CUET UG Physics Modern Physics Nuclear Physics Binding Energy Radioactive Decay Half Life

Must-Know

• The binding energy of a nucleus is the energy required to separate all nucleons from each other, measured in MeV. For example, the binding energy per nucleon for iron-56 is about 8.8 MeV, the highest of all nuclei.
• The binding energy per nucleon curve peaks around mass number A = 56 (iron), indicating maximum stability; nuclei lighter or heavier are less stable.
• Nuclear fusion occurs in stars because light nuclei (like hydrogen) combine to form heavier nuclei (helium), releasing energy due to increased binding energy per nucleon.
• Nuclear fission releases energy when heavy nuclei (like uranium-235) split into medium-mass nuclei, moving toward higher binding energy per nucleon.
• The mass defect (Δm) is the difference between the sum of masses of individual nucleons and the actual mass of the nucleus; it is related to binding energy by E = Δm × c².
• 1 atomic mass unit (u) is equivalent to 931.5 MeV of energy, used to convert mass defect into binding energy.
• The binding energy of deuterium (²H) is approximately 2.2 MeV, calculated from its mass defect of 0.00239 u.
• Radioactive decay follows the law of exponential decay: N = N₀e^(-λt), where λ is the decay constant and t is time.
• Half-life (T₁/₂) is the time taken for half the radioactive nuclei in a sample to decay; T₁/₂ = (ln 2)/λ ≈ 0.693/λ.
• The mean life (τ) of a radioactive nucleus is the reciprocal of the decay constant: τ = 1/λ, and τ = T₁/₂ / 0.693 ≈ 1.44 × T₁/₂.
• After n half-lives, the fraction of radioactive nuclei remaining is (1/2)ⁿ; after 3 half-lives, 1/8 remains.
• Activity (A) of a radioactive sample is the number of decays per second, measured in becquerel (Bq); 1 Bq = 1 decay/s.
• The SI unit of activity is becquerel (Bq); the older unit is curie (Ci), where 1 Ci = 3.7 × 10¹⁰ Bq.
• In alpha decay, the parent nucleus emits an α-particle (⁴He nucleus), reducing mass number by 4 and atomic number by 2; e.g., ²³⁸U → ²³⁴Th + α.
• In beta-minus decay, a neutron converts to a proton, emitting an electron and antineutrino; atomic number increases by 1, e.g., ¹⁴C → ¹⁴N + e⁻ + ν̄.
• In beta-plus decay (positron emission), a proton converts to a neutron, emitting a positron and neutrino; atomic number decreases by 1, e.g., ¹¹C → ¹¹B + e⁺ + ν.
• Gamma decay involves emission of high-energy photons from an excited nucleus; no change in mass or atomic number, only energy release.
• The half-life of carbon-14 is 5730 years, used in radiocarbon dating of organic materials up to about 50,000 years old.
• The half-life of uranium-238 is 4.468 × 10⁹ years, used in uranium-lead dating of rocks.
• Nuclear forces are short-range, charge-independent, and stronger than electromagnetic forces within the nucleus (range ~1–3 fm).

Difficulty Level

Intermediate — requires understanding of both mathematical relationships (e.g., decay laws) and conceptual applications (e.g., fusion/fission energy release), but formulas are standard and directly from NCERT.

Common CUET Traps

• Trap: Confusing binding energy with binding energy per nucleon. Students may pick uranium as most stable due to high total binding energy. Avoid: Iron-56 has the highest binding energy per nucleon, making it most stable.
• Trap: Assuming activity reduces to zero after two half-lives. Avoid: After two half-lives, 25% of nuclei remain; activity halves each half-life but never reaches zero.
• Trap: Thinking gamma decay changes the element. Avoid: Gamma decay only de-excites the nucleus; no change in Z or A occurs.

Practice MCQs

1. What is the approximate binding energy per nucleon for iron-56?
   A. 5.2 MeV
   B. 6.8 MeV
   C. 8.8 MeV
   D. 10.5 MeV
   Answer: C
   Explanation: Iron-56 has the highest binding energy per nucleon at about 8.8 MeV.
   Why others fail: Option D overestimates based on confusion with total binding energy (~492 MeV).

2. If the half-life of a radioactive isotope is 10 days, what fraction remains after 30 days?
   A. 1/2
   B. 1/4
   C. 1/8
   D. 1/16
   Answer: C
   Explanation: 30 days = 3 half-lives; (1/2)³ = 1/8 remains.
   Why others fail: Option B is correct for two half-lives, tempting if time is miscounted.

3. Which of the following correctly represents the relationship between mean life (τ) and half-life (T₁/₂)?
   A. τ = 0.693 × T₁/₂
   B. τ = T₁/₂ / 0.693
   C. τ = 1.44 × T₁/₂
   D. τ = T₁/₂²
   Answer: C
   Explanation: Mean life τ = 1/λ and T₁/₂ = 0.693/λ, so τ = T₁/₂ / 0.693 ≈ 1.44 × T₁/₂.
   Why others fail: Option A inverts the correct relationship, a common sign error.

4. A nucleus undergoes beta-minus decay. What happens to its atomic number?
   A. Decreases by 1
   B. Increases by 1
   C. Remains the same
   D. Decreases by 2
   Answer: B
   Explanation: In β⁻ decay, a neutron becomes a proton, increasing atomic number by 1.
   Why others fail: Option A applies to β⁺ decay, a frequent mix-up.

5. The mass defect of a nucleus is 0.03 u. What is its binding energy?
   A. 27.945 MeV
   B. 93.15 MeV
   C. 279.45 MeV
   D. 931.5 MeV
   Answer: A
   Explanation: Binding energy = Δm (in u) × 931.5 MeV/u = 0.03 × 931.5 = 27.945 MeV.
   Why others fail: Option C results from using 0.3 u instead of 0.03 u, a decimal error.

Last‑Minute Revision

• ⚠️ Binding energy per nucleon peaks at Fe-56 (~8.8 MeV), not U-235.
• ⚠️ Mass defect × 931.5 = binding energy in MeV.
• ⚠️ T₁/₂ = 0.693 / λ; λ = decay constant.
• ⚠️ After 4 half-lives, 1/16th of sample remains.
• ⚠️ Activity A = λN; decreases exponentially.
• ⚠️ 1 Ci = 3.7 × 10¹⁰ Bq — memorize this conversion.
• ⚠️ α-decay: A ↓4, Z ↓2; emits ⁴He.
• ⚠️ β⁻-decay: Z ↑1; neutron → proton + e⁻ + ν̄.
• ⚠️ β⁺-decay: Z ↓1; proton → neutron + e⁺ + ν.
• ⚠️ γ-decay: no change in A or Z — only energy release.
• ⚠️ Fusion: light → heavy nuclei (e.g., Sun: H → He).
• ⚠️ Fission: heavy → medium nuclei (e.g., ²³⁵U + n → Ba + Kr + 3n).
• ⚠️ Mean life τ = 1.44 × T₁/₂ — not equal to half-life.
• ⚠️ Carbon-14 half-life = 5730 years — used in dating fossils.
• ⚠️ Uranium-238 half-life = 4.468 × 10⁹ years — used in rock dating.
• ⚠️ Nuclear force range: ~1–3 fm; stronger than EM force inside nucleus.
• ⚠️ Binding energy curve rises rapidly for light nuclei, peaks at Fe, drops slowly.
• ⚠️ Deuterium binding energy = 2.2 MeV — verify from NCERT.
• ⚠️ 1 u = 931.5 MeV/c² — essential for mass-energy conversion.
• ⚠️ Radioactive decay is random and spontaneous — cannot predict single nucleus decay.