Half-life and Decay — Revision Notes

⚡ 30-Second Revision

Half-life (t₁/₂): — Time for 50% decay. Unique for each isotope.
Decay Constant (λ): — Probability of decay per unit time. t₁/₂ = ln(2)/λ.
Exponential Decay: — N(t) = N₀ * e^(-λt).
Activity (A): — Rate of decay (decays/sec). A = λN. Units: Bq (1 dps), Ci (3.7x10¹⁰ Bq).
Alpha Decay: — ⁴₂He emitted. Z-2, A-4.
Beta Decay: — e⁻ or e⁺ emitted. Z+1 (β⁻) or Z-1 (β⁺), A unchanged.
Gamma Decay: — γ-ray emitted. Z, A unchanged.
C-14: — t₁/₂ ~5,730 yrs. Carbon dating (organic, up to ~60k yrs).
I-131: — t₁/₂ ~8 days. Thyroid diagnosis/treatment.
Co-60: — t₁/₂ ~5.27 yrs. Radiotherapy, sterilization.
U-235: — t₁/₂ ~7x10⁸ yrs. Nuclear fuel.
U-238: — t₁/₂ ~4.5x10⁹ yrs. Parent in decay chain, breeder reactor fuel.

2-Minute Revision

Half-life (t₁/₂) is the fundamental concept describing the time required for half of a radioactive sample to decay. It's an intrinsic property of each radionuclide, independent of external conditions.

The decay process follows an exponential law, N(t) = N₀ * e^(-λt), where λ is the decay constant, inversely related to half-life (t₁/₂ = ln(2)/λ). Activity (A), the rate of decay, is A = λN, measured in Becquerel (Bq) or Curie (Ci).

Key decay types include alpha (emission of helium nucleus, reducing atomic number by 2), beta (emission of electron/positron, changing atomic number by ±1), and gamma (emission of high-energy photons, no change in atomic/mass number).

These concepts are vital for UPSC due to their applications: Carbon-14 dating for archaeology, medical isotopes like Iodine-131 (thyroid) and Cobalt-60 (radiotherapy), nuclear power generation (Uranium isotopes), and the critical challenge of nuclear waste management, where half-lives dictate storage duration and hazard levels.

Remember, shorter half-lives mean faster decay and quicker reduction in radioactivity, while longer half-lives imply prolonged hazard.

5-Minute Revision

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable one, emitting radiation. The rate of this process is quantified by the half-life (t₁/₂), the time taken for half of the radioactive nuclei in a sample to decay.

This is a constant for each specific isotope, unaffected by temperature, pressure, or chemical state. The mathematical backbone is the exponential decay law, N(t) = N₀ * e^(-λt), where N(t) is the remaining nuclei, N₀ is the initial count, λ is the decay constant, and t is time.

The decay constant (λ) represents the probability of decay per unit time and is related to half-life by t₁/₂ = ln(2)/λ. Activity (A), the rate of decay (disintegrations per second), is given by A = λN, with units of Becquerel (Bq) or Curie (Ci).

Three primary types of decay are crucial: Alpha (α) decay involves emitting a helium nucleus (⁴₂He), reducing atomic number (Z) by 2 and mass number (A) by 4. Beta (β) decay involves emitting an electron (β⁻) or positron (β⁺), changing Z by +1 or -1 respectively, while A remains constant.

Gamma (γ) decay is the emission of high-energy photons from an excited nucleus, with no change in Z or A. These decay types have distinct penetrating powers and biological effects, dictating shielding requirements.

UPSC-relevant applications are extensive: Carbon-14 (t₁/₂ ~5,730 years) is the basis for dating organic archaeological samples up to ~60,000 years. Medical isotopes like Iodine-131 (t₁/₂ ~8 days) are used for thyroid treatment, and Cobalt-60 (t₁/₂ ~5.

27 years) for radiotherapy. Uranium isotopes (U-235, U-238) with very long half-lives are critical for nuclear power. Crucially, understanding the diverse half-lives of isotopes in nuclear waste is paramount for long-term management strategies, from interim storage for short-lived fission products to deep geological repositories for long-lived actinides.

This topic integrates physics with history, medicine, energy, and environmental policy, making a comprehensive understanding indispensable for the UPSC exam.

Prelims Revision Notes

Definition of Half-life: — Time for 50% of radioactive nuclei to decay. Constant for a given isotope.
Key Formulas:

- Number of half-lives (n) = Total time / Half-life (t₁/₂) - Remaining quantity N(t) = N₀ * (1/2)^n - t₁/₂ = ln(2) / λ (where ln(2) ≈ 0.693) - Activity A = λN - Activity A(t) = A₀ * (1/2)^n

Units: — Activity: Becquerel (Bq = 1 dps), Curie (Ci = 3.7 x 10¹⁰ Bq).
Types of Decay (Properties):

- Alpha (α): ⁴₂He, +2 charge, heavy, low penetration (paper), high ionization. - Beta (β⁻): e⁻, -1 charge, light, medium penetration (thin metal), medium ionization. - Gamma (γ): Photon, 0 charge, massless, high penetration (thick lead), low ionization.

Important Isotopes & Uses:

- Carbon-14 (t₁/₂ ~5,730 yrs): Carbon dating (organic materials, ~60,000 yr limit). - Iodine-131 (t₁/₂ ~8 days): Thyroid diagnosis/treatment. - Cobalt-60 (t₁/₂ ~5.27 yrs): Radiotherapy, sterilization. - Technetium-99m (t₁/₂ ~6 hrs): Most common diagnostic imaging isotope. - Uranium-235 (t₁/₂ ~7x10⁸ yrs): Nuclear fuel (fissile). - Uranium-238 (t₁/₂ ~4.5x10⁹ yrs): Breeder reactor fuel (fertile), geological dating.

Applications: — Carbon dating, medical imaging/therapy, nuclear power, nuclear waste management, space RTGs.
Key Principle: — Radioactive decay is a nuclear process, independent of external physical or chemical conditions.

Mains Revision Notes

Conceptual Framework: — Half-life as the quantitative measure of nuclear stability and decay rate. Link to exponential decay law and decay constant. Emphasize its independence from external factors.
India's Nuclear Program (Three-Stage Strategy):

- Stage 1 (PHWRs): Reliance on U-235 (long half-life) and U-238. Half-life dictates fuel cycle and initial energy security. - Stage 2 (FBRs): Breeding Pu-239 (t₁/₂ ~24,100 years) from U-238. Half-life of Pu-239 impacts reprocessing, safeguards, and waste characteristics. - Stage 3 (AHWRs): Thorium-232 (very long half-life) to U-233. Half-life of Th-232 ensures long-term energy independence; U-233 half-life influences fuel management.

Nuclear Waste Management:

- Challenge: Diverse half-lives of fission products and actinides (seconds to millions of years). - Implications: Short-lived isotopes require interim storage; long-lived isotopes necessitate deep geological repositories for hundreds of thousands of years. - Policy: Half-life dictates long-term safety, environmental protection, and international cooperation in waste disposal strategies.

Medical Applications:

- Diagnostic: Short half-lives (e.g., Tc-99m, F-18) for minimal patient exposure and rapid imaging. - Therapeutic: Longer half-lives (e.g., I-131, Co-60) for sustained, localized radiation delivery. - Supply Chain: Short half-lives create logistical challenges; indigenous production is key for national health security.

Carbon Dating: — Principle, applications in archaeology/paleontology, and limitations (age range, calibration needs). Connect to historical reconstruction.
Strategic Implications: — Half-life of fissile materials (e.g., Pu-239) is central to non-proliferation concerns and international treaties.

Vyyuha Quick Recall

Vyyuha Quick Recall: HALF-DECAY

H - Half-life: Time for 50% decay. A - Activity: Rate of decay (Bq, Ci). L - Lambda (λ): Decay constant, t₁/₂ = ln(2)/λ. F - Formula: N(t) = N₀ * e^(-λt) or N₀ * (1/2)^n.

D - Decay Types: Alpha, Beta, Gamma (properties). E - Exponential: Decay is always exponential. C - Carbon-14: Dating organic materials. A - Applications: Medical, Power, Waste, Dating. Y - Years: Half-lives range from seconds to billions of years.

Flashcards (Q/A Pairs)

1
Q: — What is the definition of half-life (t₁/₂)?

A: The time required for half of the radioactive nuclei in a sample to decay.

1
Q: — How is the decay constant (λ) related to half-life (t₁/₂)?

A: t₁/₂ = ln(2) / λ (where ln(2) ≈ 0.693).

1
Q: — What are the SI and non-SI units for activity, and their conversion?

A: SI: Becquerel (Bq = 1 disintegration/second). Non-SI: Curie (Ci = 3.7 × 10¹⁰ Bq).

1
Q: — If a sample has a half-life of 5 days, what fraction remains after 15 days?

A: 15 days is 3 half-lives. So, (1/2)³ = 1/8 of the original sample remains.

1
Q: — Which type of radiation is stopped by a sheet of paper?

A: Alpha (α) radiation.

1
Q: — What is the primary medical application of Iodine-131?

A: Diagnosis and treatment of thyroid disorders (hyperthyroidism, thyroid cancer).

1
Q: — What is the typical dating range for Carbon-14 dating?

A: Approximately 50,000 to 60,000 years for organic materials.

1
Q: — Why is Technetium-99m (t₁/₂ ~6 hours) preferred for diagnostic imaging?

A: Its short half-life minimizes patient radiation dose, and it emits pure gamma rays suitable for external detection.

1
Q: — Name two isotopes crucial for India's three-stage nuclear power program.

A: Uranium-235 (fissile), Uranium-238 (fertile, breeds Pu-239), Thorium-232 (fertile, breeds U-233).

1
Q: — How does half-life impact nuclear waste management?

A: It determines the duration for which radioactive waste remains hazardous, necessitating different storage solutions for short-lived vs. long-lived isotopes.

Spaced-Repetition Schedule Recommendations

Day 1: — Initial study of all core concepts, formulas, and applications. Complete all flashcards.
Day 3: — Review flashcards, focusing on incorrect answers. Attempt Quiz 1.
Day 7: — Re-read 'Basics Summary' and 'Prelims Revision Notes'. Attempt Quiz 2. Review all numerical examples.
Day 14: — Review 'Mains Revision Notes' and 'Vyyuha Analysis'. Attempt Quiz 3. Practice Mains questions.
Day 28: — Comprehensive review of the entire topic. Focus on inter-topic connections and current affairs hooks.

Short Timed Quizzes

Quiz 1 (5 Questions - 5 minutes)

1
The half-life of a radioactive isotope is 2 days. What percentage of the original sample will remain after 6 days?
2
Which type of radioactive decay results in an increase in the atomic number by one, with no change in mass number?
3
The SI unit of radioactivity is the ______.
4
Carbon-14 dating is primarily used for dating ______ materials.
5
True or False: The half-life of a radionuclide is affected by temperature and pressure.

Answers to Quiz 1:

1
12.5% (6 days / 2 days = 3 half-lives; (1/2)³ = 1/8 = 12.5%)
2
Beta-minus (β⁻) decay
3
Becquerel (Bq)
4
Organic
5
False

Quiz 2 (5 Questions - 5 minutes)

1
If the decay constant (λ) of an isotope is 0.1 day⁻¹, what is its approximate half-life?
2
Which medical isotope, with a half-life of ~8 days, is used for thyroid treatment?
3
What is the main difference in penetrating power between alpha and gamma radiation?
4
Name a long-lived isotope crucial for nuclear power generation in India.
5
How many Becquerels are in 1 Curie?

Answers to Quiz 2:

1
~6.93 days (t₁/₂ = 0.693 / 0.1)
2
Iodine-131
3
Alpha has very low penetrating power (stopped by paper), while gamma has very high penetrating power (requires thick lead/concrete).
4
Uranium-235 or Uranium-238 or Thorium-232
5
3.7 × 10¹⁰ Bq

Quiz 3 (5 Questions - 5 minutes)

1
A radioactive sample's activity drops from 400 Bq to 50 Bq in 12 hours. What is its half-life?
2
What type of decay involves the emission of a positron?
3
Why is understanding half-life critical for managing high-level nuclear waste?
4
Which isotope is used in Radioisotope Thermoelectric Generators (RTGs) for deep space missions?
5
True or False: Gamma decay changes the atomic number of the nucleus.

Answers to Quiz 3:

1
4 hours (400 -> 200 -> 100 -> 50 Bq represents 3 half-lives. 12 hours / 3 = 4 hours per half-life).
2
Beta-plus (β⁺) decay
3
It determines the duration for which the waste remains hazardous, dictating storage requirements (e.g., deep geological repositories for long-lived isotopes).
4
Plutonium-238
5
False

Sample Exam-Style Questions with Model Answers

2-Mark Question (50 words): Define half-life and explain its significance in the context of medical isotopes.

Model Answer: Half-life is the time taken for half of a radioactive sample to decay. In medical isotopes, its significance is paramount: short half-lives (e.g., Technetium-99m, ~6 hrs) are preferred for diagnostics to minimize patient exposure, while longer half-lives (e.g., Iodine-131, ~8 days) are chosen for therapy to deliver a sustained, targeted dose, balancing efficacy with safety.

5-Mark Question (100 words): Discuss how the half-life of Carbon-14 enables archaeological dating and briefly mention its limitations.

Model Answer: Carbon-14 dating utilizes the known half-life of Carbon-14 (~5,730 years) to determine the age of organic artifacts. Living organisms maintain a constant ¹⁴C/¹²C ratio; upon death, ¹⁴C intake ceases, and it decays.

By measuring the remaining ¹⁴C activity, the time elapsed since death can be calculated. This provides a crucial chronological framework for archaeological studies. However, its limitations include an effective dating range of only up to ~60,000 years, applicability restricted to organic materials, and the need for calibration curves due to historical variations in atmospheric ¹⁴C levels.