Nuclei covers the structure of the atomic nucleus, including its size, constant density, mass defect and binding energy, and the binding-energy-per-nucleon curve that explains nuclear stability. It then deals with radioactivity (alpha, beta and gamma decay), the exponential decay law with half-life, mean life and activity, and the energy released in nuclear fission and fusion. The chapter carries steady weightage in NEET Physics and rewards quick numerical practice on decay and binding-energy problems.
Key Concepts
1. Composition of Nucleus
- Protons (Z): positive charge; Neutrons (N): no charge
- Mass number A = Z + N; nuclide notation: ᴬZX
- Isotopes: same Z, different A (e.g., ¹H, ²H, ³H)
- Isobars: same A, different Z (e.g., ⁴⁰₁₈Ar, ⁴⁰₂₀Ca)
Nuclear radius: R = R₀A^(1/3) where R₀ ≈ 1.2 fm
Nuclear density ≈ 2.3 × 10¹⁷ kg/m³ (same for all nuclei - incredibly dense!)
2. Mass Defect and Binding Energy
Mass defect (Δm): The mass of a nucleus is less than the sum of its constituent protons and neutrons.
Δm = [Zmp + Nmn] − M_nucleus
Binding energy: BE = Δm × c² = Δm × 931.5 MeV/u
Binding energy per nucleon (BE/A): Higher → more stable nucleus.
Peak stability: Iron-56 (BE/A ≈ 8.75 MeV). Light and heavy nuclei have lower BE/A.
3. Radioactivity
| Type | Particle | Change | Penetration |
|---|---|---|---|
| Alpha (α) | ⁴₂He | Z → Z−2, A → A−4 | Low (stopped by paper) |
| Beta-minus (β⁻) | Electron | Z → Z+1, A unchanged | Medium (stopped by aluminium) |
| Gamma (γ) | Photon | No change in Z or A | High (needs lead/concrete) |
Radioactive Decay Law
N = N₀e^(−λt)
Half-life: T₁/₂ = 0.693/λ (time for half the nuclei to decay)
After n half-lives: N = N₀/2ⁿ
4. Nuclear Fission and Fusion
| Feature | Fission | Fusion |
|---|---|---|
| Process | Heavy nucleus splits into lighter nuclei | Light nuclei combine to form heavier nucleus |
| Example | ²³⁵U + n → ¹⁴⁴Ba + ⁸⁹Kr + 3n + energy | 4¹H → ⁴He + 2e⁺ + energy (in stars) |
| Energy per event | ~200 MeV | ~24 MeV (but per nucleon: higher) |
| Application | Nuclear reactors, atomic bomb | Hydrogen bomb, stars (sun) |
| Condition | Neutron bombardment | Extremely high temperature (~10⁷ K) |
Solved Examples
Example 1
The half-life of a radioactive element is 5 years. What fraction remains after 15 years?
Answer: n = 15/5 = 3 half-lives. Fraction = 1/2³ = 1/8
Example 2
Find the binding energy per nucleon of ⁵⁶Fe (mass = 55.9349 u, Z = 26, N = 30).
Answer: Δm = 26(1.00783) + 30(1.00867) − 55.9349 = 26.2036 + 30.2601 − 55.9349 = 0.5288 u
BE = 0.5288 × 931.5 = 492.6 MeV. BE/A = 492.6/56 = 8.79 MeV/nucleon
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Quick Revision Points
- R = R₀A^(1/3); nuclear density same for all nuclei (~2.3 × 10¹⁷ kg/m³)
- BE = Δm × 931.5 MeV; BE/A peaks at Fe-56 (most stable)
- α: Z−2, A−4; β⁻: Z+1, A same; γ: no change
- N = N₀e^(−λt); T₁/₂ = 0.693/λ
- Fission: heavy → light + energy (reactors); Fusion: light → heavy + energy (stars)
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Nucleons, Z and A
The nucleus is a tiny, dense core of protons and neutrons that holds nearly all the atom’s mass.
1 u = 1.66 × 10⁻27 kg ≈ 931.5 MeV of energy.
- Protons + neutrons = nucleons; Z = protons, A = total nucleons.
- Neutron count = A − Z (a frequent slip is using A).
- Masses measured in atomic mass unit u, not kilograms.
Nuclear Radius and Density
Nuclear radius depends only on the number of nucleons, making nuclear density the same for every element.
Radius ratio = cube root of A ratio; volume ratio = A ratio.
- Volume V ∝ R3 ∝ A, so V is directly proportional to mass number.
- Mass ∝ A and volume ∝ A ⇒ density is constant for all nuclei.
- Nuclear matter is ~1014 times denser than water.
Mass Defect & Binding Energy
A bound nucleus is lighter than its free nucleons; that missing mass is the binding energy.
Use Δm in u; E_b is energy to break the nucleus into free nucleons.
- Δm is always positive for a bound nucleus (free nucleons − nucleus).
- By E = Δm c2, lost mass appears as binding energy.
- Larger E_b ⇒ more tightly bound, harder to break.
Binding Energy per Nucleon
Binding energy per nucleon measures stability, and its curve explains why nuclei split or fuse.
Average ≈ 8 MeV/nucleon; uranium ≈ 7.6 MeV/nucleon.
- Higher Ē = more stable; iron-region nuclei are the most stable.
- Heavy split (fission) and light merge (fusion) both raise Ē → release energy.
- Flat middle ⇒ nuclear force is short-range and saturates (nearest neighbours only).
Alpha, Beta & Gamma Decay
Unstable nuclei spontaneously emit one of three radiations, each shifting Z and A by fixed rules.
Penetration: γ > β > α; ionising power is the reverse: α > β > γ.
- α = helium nucleus; β⁻ = neutron → proton + electron + antineutrino.
- γ = high-energy photon from nuclear de-excitation, no Z/A change.
- Balance every decay by conserving both charge Z and mass number A.
Exponential Decay Law
Decay is random per nucleus, but the rate of a large sample is perfectly predictable.
Decay is independent of temperature, pressure or chemistry.
- λ = decay constant = probability per second a nucleus decays.
- Large λ ⇒ fast decay; small λ ⇒ long-lived sample.
- Each decay is governed by probability, not the nucleus’s age.
Half-Life & Mean Life
Half-life is the fixed time for half the nuclei to decay; mean life is a bit longer.
After n = t/T12 half-lives, fraction left = (1/2)ⁿ.
- 0.693 ≈ ln 2 ties λ, T12 and τ together — find any from one.
- Mean life τ is longer than the half-life, not shorter.
- Fraction remaining = (1/2)ⁿ; fraction decayed = 1 − (1/2)ⁿ.
Activity of a Sample
Activity is the number of decays per second and follows the same exponential law as N.
SI unit becquerel (Bq) = 1 decay/s; 1 curie (Ci) = 3.7 × 1010 Bq.
- Activity ∝ N, so it also halves every half-life.
- After 3 half-lives activity drops to 1/8 of the start value.
- Bq is decays per second; curie is the older, much larger unit.
Nuclear Fission
A heavy nucleus absorbs a neutron and splits into medium nuclei, releasing energy and neutrons.
Moderator (heavy water/graphite) slows neutrons; control rods absorb them.
- ≈ 200 MeV released per U-235 fission, mostly as fragment kinetic energy.
- Freed neutrons trigger more fissions → self-sustaining chain reaction.
- Slow (thermal) neutrons are captured by U-235 far more readily.
Nuclear Fusion
Light nuclei combine into a heavier one, releasing energy — the process that powers the Sun.
Needs ~107 K to overcome Coulomb repulsion (thermonuclear).
- Per nucleon, fusion releases more energy than fission.
- High temperature lets nuclei beat their mutual electrical repulsion.
- Both fission and fusion release energy: products have higher Ē, lower mass.
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Frequently Asked Questions
Mass defect is the difference between the total mass of the free protons and neutrons and the actual mass of the nucleus they form, since a bound nucleus is always lighter. By Einstein’s relation E equals delta m times c squared, this missing mass shows up as the binding energy, which is the energy needed to break the nucleus back into free nucleons.
The nuclear radius is given by R equals R-naught times A raised to the power one-third, where R-naught is about 1.2 femtometre and A is the mass number. Because volume is proportional to A and mass is also proportional to A, the nuclear density comes out constant for every element at roughly 2.3 times 10 to the power 17 kilograms per cubic metre.
Half-life equals 0.693 divided by the decay constant lambda, and mean life equals 1 divided by lambda, so mean life is about 1.44 times the half-life and is always longer than the half-life. After n half-lives, where n equals time divided by half-life, the fraction of nuclei remaining is one-half raised to the power n.
Yes, Nuclei is part of the NEET Physics syllabus and usually contributes one or two questions, most often from radioactive decay, half-life and binding energy. Focus on the decay law, half-life numericals, and the binding-energy-per-nucleon curve that explains fission and fusion.
In fission a heavy nucleus such as uranium-235 splits into two medium nuclei plus neutrons, releasing about 200 MeV per event and able to sustain a chain reaction. In fusion light nuclei combine into a heavier nucleus, as when deuterium and tritium form helium and release 17.6 MeV, but it needs extremely high temperature and releases more energy per nucleon than fission.