Understanding Half-Life and Radioactive Decay
A half-life calculator is an essential tool for understanding radioactive decay, exponential decay processes, and time-dependent phenomena in physics, chemistry, and biology. Half-life represents the time required for half of a substance to decay or transform.
What is Half-Life?
Half-life (t₁/₂) is the time required for exactly half of a given amount of a radioactive substance to decay. This concept applies to:
- Radioactive isotopes: Nuclear decay processes
- Chemical reactions: Degradation of compounds
- Biological processes: Drug metabolism and elimination
- Physical processes: Cooling, absorption, and other exponential changes
Mathematical Foundations
Exponential Decay Formula
The fundamental equation for exponential decay is:
N(t) = N₀ × e^(-λt)
Where:
- N(t): Amount remaining at time t
- N₀: Initial amount
- λ: Decay constant
- t: Time elapsed
- e: Euler's number (≈2.718)
Half-Life Formula
Using the half-life concept:
N(t) = N₀ × (1/2)^(t/t₁/₂)
This is equivalent to the exponential form where λ = ln(2)/t₁/₂
Relationship Between Formulas
The decay constant and half-life are related by:
λ = ln(2)/t₁/₂ ≈ 0.693/t₁/₂
Types of Radioactive Decay
Alpha Decay
Emission of alpha particles (helium nuclei):
- Characteristics: High mass, low penetration
- Examples: Uranium-238, Radium-226, Plutonium-239
- Applications: Smoke detectors, nuclear batteries
Beta Decay
Emission of beta particles (electrons or positrons):
- Beta-minus: Neutron converts to proton + electron
- Beta-plus: Proton converts to neutron + positron
- Examples: Carbon-14, Cobalt-60, Iodine-131
Gamma Decay
Emission of gamma rays (electromagnetic radiation):
- Characteristics: No mass change, high energy
- Often accompanies: Alpha or beta decay
- Applications: Medical imaging, sterilization
Practical Applications
Carbon Dating
Carbon-14 dating uses radioactive decay to determine ages:
- Half-life: 5,730 years
- Range: Up to ~50,000 years
- Applications: Archaeological artifacts, fossils
- Assumptions: Constant atmospheric C-14 levels
Medical Applications
Radioactive isotopes in medicine:
- Diagnostic imaging: Technetium-99m (6 hours)
- Cancer treatment: Cobalt-60 (5.3 years)
- Thyroid treatment: Iodine-131 (8 days)
- PET scans: Fluorine-18 (110 minutes)
Nuclear Power
Half-life considerations in nuclear energy:
- Fuel rods: Uranium-235 (704 million years)
- Waste management: Long-lived isotopes
- Safety planning: Containment duration
Geological Dating Methods
Uranium-Lead Dating
For very old rocks and minerals:
- U-238 to Pb-206: 4.47 billion years
- U-235 to Pb-207: 704 million years
- Applications: Age of Earth, meteorites
Potassium-Argon Dating
For volcanic rocks and minerals:
- K-40 to Ar-40: 1.25 billion years
- Applications: Volcanic deposits, human evolution
Non-Radioactive Applications
Pharmacokinetics
Drug elimination from the body:
- Drug half-life: Time for 50% elimination
- Dosing intervals: Based on half-life
- Steady state: Reached after ~5 half-lives
Environmental Science
Pollutant degradation and persistence:
- Chemical breakdown: Pesticide degradation
- Atmospheric processes: Greenhouse gas lifetime
- Biological systems: Biodegradation rates
Calculation Examples
Example 1: Remaining Amount
Given: 100g of C-14, half-life = 5,730 years, time = 11,460 years
Solution: N(t) = 100 × (1/2)^(11,460/5,730) = 100 × (1/2)² = 25g
Example 2: Finding Half-Life
Given: 100g → 50g in 5,730 years
Solution: t₁/₂ = 5,730 × ln(2)/ln(100/50) = 5,730 years
Example 3: Time to Decay
Given: 100g → 12.5g, half-life = 5,730 years
Solution: t = 5,730 × log₂(100/12.5) = 5,730 × 3 = 17,190 years
Advanced Concepts
Decay Chains
Series of consecutive decays:
- Uranium series: U-238 → ... → Pb-206
- Thorium series: Th-232 → ... → Pb-208
- Actinium series: U-235 → ... → Pb-207
Branching Ratios
When multiple decay paths exist:
- Partial half-lives: For each decay mode
- Effective half-life: Combined decay rate
- Branch fractions: Probability of each path
Safety and Radiation Protection
Exposure Limits
Understanding radiation safety through half-life:
- 10 half-lives rule: ~99.9% decay (safe level)
- ALARA principle: As Low As Reasonably Achievable
- Biological half-life: Elimination from body
Waste Management
Long-term storage considerations:
- Short-lived: Months to years storage
- Intermediate: Decades to centuries
- Long-lived: Thousands to millions of years
Common Misconceptions
- Linear vs. exponential: Decay is not linear
- Zero endpoint: Never reaches exactly zero
- Temperature independence: Nuclear decay unaffected by temperature
- Half-life constancy: Half-life is constant for each isotope
Master radioactive decay calculations with our comprehensive half-life calculator, designed for students, researchers, and professionals in nuclear science, geology, medicine, and environmental studies.