Half-Life Calculator
Calculate radioactive decay, remaining quantity, half-life duration, or elapsed time with exponential decay formulas.
Decay Parameters
Decay Results
Radioactive Decay Law & Exponential Mathematics
Radioactive decay is a first-order kinetics process. Because radioactive disintegration is completely stochastic, the probability that any single atomic nucleus will decay per unit of time is constant ($\lambda$):
For deeper analysis and related planning, you can also explore our Area Calculator and Average Calculator.
$$\frac{dN}{dt} = -\lambda N$$
Integrating this differential equation yields the fundamental exponential decay law:
$$N(t) = N_0 \, e^{-\lambda t} = N_0 \left(\frac{1}{2}\right)^{t / t_{1/2}}$$
where the relationship between the half-life $t_{1/2}$ and the decay constant $\lambda$ is governed by the natural logarithm of 2:
$$\lambda = \frac{\ln(2)}{t_{1/2}} \approx \frac{0.693147}{t_{1/2}}$$
Key Radioactive Isotopes and Half-Lives
| Isotope | Half-Life | Decay Mode | Primary Application |
|---|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | Beta (β⁻) | Archaeological and organic radiocarbon dating |
| Iodine-131 (¹³¹I) | 8.02 days | Beta / Gamma | Thyroid cancer and hyperthyroidism therapy |
| Technetium-99m (⁹⁹ᵐTc) | 6.01 hours | Isomeric (γ) | Nuclear medicine organ perfusion imaging |
| Uranium-238 (²³⁸U) | 4.468 billion years | Alpha (α) | Geological dating and nuclear power fuel |
| Cesium-137 (¹³⁷Cs) | 30.17 years | Beta / Gamma | Industrial radiography and radiation gauges |
| Cobalt-60 (⁶⁰Co) | 5.27 years | Beta / Gamma | Industrial food irradiation and cancer radiotherapy |
| Radon-222 (²²²Rn) | 3.82 days | Alpha (α) | Naturally occurring radioactive basement gas |
Frequently Asked Questions
What is half-life in radioactive decay?
The half-life (t1/2) is the precise duration of time required for exactly one-half (50%) of the radioactive nuclei in an unstable isotope sample to undergo spontaneous decay into a more stable daughter element.
What is the formula for radioactive decay?
The standard decay formula is N(t) = N0 × (1/2)^(t / t1/2) = N0 × e^(-λt), where N0 is the initial quantity, N(t) is the remaining quantity, t is elapsed time, t1/2 is half-life, and λ = ln(2) / t1/2 is the decay constant.
How is carbon-14 dating calculated?
Living organisms maintain a steady equilibrium ratio of radioactive carbon-14 (half-life of 5,730 years) to stable carbon-12. Upon death, C-14 intake stops and begins decaying. By measuring the remaining C-14 fraction, scientists determine the specimen's age: t = 5,730 × [ln(N / N0) / ln(0.5)].
What is the difference between half-life and mean lifetime?
The half-life ($t_{1/2}$) is the time required for half the original atoms to decay. The mean lifetime ($\tau = 1/\lambda$) is the average survival time of a single nucleus before disintegration. Mathematically, $\tau = t_{1/2} / \ln(2) \approx 1.4427 \times t_{1/2}$.
Does temperature or pressure affect an isotope's half-life?
No. Radioactive half-life is a fundamental nuclear property determined by the strong nuclear force and quantum tunneling probability. Chemical reactions, extreme temperatures, or high pressures do not alter nuclear decay rates.
How many half-lives until a radioactive substance is safe?
In health physics and nuclear medicine, a sample is generally considered to have reached background levels after 10 half-lives, when remaining activity drops below $0.1\%$ ($1/2^{10} = 1/1024 \approx 0.098\%$).