Nuclei
'The nucleus is the powerhouse of the atom — it contains 99.9% of the mass in 10⁻¹⁵ of the volume. And it is held together by the STRONGEST force in nature.'
1. Chapter Overview
This chapter explores the NUCLEUS — its composition, properties, and behaviour. Topics include: NUCLEAR COMPOSITION (protons and neutrons — collectively called NUCLEONS), ATOMIC MASS UNIT and mass defect, BINDING ENERGY (the energy holding the nucleus together), RADIOACTIVITY (alpha, beta, gamma decay — the spontaneous disintegration of unstable nuclei), HALF-LIFE and mean life, NUCLEAR FISSION (splitting of heavy nuclei), and NUCLEAR FUSION (combining light nuclei).
2. Nuclear Composition and Size
- Nucleus: Z protons + N neutrons. A = Z + N (mass number).
- Isotopes: Same Z, different A. Isobars: Same A, different Z. Isotones: Same N, different Z.
- Nuclear size: R = R₀ A¹/³, where R₀ ≈ 1.2 × 10⁻¹⁵ m (1.2 fm).
- Nuclear density: Approximately 2.3 × 10¹⁷ kg/m³ — CONSTANT for all nuclei. 'If the Earth were compressed to nuclear density, it would be about 200 m in diameter!'
3. Mass Defect and Binding Energy
Mass Defect
- Δm = [Zm_p + (A−Z)m_n] − M_nucleus — the difference between the mass of the SEPARATE nucleons and the mass of the NUCLEUS.
- 'The mass of a nucleus is LESS than the sum of masses of its constituent nucleons. Where does the missing mass go? It is released as ENERGY.'
Binding Energy
- E_b = Δm × c² — the energy equivalent of the mass defect.
- 'Binding energy is the energy REQUIRED to separate the nucleus into its individual nucleons.'
- Binding energy per nucleon (BE/A) : Measure of NUCLEAR STABILITY. Higher BE/A → more stable nucleus.
- Peak: BE/A ≈ 8.8 MeV for iron (A ≈ 56). 'Iron is the MOST STABLE nucleus.'
- Light nuclei (fusion) and heavy nuclei (fission) both move TOWARDS iron — releasing energy.
Worked Example 1
Problem: Find the binding energy of helium-4 (2p+2n). Given: m_p = 1.007825 u, m_n = 1.008665 u, m_He = 4.002603 u. Solution: Δm = 2(1.007825) + 2(1.008665) − 4.002603 = 2.015650 + 2.017330 − 4.002603 = 0.030377 u. E_b = 0.030377 × 931.5 = 28.3 MeV. BE/A = 28.3/4 = 7.07 MeV.
4. Radioactivity
Types of Radioactive Decay
| Decay Type | What is Emitted | Change in A | Change in Z | Example |
|---|---|---|---|---|
| Alpha (α) | Helium nucleus (²He⁴) | A → A−4 | Z → Z−2 | ²²⁶Ra⁸⁸ → ²²²Rn⁸⁶ + α |
| Beta (β⁻) | Electron + antineutrino | Unchanged | Z → Z+1 | ¹⁴C₆ → ¹⁴N₇ + β⁻ + ν̄ |
| Beta (β⁺) | Positron + neutrino | Unchanged | Z → Z−1 | — |
| Gamma (γ) | High-energy photon | Unchanged | Unchanged | Follows α/β decay to shed excess energy |
Properties
- Alpha: Low penetration (stopped by paper), HIGH ionisation. 'Alpha is the HEAVY lifter — strong ioniser, short range.'
- Beta: Medium penetration (stopped by 3 mm Al), medium ionisation.
- Gamma: HIGH penetration (several cm of lead), LOW ionisation. 'Gamma is the DEEP penetrator — weak ioniser, long range.'
5. Law of Radioactive Decay
- N = N₀ e^(−λt) — exponential decay. λ = decay constant.
- Half-life (T_½) : T_½ = ln 2/λ = 0.693/λ.
- Mean life (τ) : τ = 1/λ = T_½/ln 2 ≈ 1.44 T_½.
- Activity: A = dN/dt = λN = λN₀ e^(−λt). Unit: Becquerel (Bq) = 1 decay/s.
- 'After one half-life, HALF the nuclei remain. After two half-lives, ONE-QUARTER. After n half-lives, (1/2)ⁿ remain.'
Worked Example 2
Problem: The half-life of ¹⁴C is 5730 years. A sample has activity 15 decays/min. After how long will the activity be 3.75 decays/min? Solution: 15 × (1/2)ⁿ = 3.75 ⇒ (1/2)ⁿ = 0.25 ⇒ n = 2 half-lives. Time = 2 × 5730 = 11460 years.
6. Nuclear Fission
- 'A heavy nucleus (like ²³⁵U) SPLITS into two lighter nuclei when struck by a neutron — releasing ENERGY and more NEUTRONS.'
- ²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n + Energy (~200 MeV) .
- Chain reaction: The neutrons released can trigger MORE fission events — leading to a SELF-SUSTAINING reaction.
| Type | Description | Application |
|---|---|---|
| Controlled chain reaction | Neutron-absorbing control rods regulate the rate | Nuclear POWER REACTOR |
| Uncontrolled chain reaction | No regulation — exponential increase | Nuclear WEAPON |
| Critical mass | Minimum mass needed to sustain a chain reaction | Design parameter |
7. Nuclear Fusion
- 'TWO LIGHT nuclei COMBINE to form a HEAVIER nucleus — releasing enormous energy.'
- Example: H¹ + H¹ → H² + e⁺ + ν (proton-proton chain in stars).
- H² + H³ → He⁴ + n + 17.6 MeV (potential fusion reactor reaction).
Challenges of Fusion Power
- Temperature: Requires ~10⁸ K to overcome Coulomb repulsion (Tokamak, stellarator).
- Containment: No material can withstand such temperatures — magnetic confinement needed.
- Energy gain: Must produce MORE energy than consumed — the 'breakeven' point (recently achieved in labs).
Fission vs Fusion
| Aspect | Fission | Fusion |
|---|---|---|
| Process | SPLITTING heavy nuclei | COMBINING light nuclei |
| Fuel | Uranium, Plutonium | Hydrogen isotopes (deuterium, tritium) |
| Energy per reaction | ~200 MeV | ~17.6 MeV (but per unit mass: fusion is HIGHER) |
| Waste | Radioactive (long half-lives) | Helium (non-radioactive) |
| Containment | Natural (subcritical assembly) | Magnetic confinement required |
| Status | COMMERCIAL — operational power plants | Experimental — NOT yet commercial |
8. Common Mistakes
- Mass defect is POSITIVE: Δm = Σm_nucleons − M_nucleus > 0. The nucleus has LESS mass.
- Binding energy is energy RELEASED: When nucleons combine, energy is released — equal to the binding energy. To separate them, the SAME energy must be SUPPLIED.
- Half-life is CONSTANT: T_½ is INDEPENDENT of temperature, pressure, chemical state — an intrinsic property of the isotope.
- Fission vs fusion products: Fission produces RADIOACTIVE waste. Fusion produces HELIUM (non-radioactive) — but creates radioactive reactor components through neutron activation.
9. CBSE Exam Focus
- Mass defect and binding energy — Δm, E_b = Δm × 931.5 MeV, BE/A curve
- Radioactive decay — N = N₀e^(−λt), half-life (T_½ = 0.693/λ), mean life (τ = 1/λ)
- Alpha, beta, gamma — properties, penetration, ionisation
- Nuclear fission — chain reaction, critical mass, nuclear reactor
- Nuclear fusion — conditions, energy release, comparison with fission
10. Self-Test
Q1: The half-life of a radioactive substance is 10 days. How much of a 1 g sample remains after 30 days? A1: n = 30/10 = 3 half-lives. Remaining = 1 × (1/2)³ = 1/8 = 0.125 g.
Q2: A sample has activity 240 Bq. After 6 hours, activity is 30 Bq. Find the half-life. A2: 240 × (1/2)ⁿ = 30 ⇒ (1/2)ⁿ = 30/240 = 1/8 ⇒ n = 3 half-lives in 6 hours. T_½ = 2 hours.
Q3: Find the energy released when 1 g of ²³⁵U undergoes fission (approx. 200 MeV per fission). A3: Number of nuclei = (1/235) × 6.022×10²³ = 2.56×10²¹. Energy = 2.56×10²¹ × 200 × 1.6×10⁻¹³ = 8.2×10¹⁰ J. 'One gram of uranium releases about the same energy as 2.5 TONNES of coal.'
Q4: The binding energy per nucleon of ⁵⁶Fe (mass = 55.9349 u, m_p = 1.007825 u, m_n = 1.008665 u) is? A4: Δm = 26(1.007825) + 30(1.008665) − 55.9349 = 26.20345 + 30.25995 − 55.9349 = 0.5285 u. E_b = 0.5285×931.5 = 492.3 MeV. BE/A = 492.3/56 = 8.79 MeV.
Q5: A radioactive element decays to 1/16 of its initial amount in 20 days. Find its half-life and decay constant. A5: (1/2)ⁿ = 1/16 ⇒ n = 4 half-lives. T_½ = 20/4 = 5 days. λ = 0.693/5 = 0.1386 day⁻¹.
11. Conclusion
Nuclear physics revealed the INNERMOST structure of matter:
- NUCLEUS: 'A tiny, dense core held together by the STRONG NUCLEAR force — stronger than electromagnetism but very short range.'
- BINDING ENERGY: 'The "glue" holding the nucleus together — measured by the MISSING MASS.'
- RADIOACTIVITY: 'Unstable nuclei decay — spontaneously transforming into other elements.'
- FISSION: 'Splitting atoms to RELEASE energy — the principle behind nuclear power.'
- FUSION: 'Combining atoms — the energy of the SUN. The holy grail of clean energy on Earth.'
'Nuclear physics has given us both the ATOMIC BOMB and NUCLEAR POWER — the same science, different applications. Understanding it is essential for an informed citizen of the nuclear age.'
