Calculate radioactive decay, remaining quantity, elapsed time, and half-life values using the exponential decay formula. Supports scientific notation for precise isotope calculations in nuclear physics, chemistry, and radiometric dating.
Select which value you want to calculate, then enter the known values. Leave the field you want to calculate empty.
See how the quantity decreases over successive half-lives.
| Half-Life # | Time Elapsed | Remaining Quantity | % Remaining |
|---|
Where:
The decay constant λ represents the probability of decay per unit time. It is related to the half-life by the natural logarithm of 2.
This is the continuous exponential decay form, where e is the base of natural logarithms and λ is the decay constant.
The mean lifetime τ is the average time a particle or nucleus exists before decaying. It is longer than the half-life by a factor of approximately 1.4427.
Reference table of half-lives for common radioactive isotopes used in science, medicine, and industry.
| Isotope | Symbol | Half-Life | Decay Mode | Application |
|---|---|---|---|---|
| Uranium-238 | ²³⁸U | 4.468 billion years | Alpha | Geological dating |
| Uranium-235 | ²³⁵U | 703.8 million years | Alpha | Nuclear power, dating |
| Potassium-40 | ⁴⁰K | 1.248 billion years | Beta+, EC | Rock dating |
| Carbon-14 | ¹⁴C | 5,730 years | Beta- | Radiocarbon dating |
| Plutonium-239 | ²³⁹Pu | 24,110 years | Alpha | Nuclear weapons, fuel |
| Radium-226 | ²²⁶Ra | 1,600 years | Alpha | Medical radiotherapy |
| Cobalt-60 | ⁶⁰Co | 5.27 years | Beta- | Medical sterilization |
| Strontium-90 | ⁹⁰Sr | 28.8 years | Beta- | Nuclear fallout |
| Caesium-137 | ¹³⁷Cs | 30.17 years | Beta- | Medical, industrial |
| Tritium (H-3) | ³H | 12.32 years | Beta- | Biolabeling, fusion |
| Iodine-131 | ¹³¹I | 8.02 days | Beta- | Thyroid treatment |
| Phosphorus-32 | ³²P | 14.26 days | Beta- | Biological research |
| Technetium-99m | ⁹⁹ᵐTc | 6.01 hours | Gamma | Medical imaging |
| Polonium-212 | ²¹²Po | 299 nanoseconds | Alpha | Research |
Problem: An archaeological sample contains 25% of its original Carbon-14. Carbon-14 has a half-life of 5,730 years. How old is the sample?
Solution: After 1 half-life (5,730 yr): 50% remains. After 2 half-lives (11,460 yr): 25% remains. The sample is approximately 11,460 years old.
Problem: A patient receives 200 MBq of Technetium-99m (half-life = 6.01 hours). How much remains after 18 hours?
Solution: 18 hours = 3 half-lives. N = 200 × (½)³ = 200 × ⅛ = 25 MBq remains.
Problem: A 500 g sample decays to 125 g in 24 years. What is the half-life?
Solution: 500 → 250 → 125 = 2 half-lives in 24 years. Half-life = 24 ÷ 2 = 12 years.
Problem: Iodine-131 has a half-life of 8.02 days. If you start with 100 mg, how much remains after 20 days?
Solution: N = 100 × (½)^(20/8.02) = 100 × (½)^2.494 ≈ 100 × 0.177 = 17.7 mg remains.
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Half-life (T₁/₂) is the time required for a quantity of a radioactive substance to reduce to half of its initial value. This concept is fundamental in nuclear physics, chemistry, and radiometric dating. The key insight is that radioactive decay follows an exponential pattern — the amount of substance decreases by a constant factor (half) over each equal time interval. This means that after one half-life, 50% remains; after two, 25%; after three, 12.5%; and so on.
The mathematical foundation of half-life calculations is the exponential decay formula: N(t) = N₀ × (½)^(t/T₁/₂). This formula can be used to solve for any variable — the remaining quantity, the half-life, the elapsed time, or the initial quantity — as long as the other three values are known. The decay constant λ = ln(2)/T₁/₂ provides an alternative formulation: N(t) = N₀ × e^(-λt), which is particularly useful in differential equation models of decay processes.
Half-life calculations have wide-ranging practical applications. In archaeology, carbon-14 dating uses the 5,730-year half-life of ¹⁴C to determine the age of organic materials up to about 50,000 years old. In nuclear medicine, isotopes like technetium-99m (6-hour half-life) are used for diagnostic imaging because they decay quickly enough to minimize patient radiation exposure while providing clear images. In geology, uranium-lead dating relies on the 4.5-billion-year half-life of ²³⁸U to date the oldest rocks on Earth and in the solar system. In environmental science, understanding the half-lives of nuclear fallout isotopes like ¹³⁷Cs (30 years) and ⁹⁰Sr (29 years) is crucial for assessing long-term contamination risks.
Half-life and doubling time are mathematically related concepts. While half-life describes exponential decay (quantities decreasing by half), doubling time describes exponential growth (quantities doubling). The formulas are structurally identical: N(t) = N₀ × 2^(t/T_d) for doubling, compared to N(t) = N₀ × (½)^(t/T_½) for halving. The doubling time T_d can be calculated from a growth rate using the rule of 70 (T_d ≈ 70/growth rate %), while half-life uses the same mathematics in reverse. This symmetry makes the same calculator useful for modeling both radioactive decay and population growth when adjusted appropriately.
Our Half-Life Calculator is a comprehensive tool for modeling radioactive decay and exponential decay processes. Whether you're a student studying nuclear physics, a researcher working with radioactive isotopes, or a medical professional calculating dosage decay, this calculator provides accurate results using the standard exponential decay formula N(t) = N₀ × (½)^(t/T₁/₂).
Calculate remaining quantity, half-life, elapsed time, or initial quantity from any three known values. Complete flexibility for any decay problem.
View complete decay curve data showing remaining quantity and percentage after each successive half-life, helping visualize the exponential pattern.
Full mathematical reference including exponential decay formula, decay constant (λ), mean lifetime (τ), and their relationships with worked examples.
Built-in reference table of common radioactive isotopes with their half-lives, decay modes, and practical applications in science and medicine.
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Important Disclaimer: This Half-Life Calculator provides mathematical estimates based on the exponential decay model. Results should be verified against authoritative sources for critical applications in nuclear medicine, radiation safety, or scientific research. This tool is for educational and informational purposes only and should not replace professional consultation in matters involving radioactive materials or health and safety decisions.