explain what this value tells us about conditions during the first three minutes.
Explain how the BBN model predicts this value and why no alternative stellar model can reproduce a 25% helium abundance across all environments.
Identify at least two uncertainties or limitations in using this method as evidence for BBN.
State an independent piece of evidence (not helium abundance) that also supports the Big Bang model.

Stuck? Plan: why low-metallicity? (minimal stellar processing, closer to primordial) → Y_p ≈ 0.245 (about 25% He by mass) → BBN prediction: p+n fusion in first 3 min at baryon density ρ≈1 kg/m³; stellar sources cannot generate 25% universally → limitations: spectral line measurement uncertainty, stellar He contamination even in low-Z galaxies → independent evidence: cosmic microwave background, deuterium abundance.

Answers — Do not peek before attempting

Q1 — Sample Band 6 response (8 marks), annotated

(a) Mass defect (2 marks): Δm = 4 × m_p − m(⁴He) = 4 × 1.007276 − 4.001506 = 4.029104 − 4.001506 = 0.027598 u [1 — correct substitution]. Δm = 0.0276 u (to 3 sig. fig.) [1 — correct answer with units]. Accept 0.027598 u or rounded values.

(b) Energy in MeV (2 marks): E = Δm × 931.5 = 0.027598 × 931.5 = 25.7 MeV [1 — correct formula used]. E ≈ 25.7 MeV [1 — correct answer with units]. Accept 25.6–25.8 MeV for rounding.

(c) Reactions per second (2 marks): Convert: 25.7 MeV × 1.602 × 10⁻¹³ J MeV⁻¹ = 4.12 × 10⁻¹² J per reaction [1 — correct conversion]. Number of reactions = P / E = 3.85 × 10²⁶ / 4.12 × 10⁻¹² = 9.3 × 10³⁷ reactions per second [1 — correct calculation with correct order of magnitude]. Accept 9.0–9.6 × 10³⁷.

(d) Hydrogen fusion and binding energy (2 marks): The large energy release per reaction (~25.7 MeV) means hydrogen fusion is highly efficient per unit mass consumed [1 — energy efficiency]. On the binding energy curve, hydrogen (A=1) lies at essentially zero binding energy per nucleon, while ⁴He sits at ~7.1 MeV/nucleon; the large rise from H to He represents a substantial increase in nuclear stability, releasing significant energy. Because the Sun contains ~70% hydrogen by mass, this process can sustain the Sun’s luminosity for ~10 billion years [1 — links large fuel reservoir to longevity on binding energy curve].

Marking criteria summary (8 marks): (a) 1 = correct formula (Δm = 4m_p − m_He); 1 = correct numerical answer ~0.0276 u. (b) 1 = uses E = Δm × 931.5; 1 = correct answer ~25.7 MeV. (c) 1 = converts MeV to J; 1 = divides luminosity by energy per reaction to get ~9.3 × 10³⁷ s⁻¹. (d) 1 = links large energy release to efficiency/longevity; 1 = correctly interprets rise of H to He on binding energy curve.

Q2 — Sample Band 6 response (8 marks), annotated

Why low-metallicity galaxies: Astronomers use metal-poor galaxies because they have experienced minimal stellar processing — few generations of stars have formed and died — so their interstellar medium is closest to the primordial composition. In the Milky Way or the Sun, stellar nucleosynthesis has added heavy elements and recycled helium, raising the He abundance above its primordial level and obscuring the BBN signal [1 — correctly explains stellar contamination problem].

Y_p value and what it means: Extrapolating the three measurements to zero metallicity gives a primordial helium mass fraction Y_p ≈ 0.245 (±0.004), i.e. approximately 24.5% of baryonic matter by mass [1 — reads data correctly]. This value is consistent with BBN theory, which predicts that at the baryon-to-photon ratio measured from the cosmic microwave background, about 25% of hydrogen and neutrons should have fused into helium-4 in the first three minutes when the temperature was ~10⁹–10¹⁰ K [1 — links to BBN conditions].

BBN predicts 25%; stellar sources cannot: BBN theory uniquely predicts this universal helium floor because the conditions (temperature, density, baryon-to-photon ratio) in the early universe are constrained by the CMB and are not arbitrary. Even the most prolific stellar nucleosynthesis accounts for only ~5–10% additional helium over cosmic time; no stellar model can produce a 25% helium abundance uniformly across all ancient, metal-poor regions simultaneously [1 — distinguishes BBN prediction from stellar origin; 1 — quantitative argument that stellar He is insufficient].

Limitations: (1) Even low-metallicity galaxies have undergone some stellar processing, so the measured He abundance is a slight overestimate of the true primordial value; extrapolation to Z=0 introduces model-dependent uncertainty [1 — stellar contamination remains]. (2) Helium abundance is determined from hydrogen and helium recombination spectral lines, which depend on temperature and ionisation conditions of the nebula; systematic errors in these assumptions propagate into Y_p [1 — observational/spectroscopic uncertainty].

Independent evidence: The cosmic microwave background (CMB): the Big Bang model predicts a near-perfect blackbody radiation spectrum with temperature ~2.7 K permeating the universe and specific angular anisotropies; these have been measured by COBE, WMAP, and Planck to extraordinary precision and are fully consistent with BBN parameters [1 — names CMB or deuterium abundance as independent evidence and explains]. Accept also: primordial deuterium abundance; large-scale structure of the universe; Hubble expansion.

Marking criteria summary (8 marks): 1 = explains why low-metallicity galaxies (minimal stellar processing, closest to primordial). 1 = correctly reads Y_p ≈ 0.245 from extrapolation. 1 = links Y_p ≈ 25% to BBN temperature/density conditions. 1 = argues BBN uniquely predicts this floor. 1 = quantitative or qualitative argument that stellar nucleosynthesis cannot explain a universal 25% He floor. 1 = names and explains at least one valid limitation (stellar contamination). 1 = names and explains a second valid limitation (spectroscopic/model uncertainty). 1 = names and explains at least one independent piece of evidence supporting the Big Bang.