Radioactive Decay Formula
Exponential decay of nuclei with time
Understanding Radioactive Decay
Radioactive decay is a first-order kinetic process where unstable atomic nuclei spontaneously transform into more stable configurations by emitting radiation. This phenomenon, discovered by Henri Becquerel in 1896 and extensively studied by Marie and Pierre Curie, follows exponential decay kinetics characterized by a constant decay probability per unit time. Unlike chemical reactions, radioactive decay is unaffected by temperature, pressure, or chemical environment, making it an ideal natural clock for applications ranging from carbon dating to medical diagnostics.
The exponential nature of radioactive decay means that a fixed fraction of remaining nuclei decays per unit time, regardless of how many nuclei are present. This leads to the concept of half-life—the time required for exactly half of the radioactive material to decay. Half-lives vary enormously: carbon-14 has a half-life of 5,730 years used in archaeology, iodine-131 (8 days) is used in thyroid treatment, and uranium-238 (4.5 billion years) helps geologists date Earth's oldest rocks.
Understanding radioactive decay formulas is essential for nuclear medicine, where radioisotopes are used for both diagnosis and treatment. Radiation safety officers use these equations to calculate shielding requirements and safe handling times. Environmental scientists apply them to model the transport and fate of radioisotopes in ecosystems. In nuclear power generation, decay calculations predict fuel rod lifetimes and manage radioactive waste storage requirements over thousands of years.
The Formula
N = N₀ e^(−λt)
λ = ln(2) / t1/2
Exponential decay relationship for radioactive nuclei over time.
Alternative Forms:
- N = N₀ (1/2)^(t/t1/2) — using half-life directly
- Activity: A = A₀ e^(−λt) — decay rate in becquerels (Bq)
- ln(N/N₀) = −λt — linearized form useful for graphing
Step-by-Step Example
Problem: A medical isotope has a half-life of 5.0 hours. If you start with 120 mg, how much remains after 8.0 hours?
Given: t1/2 = 5.0 h, Nâ‚€ = 120 mg, t = 8.0 h
Step 1: Calculate Decay Constant
λ = ln(2) / t1/2 = 0.693147 / 5.0 h = 0.1386 hâ»Â¹
Step 2: Calculate Exponent
−λt = −0.1386 hâ»Â¹ × 8.0 h = −1.1088
Step 3: Apply Exponential Function
e^(−1.1088) = 0.3299
Step 4: Calculate Remaining Amount
N = 120 mg × 0.3299 = 39.6 mg
Answer: 39.6 mg remains after 8.0 hours
This represents 33% of the original amount (slightly less than 1.6 half-lives)
Key Applications
1. Carbon-14 Dating
Archaeologists use carbon-14 decay (t1/2 = 5,730 years) to determine the age of organic materials up to 50,000 years old. By measuring the ratio of C-14 to C-12 in ancient wood, bone, or fabric, scientists calculate how long ago the organism died, revolutionizing archaeology and paleontology.
2. Nuclear Medicine
Hospitals use radioisotopes like technetium-99m (t1/2 = 6 hours) for diagnostic imaging and iodine-131 for thyroid cancer treatment. Decay calculations ensure optimal imaging times and minimize patient radiation exposure, balancing diagnostic quality with safety.
3. Radiometric Dating of Rocks
Geologists use uranium-lead dating (U-238 t1/2 = 4.5 billion years) to determine Earth's age and date geological events. The decay series provides multiple independent age estimates, confirming the 4.54 billion year age of our planet.
4. Radiation Safety and Waste Management
Nuclear facilities calculate required storage times for radioactive waste using decay formulas. Materials must be isolated until activity decreases to safe levels, which for some isotopes means thousands of years of secure containment in geological repositories.
Common Radioisotopes Comparison
| Isotope | Half-Life | Application |
|---|---|---|
| Technetium-99m | 6.0 hours | Medical imaging scans |
| Iodine-131 | 8.0 days | Thyroid cancer treatment |
| Carbon-14 | 5,730 years | Archaeological dating |
| Plutonium-239 | 24,100 years | Nuclear fuel/waste |
| Uranium-238 | 4.5 billion years | Geological dating |
Common Mistakes to Avoid
Mistake 1: Inconsistent Time Units
The decay constant λ and time t must have reciprocal units. If λ is in hâ»Â¹, t must be in hours. Mixing hours and seconds produces completely incorrect results.
Mistake 2: Forgetting the Negative Sign
The exponent must be negative (−λt), not positive. A positive exponent would incorrectly predict exponential growth instead of decay.
Mistake 3: Using Wrong Base for Logarithm
Always use natural logarithm ln(2) ≈ 0.693, not logâ‚â‚€(2) ≈ 0.301. The exponential decay formula requires the natural base e.
Mistake 4: Confusing Activity and Amount
Both amount (N) and activity (A = λN) follow exponential decay with the same λ, but they have different units and meanings. Activity is measured in becquerels (decays per second).