Nuclei

Nuclei

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

• Nucleus: dense core of an atom containing Z protons and N neutrons (nucleons); mass number A = Z + N.
• 1 u = 931.5 MeV/c² — use this to convert mass defect into binding energy.
• Mass defect Δm = [Z·m_p + N·m_n] − m_nucleus; binding energy E_b = Δm·c².
• Binding energy per nucleon = E_b/A; peaks at A ≈ 56 (Fe-56), explaining energy release in both fission and fusion.
• Half-life: T₁/₂ = ln2/λ = 0.693/λ; after n half-lives, remaining = N₀·(½)ⁿ.
• Activity: A = λN = A₀·e^(−λt); SI unit is becquerel (Bq).
• Alpha decay: A → A−4, Z → Z−2; Beta⁻ decay: Z → Z+1, N → N−1; Gamma: no change in A or Z.
• Nuclear radius: R = R₀·A^(1/3) with R₀ ≈ 1.2 fm; nuclear density ≈ 10¹⁷ kg/m³ (constant for all nuclei).

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Definitions and Key Quantities

The atomic mass unit (u) equals 1.66 × 10⁻²⁷ kg, and by mass–energy equivalence, 1 u corresponds to 931.5 MeV/c². The mass defect Δm is the difference between the sum of individual nucleon masses and the actual nuclear mass. Multiplying Δm by c² gives the binding energy E_b — the energy released when the nucleus forms from free nucleons. Dividing E_b by the mass number A yields the binding energy per nucleon, a direct measure of nuclear stability.

Nuclear Radius and Density

The empirical nuclear radius formula R = R₀·A^(1/3) with R₀ ≈ 1.2 fm shows that nuclear volume scales linearly with A. Substituting: for A = 56 (iron), R ≈ 1.2 × 56^(1/3) ≈ 3.6 fm. Because volume ∝ A and mass ∝ A, nuclear density ρ = mass/volume ≈ 10¹⁷ kg/m³ — essentially identical for all nuclides. This constancy supports the liquid drop model analogy.

Radioactive Decay Law

Decay follows N = N₀·e^(−λt), a first-order process independent of temperature, pressure, or chemical environment. The decay constant λ has units s⁻¹. Half-life T₁/₂ = 0.693/λ relates to mean life τ = 1/λ. The activity A = λN also decays exponentially: A = A₀·e^(−λt).

Q-Value of Nuclear Reactions

The energy released or absorbed is Q = (mass_initial − mass_final)·c². For α-decay: Q = [M_parent − M_daughter − M_He]·c². A positive Q means the decay is energetically allowed.

Binding Energy Per Nucleon Curve

The curve rises to a maximum of ~8.8 MeV/nucleon near A ≈ 56 (Fe-56) and then gradually falls. This shape tells us: (1) fission of heavy nuclei (A > 120) toward the peak releases energy, and (2) fusion of light nuclei (A < 20) toward the peak also releases energy. No energy is released fusing nuclei beyond the peak.

Nuclide	Binding Energy per Nucleon (MeV)
²H	~1.1
⁴He	~7.1
¹²C	~7.7
⁵⁶Fe	~8.8 (peak)
²³⁵U	~7.6

CUET exam pointers: MCQs frequently test the decay law formula, unit conversion between MeV and joules, and identification of decay-type changes to Z and A. Memorise T₁/₂ = 0.693/λ and R = R₀·A^(1/3) exactly.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Liquid Drop Model

The semi-empirical mass formula treats nuclear binding as a sum of five terms: volume (∝ A), surface (∝ −A^(2/3)), Coulomb (∝ −Z(Z−1)/A^(1/3)), asymmetry (∝ −(N−Z)²/A), and pairing (±δ/A^(3/4)). The volume term dominates for large A; the Coulomb term (repulsion between protons) becomes increasingly destabilising in heavy nuclei, ultimately causing the strong-force attraction to lose dominance beyond lead (Z ≈ 82). This is why nuclei beyond the valley of stability undergo β-decay.

Mass Defect in a Worked Calculation

For ⁵⁶Fe₂₆: Z = 26, N = 30, A = 56. Approximate masses: m_p = 1.00728 u, m_n = 1.00867 u, m_nucleus(⁵⁶Fe) ≈ 55.9349 u. Δm = (26 × 1.00728 + 30 × 1.00867) − 55.9349 = 56.4635 − 55.9349 = 0.5286 u. E_b = 0.5286 × 931.5 = 492.4 MeV. Binding energy per nucleon = 492.4/56 ≈ 8.8 MeV — the highest value for any stable nuclide.

Half-Life Calculation — Carbon-14 Dating

For carbon-14 (T₁/₂ = 5730 yr), find remaining activity after 11,460 yr: n = 11,460/5730 = 2 half-lives. Remaining fraction = (½)² = ¼. If A₀ = 800 Bq initially, remaining A = 800 × 0.25 = 200 Bq.

Chain Reactions and Critical Mass

In uranium-235 fission, each event releases ~3 neutrons. A chain reaction sustains when at least one neutron induces another fission. Critical mass is the minimum fissile material needed to maintain a self-sustaining reaction. In a nuclear reactor, control rods absorb excess neutrons to keep the reaction exactly critical. In an atomic bomb, multiple subcritical masses are assembled supercritically for an uncontrolled chain reaction.

Common Mistakes to Avoid

• Confusing atomic mass with nuclear mass: atomic mass includes orbital electrons; subtract ~Z × 0.00055 u for nuclear mass in precise calculations.
• Unit errors: λ is often quoted in s⁻¹ or yr⁻¹; convert consistently before applying T₁/₂ = 0.693/λ.
• Forgetting the exponential nature: halving the sample does not halve the decay constant — λ is a property of the nuclide, not the quantity.
• Q-value sign conventions: positive Q means exothermic (energy released); a negative Q means the reaction or decay is endothermic and will not occur spontaneously without additional energy input.
• Alpha decay atomic number confusion: the daughter nuclide has Z−2 protons, but if you use atomic masses in the Q-formula, account for the electron differences correctly.

Practice Prompts

1. A ²³⁸U nucleus undergoes successive α and β⁻ decays to form ²⁰⁶Pb. Determine how many α and β⁻ decays occur.
2. The activity of a radioactive sample drops from 6400 Bq to 800 Bq in 30 minutes. Calculate the decay constant and half-life of the sample.

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