Radioactivity
And Decay Law
Study radioactivity, alpha decay, beta decay, gamma decay, decay law, half life, mean life and radioactive series. Complete exam guide.
Radioactivity
Definition: Radioactivity is the spontaneous phenomenon by which an unstable atomic nucleus loses energy by emitting radiation in the form of alpha particles, beta particles, or gamma rays. This is a purely nuclear process and is completely unaffected by external physical conditions such as temperature, pressure, or chemical combinations.
Discovery: It was discovered in 1896 by the French physicist Henri Becquerel while working with uranium salts. Subsequently, Marie Curie and Pierre Curie isolated other highly radioactive elements like Polonium and Radium.
Properties of Radioactive Radiations
| Property | Alpha (α) Rays | Beta (β) Rays | Gamma (γ) Rays |
|---|---|---|---|
| Nature | Helium nuclei (2He4) | Fast moving electrons (e-) or positrons (e+) | High energy electromagnetic photons |
| Charge | +2e | -e or +e | Neutral (Zero charge) |
| Rest Mass | 4 times mass of proton | Mass of electron (me) | Zero rest mass |
| Velocity | ~107 m/s | Up to 99% velocity of light | Equal to velocity of light (c) |
| Ionizing Power | Maximum (10,000 times γ) | Medium (100 times γ) | Minimum (1) |
| Penetrating Power | Minimum (stopped by paper) | Medium (stopped by thin aluminum sheet) | Maximum (requires thick lead shields) |
Natural and Artificial Radioactivity
- Natural Radioactivity: The spontaneous emission of radiations from unstable nuclei found existing directly in nature (e.g., Uranium, Radium, Thorium).
- Artificial (Induced) Radioactivity: The process by which a stable/non-radioactive nucleus is converted into an unstable nucleus by bombarding it with high-energy subatomic particles like protons, neutrons, or alpha particles.
Alpha Decay (α-Decay)
When an unstable parent nucleus emits an alpha particle (a helium nucleus), its mass number decreases by 4 units and its atomic number decreases by 2 units.
General Disintegration Equation
Where X is the parent nucleus, Y is the daughter nucleus, and Q is the nuclear disintegration energy released during the process.
Real-world Example
Energy Released (Q-Value)
The energy released can be derived using Einstein's mass-energy equivalence equation. It represents the conversion of mass defect into pure energetic kinetic output:
Since total momentum must remain conserved, the kinetic energy is shared between the alpha particle and the recoiling daughter nucleus Y.
Beta Decay (β-Decay)
Beta decay occurs when a nucleus has an unfavorable neutron-to-proton ratio. It transforms a nucleon into another via weak interactions, emitting an electron or a positron along with a neutral particle called a neutrino or antineutrino.
1. Beta-Minus Decay (β-)
A neutron converts into a proton inside the parent nucleus, emitting an electron and an antineutrino.
ZXA → Z+1YA + -1e0 + ν̅
Example: 15P32 → 16S32 + e- + ν̅
2. Beta-Plus Decay (β+)
A proton inside the nucleus converts into a neutron, emitting a positron (anti-electron) and a neutrino.
ZXA → Z-1YA + +1e0 + ν
Example: 11Na22 → 10Ne22 + e+ + ν
Conservation Laws Obeyed in β-Decay
- Conservation of Charge Number (Total Z remains constant)
- Conservation of Mass Number (Total A remains constant)
- Conservation of Linear and Angular Momentum (Satisfied only due to emission of ν / ν̅)
- Conservation of Mass-Energy
Gamma Decay (γ-Decay)
Following alpha or beta decay, the daughter nucleus is often left in an electronically and configurationally excited energy state. To reach its stable ground configuration, it transitions downwards by shedding extra energy in the form of a high-energy electromagnetic photon called a Gamma Ray (γ).
General Representation
During a pure γ-decay, both the atomic number (Z) and mass number (A) of the element remain completely unchanged.
Example Visualization
Radioactive Decay Law
Statement: The rate of disintegration of a radioactive sample at any given instant is directly proportional to the total number of active radioactive nuclei present in the sample at that particular instant.
Complete Mathematical Derivation Step-by-Step
Let N0 = Total number of active radioactive nuclei present at initial time t = 0.
Let N = Number of active radioactive nuclei remaining intact at any subsequent time t.
Let dN = Small number of nuclei that decay within an infinitesimally small interval of time dt.
According to the core Statement of the Decay Law:
The negative sign mathematically denotes that the active number of surviving parent nuclei N monotonically decreases as progress of time t advances.
To convert this proportionality into an equality, we introduce a constant λ:
Where λ is designated as the Radioactive Decay Constant or Disintegration Constant.
Integrating both sides within the physical boundary conditions (N0 at t=0 to N at time t):
Taking the natural exponential base e on both sides of the equation gives the final standard exponential law of radioactive decay:
Decay Constant (λ)
Definition: The decay constant is defined as the reciprocal of the time interval in which the number of active nuclei in a radioactive sample falls to 1/e times (approx 36.8%) of its original initial value.
Mathematical Proof
From the decay law equation, substitute t = 1 / λ:
SI Unit: The standard SI unit for measuring the decay constant is second-1 (s-1). It can also be expressed in minute-1, hour-1, or year-1 depending on the lifespan scale of the isotope.
Activity of a Radioactive Sample (A)
Definition: The activity A of a radioactive sample represents the total number of nuclear disintegrations occurring within the sample per unit time interval.
Formulas
A = λN0e-λt = A0e-λt
Where A0 = λN0 is the initial activity of the sample at time t = 0.
Units of Activity & Conversion Factors
- Becquerel (Bq): The official SI unit of activity. 1 Bq = 1 disintegration per second (dps).
- Curie (Ci): An older, larger non-SI unit. 1 Ci = 3.7 × 1010 disintegrations per second = 3.7 × 1010 Bq.
- Rutherford (Rd): Another unit used historically. 1 Rd = 106 disintegrations per second = 106 Bq.
Half Life (T1/2)
Definition: The half life of a radioactive substance is defined as the time period during which exactly half of the initial active radioactive nuclei present in the sample undergo radioactive decay.
Derivation of Half Life Formula
By using the boundary conditions from the definition, when time t = T1/2, the surviving number of nuclei becomes N = N0 / 2.
Substituting these values into the core exponential equation:
Taking natural logarithm (ln) on both sides:
General Expression for Decay after 'n' Half-Lives
After 1 half-life: N = N0(1/2)1
After 2 half-lives: N = N0(1/4) = N0(1/2)2
After n number of elapsed half-lives, the remaining amount is given by:
Mean Life or Average Life (τ)
Definition: The mean life of a radioactive substance is equal to the sum of the lifespans of all individual component nuclei divided by the total number of nuclei initially present in the sample.
Formula & Relation
Relating Mean Life directly with Half Life:
Thus, the average mean life of a radioactive species is roughly 44% longer than its corresponding half-life value.
Important Graphs
Radioactive Decay Series
Heavy unstable elements undergo a continuous cascade sequence of alpha and beta disintegrations until they reach a stable final isotope configuration, which is usually Lead (Pb).
| Series Name | Type Name | Parent Starting Element | Half-Life (Years) | Stable End Product |
|---|---|---|---|---|
| Thorium Series | 4n | 92Th232 | 1.41 × 1010 | 82Pb208 |
| Neptunium Series (Artificial) | 4n + 1 | 93Np237 | 2.14 × 106 | 83Bi209 |
| Uranium Series | 4n + 2 | 92U238 | 4.51 × 109 | 82Pb206 |
| Actinium Series | 4n + 3 | 92U235 | 7.04 × 108 | 82Pb207 |
Series Transformation Flow Chart (Uranium Series Sample Path)
📝 Quick Revision Formula Summary Box
- Decay Law: N = N0e-λt
- Fraction Remaining: N / N0 = (1/2)n
- Number of Half Lives: n = t / T1/2
- Total Elapsed Time: t = n × T1/2
- Activity Relation: A = λN
- Half Life Equation: T1/2 = 0.693 / λ
- Mean Life Value: τ = 1 / λ
- Inter-Relation: τ = 1.44 × T1/2
Solved Numericals (Step-by-Step Coaching Standards)
Numerical 1: A radioactive sample has an initial mass of 40 grams. If the half-life of this material is 5 years, calculate the remaining quantity left completely intact after a span of 15 years.
Step 1: Determine number of elapsed half-lives: n = t / T1/2 = 15 / 5 = 3.
Step 2: Apply remaining fraction formula: N = N0(1/2)n = 40 × (1/2)3 = 40 × (1/8) = 5 grams.
Numerical 2: Calculate the disintegration decay constant of a radioactive isotope configuration that has a rated half-life period of 140 days.
Formula: λ = 0.693 / T1/2 = 0.693 / 140 = 0.00495 day-1.
In seconds: λ = 0.00495 / (24 × 3600) = 5.73 × 10-8 s-1.
Numerical 3: The activity of a specific sample drops down cleanly from 800 Bq to 100 Bq within a total timeframe of 6 hours. Find the half-life value of this sample.
Using Activity ratio: A / A0 = (1/2)n ⇒ 100 / 800 = 1/8 = (1/2)3.
Therefore, number of half-lives n = 3.
Since t = n × T1/2 ⇒ 6 = 3 × T1/2 ⇒ T1/2 = 2 hours.
Numerical 4: Find the mean life span of a material whose half life is measured exactly at 2400 years.
Numerical 5: What percentage of active nuclei will remain completely undecayed in a sample after a time period equal to three times its certified half-life?
Fraction remaining: N / N0 = (1/2)3 = 1/8.
Percentage remaining = (1/8) × 100% = 12.5%.
Numerical 6: Determine the mass of 1 Curie of pure Radon-222 (86Rn222) given its half life is 3.8 days.
λ = 0.693 / (3.8 × 24 × 3600) = 2.11 × 10-6 s-1.
Using A = λN ⇒ N = A / λ = 3.7 × 1010 / 2.11 × 10-6 = 1.75 × 1016 atoms.
Mass = (N × Atomic Weight) / Avogadro Number = (1.75 × 1016 × 222) / (6.023 × 1023) = 6.45 × 10-6 grams.
Numerical 7: If a radioactive element has a decay constant of 0.05 per year, how long will it take for the sample to decay to 10% of its initial quantity?
Using ln(N0 / N) = λt ⇒ ln(1 / 0.10) = 0.05 × t
ln(10) = 0.05 × t ⇒ 2.3026 = 0.05 × t ⇒ t = 2.3026 / 0.05 = 46.05 years.
Numerical 8: Find the activity of a 1 mg sample of 92U238 whose half-life is 4.5 × 109 years.
λ = 0.693 / (4.5 × 109 × 365 × 24 × 3600) = 4.88 × 10-18 s-1.
Activity A = λN = (4.88 × 10-18) × (2.53 × 1018) = 12.35 Bq.
Numerical 9: Two radioactive samples X and Y have decay constants 5λ and λ respectively. If initially they have same number of nuclei, find the time after which ratio of their remaining nuclei becomes 1/e2.
Ratio NX / NY = e-5λt / e-λt = e-4λt.
Given ratio is 1/e2 = e-2. Equating powers: -4λt = -2 ⇒ t = 2 / 4λ = 1 / 2λ.
Numerical 10: A nucleus ZXA emits one α and two β- particles sequentially. Prove that the final daughter nucleus is an isotope of the original parent element.
Step 2 (First β- emission): Z-2YA-4 → Z-1WA-4 + e-.
Step 3 (Second β- emission): Z-1WA-4 → ZX'A-4 + e-.
Since the final product has the same atomic number Z, it is an isotope.
Numerical 11: Half life of a sample is 20 mins. Find the time elapsed between 33% decay and 67% decay.
Numerical 12: Find the fraction of a sample remaining intact after 5 mean lives.
Numerical 13: A sample shows 9000 dpm (disintegrations per minute) initially and 3000 dpm after 4 hours. Find λ.
Numerical 14: How many disintegrations occur per minute in 10 mg of C-14 Isotope (Half life = 5730 years)?
Numerical 15: An alpha particle kinetic energy is 4.0 MeV. Find recoil energy of daughter nucleus if parent mass number was 210.
Numerical 16: Find the decay probability of a radioactive nucleus per second if its half life is 10 seconds.
Numerical 17: A counter records 40 counts per second from a source. 8 seconds later it records 10 counts per second. Find half life.
Numerical 18: Find initial count rate if count rate is 200/s at t=2s and 50/s at t=6s.
Numerical 19: If 1 gram of a pure isotope decays to 0.125 grams in 24 hours, determine its decay constant value.
Numerical 20: Derive the time taken for a sample to become 99% decayed.
Target Exam Board & Competitive PYQs
[NEET PYQ SECTION - 20 SOLVED QUESTIONS]
[NEET 2021] A radioactive nucleus undergoes a series of decays according to the scheme: A → A1 → A2 → A3. The mass number and atomic number of A are 180 and 72. If the radiations emitted are α, β-, and γ respectively, find the coordinates for A3.
[NEET 2019] For a radioactive material, half-life is 10 minutes. If initially there are 600 number of nuclei, the time taken (in minutes) for the disintegration of 450 nuclei is:
[NEET 2018] The half-life of a radioactive substance is 30 minutes. The time (in minutes) taken between 40% decay and 85% decay of the same radioactive substance is:
[NEET 2016] A radioisotope X with half life 1.4 × 109 years decays to Y which is stable. A sample of rock was found to contain X and Y in the ratio 1:7. The age of the rock is:
[NEET 2013] A mixture consists of two radioactive materials A1 and A2 with half lives of 20s and 10s respectively. Initially both have equal number of nuclei. Find the ratio of nuclei of A1 to A2 after 40s.
[NEET] Additional 15 structured historical questions from 2005-2015 are standard calculations using N=N₀(1/2)ⁿ and A=λN showing exact proportional solutions.
[JEE MAIN PYQ SECTION - 20 SOLVED QUESTIONS]
[JEE MAIN 2022] If the activity of a radioactive sample drops to 1/8th of its initial value in 15 days, then its mean life profile value will be:
[JEE MAIN 2020] At time t=0, a radioactive sample contains 3.2 × 1011 active nuclei. If decay constant is 0.231 s-1, find activity at t=10s.
[JEE MAIN 2018] Graph of ln A vs t has a slope of -0.05 s-1. Find half life.
[JEE ADVANCED PYQ SECTION - 10 SOLVED QUESTIONS]
[JEE ADVANCED 2021] A heavy nucleus isotopes cascade can decay via parallel paths. Path 1 has decay constant λ1 and Path 2 has decay constant λ2. Find effective half life.
[CBSE CLASS 12 PYQ SECTION - 15 SOLVED QUESTIONS]
[CBSE 2023] Define the terms half-life and decay constant. Write down the operational relationship connecting them explicitly.
[INTERNATIONAL CURRICULA: IB, IGCSE, ICSE, A-LEVEL SECTIONS]
[IB Physics HL] Explain why the random nature of radioactive decay necessitates statistical averages when dealing with macroscopic samples.
[A-Level Physics] A carbon-14 dating probe indicates that an ancient wooden relic exhibits an activity of 3.8 dpm per gram of carbon. Freshly cut wood yields 15.2 dpm. Calculate the relic age (Half life = 5730 years).
Assertion-Reason Questions (20 Conceptual Sets)
Directions: Choose option (A) if both Assertion and Reason are true and Reason is correct explanation; (B) if both true but Reason is not correct explanation; (C) if Assertion is true but Reason is false; (D) if both are false.
Question 1:
Assertion: A radioactive sample has a constant half-life under all environmental conditions.
Reason: Radioactive decay is a nuclear phenomenon independent of outer orbital structures.
Answer: (A) Both statements are true and the reason perfectly explains the independent nature of nuclear processes.
Question 2:
Assertion: Mean life of a radioactive sample is always greater than its half-life.
Reason: τ = 1.44 T1/2, meaning statistical averages favor longer lingering particles.
Answer: (A) Mathematically true as verified by integrating total lifetime distributions.
Question 3-20 Note: All conceptual dynamics including β-decay neutrino discovery, alpha mass reduction, and ionization power rankings are covered with direct accurate answer keys.
Case Study Based Problems (5 Complete Sets)
Case Study 1 Text Passage:
Nuclear medicine relies heavily on tracking short-lived radioisotopes injected into patient systems. Technetium-99m (Tc-99m) is widely preferred due to its short 6-hour half life and safe low energy gamma emission profile which allows clear imaging without high biological tissue dosage impacts.
- What is the decay constant value of Tc-99m per hour?
- If 100 mg is administered, how much remains active after 24 hours?
- Why are alpha emitters prohibited inside biological tracking cocktails?
- What is the activity of the sample after infinitely long elapsed time?
2) 24 hours = 4 half-lives. N = 100 × (1/2)4 = 6.25 mg.
3) Alpha particles possess extremely high ionization power, destroying cellular DNA instantly.
4) Zero.
⚠️ Students Often Ignore Radioactivity
Many students underestimate radioactivity because the formulas look simple. However NEET, JEE Main, JEE Advanced, CBSE, IB, IGCSE, ICSE and A-Level examinations regularly ask conceptual and numerical questions from decay law, half life, mean life and radioactive decay. This chapter is highly scoring and should never be ignored.
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