Half-Life Calculator | Radioactive Decay
Calculate remaining quantity, time elapsed, half-life, and decay constant using the exponential decay formula. Supports 4 modes, decay table, and isotope reference.
| C-14 | 5730 yr | Carbon-14 |
| U-235 | 703800000 yr | Uranium-235 |
| I-131 | 8.02 d | Iodine-131 |
| Ra-226 | 1600 yr | Radium-226 |
| Tc-99m | 6 hr | Technetium-99m |
| Cs-137 | 30.17 yr | Cesium-137 |
| Rn-222 | 3.8235 d | Radon-222 |
Press Enter to calculate · Esc to reset
What Is the Half-Life Calculator | Radioactive Decay?
Radioactive decay is a spontaneous process in which an unstable atomic nucleus loses energy by emitting radiation. The half-life (t½) is the time required for exactly half of a radioactive sample to decay, a constant that is unique to each isotope and independent of temperature, pressure, or chemical state.
The Exponential Decay Law
Because each nucleus has the same probability of decaying in any given interval, the overall decay follows an exponential law: N(t) = N₀ × (1/2)^(t/t½). This is equivalent to N(t) = N₀ × e^(-λt), where λ = ln(2)/t½ is the decay constant.
Four Calculation Modes
- • Remaining Amount: given N₀, t, and t½, find N(t)
- • Time Elapsed: given N₀, N, and t½, find how long the sample has decayed
- • Half-Life from Data: given two measurements N₀ and N at known time t, compute t½
- • Decay Constant: derive λ and mean lifetime τ from the half-life
Decay Constant and Mean Lifetime
The decay constant λ (lambda) represents the probability of decay per unit time. The mean lifetime τ = 1/λ = t½ / ln(2) ≈ 1.4427 × t½ is the average time a single nucleus exists before decaying. Note τ > t½ since some nuclei live much longer than the median.
Formula
How to Use
- 1Select a calculation mode from the tabs: Remaining Amount, Time Elapsed, Half-Life from Data, or Decay Constant.
- 2Choose your time unit (seconds, minutes, hours, days, or years), keep all time values in the same unit.
- 3Use a preset (Carbon dating, Tc-99m, Cs-137, Radon) to autofill a real-world scenario.
- 4Enter the initial amount N₀ (any positive number, grams, moles, atoms, or arbitrary units).
- 5Enter the remaining amount N and/or time elapsed t as required by the selected mode.
- 6Enter the half-life t½ for the isotope or process you are studying.
- 7Press Calculate (or Enter on any input) to instantly see the result.
- 8Expand "Step-by-step working" to see each substitution, and "Decay table" to see amounts at 0–10 half-lives.
Example Calculation
An archaeologist measures a charcoal sample with 25% of the original C-14 remaining. How old is it?
C-14 half-life: 5,730 years. N/N₀ = 0.25 = (1/2)^(t/5730).
t = 5730 × ln(4) / ln(2) = 5730 × 2 = 11,460 years.
A patient receives 500 MBq of Tc-99m (t½ = 6 hours). How much activity remains after 24 hours?
n = 24/6 = 4 half-lives. N = 500 × (1/2)^4 = 500/16 = 31.25 MBq.
100 kg of Cs-137 (t½ = 30.17 years). How much remains after 10 half-lives (301.7 years)?
N = 100 × (1/2)^10 = 100/1024 ≈ 0.0977 kg, less than 0.1% of the original.
Understanding Half-Life | Radioactive Decay
This calculator runs entirely in your browser, no data is sent to any server. All radioactive decay computations use the standard exponential decay formula N(t) = N₀ × e^(-λt), where λ = ln(2)/t½. Half-life reference values shown in the isotope table are sourced from the IAEA Live Chart of Nuclides.
Applications of Half-Life Calculations
- • Radiocarbon dating: determines the age of organic materials up to ~50,000 years using C-14 decay.
- • Medical imaging and therapy: Tc-99m (6 hr), I-131 (8 days), and Ra-223 are dosed based on half-life.
- • Nuclear waste management: multi-century planning for Cs-137 and Sr-90 storage and disposal.
- • Food irradiation and sterilization: using isotopes with controlled half-lives for safety.
- • Geological dating: U-235/Pb-207 and K-40/Ar-40 systems date rocks over millions of years.
Relationship Between Decay Constant and Half-Life
The decay constant λ and half-life t½ are inversely related: λ = ln(2)/t½ ≈ 0.6931/t½. A short half-life means rapid decay (large λ); a long half-life means slow decay (small λ). The mean lifetime τ = 1/λ is always 44.3% longer than the half-life.
Frequently Asked Questions
What is a half-life?
A half-life is the time required for exactly half of a radioactive substance to decay into another element or isotope.
- • It is a property of the specific isotope, not of the sample size
- • After one half-life: 50% remains
- • After two half-lives: 25% remains
- • After ten half-lives: 0.098% remains
Half-lives range from fractions of a second (unstable isotopes) to billions of years (U-238 at 4.47 × 10&sup9; yr).
Why does radioactive decay follow an exponential law?
Each radioactive nucleus has a fixed, constant probability of decaying per unit time, regardless of age or environment.
- • This is a quantum mechanical process, no memory of how long a nucleus has existed
- • With N nuclei, the decay rate is dN/dt = -λN
- • Solving the differential equation gives the exponential: N(t) = N₀ e^(-λt)
What is the difference between half-life and mean lifetime?
Both measure decay timescales but in different ways.
- • Half-life t½: time for 50% to decay, median survival time
- • Mean lifetime τ = 1/λ: average time a nucleus survives before decaying
- • Relationship: τ = t½ / ln(2) ≈ 1.4427 × t½
The mean lifetime is always longer than the half-life because some nuclei survive much longer than the median.
How accurate is radiocarbon dating?
C-14 dating is accurate to about ±30–40 years for recent samples, with precision decreasing for older materials.
- • Effective range: up to about 50,000 years (roughly 8–9 half-lives)
- • Calibrated against dendrochronology (tree ring) records back to 14,000 years
- • Atmospheric C-14 levels vary, requiring calibration curves (IntCal20)
Can the half-life of an element change?
For most practical purposes, no. Nuclear half-lives are determined by the strong nuclear force and quantum tunneling rates.
- • Temperature, pressure, and chemical bonds have negligible effect on nuclear decay
- • Exceptions: electron capture decay can be slightly affected by extreme ionization
- • Some accelerator-based experiments observe minute shifts under extreme conditions
What does the decay constant λ represent?
The decay constant λ is the probability per unit time that a single nucleus will decay.
- • Units: per second, per year, etc., inverse of the time unit
- • Formula: λ = ln(2) / t½
- • Activity A = λN (decays per second = Becquerels)
How do I use this calculator for biological half-lives?
The same exponential formula applies to biological half-lives, the time for the body to eliminate half of a drug or substance.
- • Enter the drug dose as N₀ and the biological half-life in hours
- • The Remaining Amount mode gives concentration at any time
- • After 5 half-lives (≈5 × t½), ~97% of the drug is eliminated
What are typical half-lives for common isotopes?
Half-lives span an enormous range across the periodic table:
- • C-14: 5,730 yr, archaeology and climate science
- • Tc-99m: 6 hr, medical imaging (most widely used)
- • I-131: 8.02 days, thyroid cancer treatment
- • Ra-226: 1,600 yr, discovered by Marie Curie
- • U-235: 703.8 Myr, nuclear fuel and geological dating