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Radioactivity & Nuclear Decay in Physics: Alpha, Beta, Gamma Rays Guide
Discovered accidentally by French physicist Henri Becquerel in 1896 and pioneered by Marie Curie and Pierre Curie, **Radioactivity** is the spontaneous disintegration of unstable atomic nuclei accompanied by the emission of ionizing radiation.
This 4,000+ word comprehensive exam guide covers **Rutherford's Discovery**, **Radioactive Decay Law ($N(t) = N_0 e^{-\lambda t}$)**, **Half-Life ($T_{1/2} = \frac{0.693}{\lambda}$)**, **Mean Life ($\tau = 1/\lambda$)**, comparison of **Alpha ($\alpha$), Beta ($\beta$), and Gamma ($\gamma$) Radiation**, **Soddy-Fajans Displacement Laws**, **Units of Radioactivity (Becquerel, Curie, Rutherford)**, **Carbon-14 Radiocarbon Dating**, **Medical Radioisotopes**, and solved numerical problems for SSC CGL, RRB NTPC, and UPSC Prelims.
Table of Contents
- 1. Discovery of Radioactivity & Natural vs Artificial Radioactivity
- 2. Rutherford-Soddy Law of Radioactive Decay ($N(t) = N_0 e^{-\lambda t}$)
- 3. Half-Life ($T_{1/2}$) & Mean Life ($\tau$) Formulas
- 4. Comparison of Alpha ($\alpha$), Beta ($\beta$) & Gamma ($\gamma$) Rays
- 5. Soddy-Fajans Group Displacement Laws
- 6. SI Units of Radioactivity: Becquerel (Bq), Curie (Ci) & Rutherford
- 7. Radiocarbon Dating ($^{14}\text{C}$) & Uranium-Lead Geological Dating
- 8. Applications of Radioisotopes in Medicine & Industry
- 9. Solved Numerical Examples for Competitive Exams
- 10. Must Remember Points for Quick Revision
- 11. Frequently Asked Questions (FAQ)
Key Takeaways & Core Highlights
- Radioactive Decay Law: $N(t) = N_0 e^{-\lambda t}$. Rate of decay $\frac{dN}{dt} = -\lambda N$. Exponential spontaneous process unaffected by external temperature or pressure.
- Half-Life Formula: $T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.69315}{\lambda}$. Remaining fraction after $n$ half-lives: $N = N_0 \left(\frac{1}{2}\right)^n$.
- Mean Life ($\tau$): $\tau = \frac{1}{\lambda} = \frac{T_{1/2}}{0.693} \approx 1.44 \cdot T_{1/2}$.
- Ionizing vs Penetrating Power:
- Ionizing Power: $\boldsymbol{\alpha > \beta > \gamma}$ ($\alpha$ is $100\times$ more ionizing than $\beta$).
- Penetrating Power: $\boldsymbol{\gamma > \beta > \alpha}$ ($\gamma$ penetrates lead; $\alpha$ stopped by paper sheet).
- Units of Radioactivity:
- SI Unit: **Becquerel ($\text{Bq} = 1 \text{ decay/second}$)**.
- Curie ($\text{Ci}$): $1 \text{ Ci} = 3.7 \times 10^{10} \text{ Bq}$.
- Rutherford ($\text{Rd}$): $1 \text{ Rd} = 10^6 \text{ Bq}$.
1. Discovery of Radioactivity & Natural vs Artificial Radioactivity
- Natural Radioactivity: Spontaneous emission of radiation by heavy elements ($Z > 82$, e.g. Uranium, Thorium, Radium). Discovered by Henri Becquerel (1896).
- Artificial (Induced) Radioactivity: Bombardment of stable nuclei with alpha particles or neutrons to produce radioactive isotopes (Discovered by Irène Joliot-Curie and Frédéric Joliot in 1934).
2. Rutherford-Soddy Law of Radioactive Decay ($N(t) = N_0 e^{-\lambda t}$)
The rate of disintegration ($-\frac{dN}{dt}$) of a radioactive sample at any instant is directly proportional to the number of active nuclei ($N$) present at that instant:
$$-\frac{dN}{dt} = \lambda \cdot N \implies N(t) = N_0 \cdot e^{-\lambda t}$$Where $\lambda$ is the **Decay Constant (Disintegration Constant)** ($\text{s}^{-1}$).
3. Half-Life ($T_{1/2}$) & Mean Life ($\tau$) Formulas
$$\text{Half-Life } T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.69315}{\lambda}$$ $$\text{Mean Life } \tau = \frac{1}{\lambda} = \frac{T_{1/2}}{0.693} \approx 1.443 \cdot T_{1/2}$$4. Comparison of Alpha ($\alpha$), Beta ($\beta$) & Gamma ($\gamma$) Rays
| Property | Alpha ($\alpha$) Rays | Beta ($\beta$) Rays | Gamma ($\gamma$) Rays |
|---|---|---|---|
| Nature | Helium nuclei ($^4_2\text{He}^{2+}$) | Fast electrons ($e^-$) or positrons ($e^+$) | High-frequency Electromagnetic Photons |
| Charge | $+2 e$ | $-1 e$ (or $+1 e$) | Neutral ($0$) |
| Rest Mass | $4 \text{ amu}$ ($6.64 \times 10^{-27}\text{ kg}$) | $1/1836 \text{ amu}$ ($9.1 \times 10^{-31}\text{ kg}$) | Zero rest mass |
| Speed | $\sim 10^7 \text{ m/s}$ ($5\text{--}10\% \text{ of } c$) | Up to $99\% \text{ of } c$ | Speed of Light ($c = 3 \times 10^8\text{ m/s}$) |
| Ionizing Power | Highest ($10,000 \times \gamma$) | Moderate ($100 \times \gamma$) | Lowest ($1$) |
| Penetrating Power | Lowest (Stopped by paper / skin) | Moderate (Stopped by $5\text{ mm}$ Aluminum) | Highest (Requires thick Lead / Concrete) |
5. Soddy-Fajans Group Displacement Laws
- Alpha Decay ($\alpha$): Mass number drops by $4$, Atomic number drops by $2$: $${}^A_Z\text{X} \xrightarrow{\alpha} {}^{A-4}_{Z-2}\text{Y} + {}^4_2\text{He}$$
- Beta Minus Decay ($\beta^-$): Mass number remains unchanged, Atomic number increases by $1$: $${}^A_Z\text{X} \xrightarrow{\beta^-} {}^A_{Z+1}\text{Y} + {}^0_{-1}e + \bar{\nu}_e$$
- Gamma Decay ($\gamma$): Both Mass and Atomic numbers remain unchanged (nucleus sheds excited energy).
8. Applications of Radioisotopes in Medicine & Industry
| Radioisotope | Primary Medical / Industrial Application |
|---|---|
| Cobalt-60 ($^{60}\text{Co}$) | Radiotherapy for Cancer Tumor treatment |
| Iodine-131 ($^{131}\text{I}$) | Diagnosis and treatment of Thyroid gland disorders |
| Sodium-24 ($^{24}\text{Na}$) | Detecting blood clots and circulation blockages |
| Phosphorus-32 ($^{32}\text{P}$) | Leukemia treatment and agricultural fertilizer tracking |
| Carbon-14 ($^{14}\text{C}$) | Radiocarbon dating of organic fossils |
9. Solved Numerical Examples for Competitive Exams
Numerical Problem 1 (Half-Life Decay Calculation):
Question: A radioactive element has a half-life of $10\text{ days}$. What percentage of the original sample remains undecayed after $30\text{ days}$?
Solution:
$$\text{Number of half-lives } n = \frac{\text{Total Time } t}{T_{1/2}} = \frac{30}{10} = 3$$ $$\text{Remaining fraction } \frac{N}{N_0} = \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^3 = \frac{1}{8} = 0.125 = 12.5\%$$Answer: $12.5\%$ of the original sample remains.
Numerical Problem 2 (Curie to Becquerel Conversion):
Question: Express an activity of $5\text{ Curies}$ in Becquerels.
Solution:
$$1\text{ Ci} = 3.7 \times 10^{10} \text{ Bq}$$ $$\text{Activity } A = 5 \times 3.7 \times 10^{10} \text{ Bq} = 1.85 \times 10^{11} \text{ Bq}$$Answer: Activity is $1.85 \times 10^{11}\text{ Bq}$.
10. Must Remember Points for Quick Revision
Exam Revision Cheat Sheet:
- Decay Law: $N(t) = N_0 e^{-\lambda t}$. Exponential disintegration.
- Half-Life: $T_{1/2} = 0.693/\lambda$. $N = N_0 (1/2)^n$.
- Ionizing Power: $\alpha > \beta > \gamma$.
- Penetrating Power: $\gamma > \beta > \alpha$.
- SI Unit: Becquerel ($\text{Bq}$). $1\text{ Ci} = 3.7 \times 10^{10}\text{ Bq}$.
- Carbon-14 Dating: $T_{1/2} = 5,730\text{ years}$ (Organic fossils up to 50k years).
- Cobalt-60: Cancer therapy. Iodine-131: Thyroid treatment.
11. Frequently Asked Questions (FAQ)
What is the difference between Alpha (α), Beta (β), and Gamma (γ) radiation?
Alpha particles are Helium nuclei (⁴₂He²⁺) with high ionizing power but low penetrating power (stopped by paper). Beta particles are fast electrons (e⁻) or positrons (e⁺) with moderate ionizing and penetrating power (stopped by aluminum). Gamma rays are high-energy photons (EM radiation) with very low ionizing power but extreme penetrating power (requires thick lead/concrete).
What is the Half-Life (T₁/₂) of a radioactive substance?
Half-life is the time required for half of the initial number of radioactive nuclei in a sample to decay: T₁/₂ = (ln 2)/λ ≈ 0.693/λ. After n half-lives, remaining amount N = N₀ (1/2)ⁿ.
How does Carbon-14 Radiocarbon Dating work?
Living organisms absorb Carbon-14 (half-life T₁/₂ = 5,730 years) from the atmosphere. Upon death, C-14 intake stops and its activity decays. By measuring the remaining ratio of C-14 to stable C-12, scientists accurately date organic artifacts up to ~50,000 years old.
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