Math · Live
Half-life & exponential
decay calculator.
Solve any exponential decay problem in four directions — find the remaining quantity, initial quantity, elapsed time, or half-life itself. Supports all time units, includes a smooth decay curve, and covers everything from radioactive isotopes to drug pharmacokinetics and chemical reactions.
Inputs
Decay parameters
Solve for
N = 1,000 × (½)^1.745 = 298.29
Half-life (t½)
Elapsed time (t)
Common half-lives
Find remaining → N
29.8292% of the original remains
Remaining
29.829%
of N₀
Decay curve
Remaining quantity vs. half-lives elapsed
Reference
Fraction remaining after n half-lives
| Half-lives (n) | Fraction | % remaining |
|---|---|---|
| 0 | 1 | 100% |
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 4 | 1/16 | 6.25% |
| 5 | 1/32 | 3.13% |
| 7 | 1/128 | 0.78% |
| 10 | 1/1024 | 0.1% |
Decay guide
What is half-life and how is it calculated?
The half-life of a substance is the time it takes for exactly half of a quantity to decay, disintegrate, or be eliminated. It is the fundamental unit of exponential decay — a mathematical process in which the rate of decrease at any moment is proportional to the current amount. Exponential decay governs radioactive nuclei, drug concentrations in the bloodstream, atmospheric pollutants, charging capacitors, and many other physical and biological processes.
The exponential decay formula
The general law of exponential decay is:
The two forms are equivalent. The first uses the natural exponential with decay constant λ; the second uses base-½ with the half-life t½ directly. The relationship between them is:
where ln(2) is the natural logarithm of 2. Given a half-life of 5,730 years (Carbon-14), the decay constant is 0.693147 / 5730 ≈ 1.21 × 10⁻⁴ yr⁻¹.
Solving the four unknowns
The decay equation has four variables — N₀, N, t½, and t. Given any three, the fourth can be solved algebraically:
| Solving for | Formula | Requires |
|---|---|---|
| N: Remaining | N = N₀ × (½)^(t / t½) | N₀, t½, t |
| N₀: Initial | N₀ = N ÷ (½)^(t / t½) | N, t½, t |
| t — Elapsed time | t = t½ × log₂(N₀ / N) | N₀, N, t½ |
| t½: Half-life | t½ = t ÷ log₂(N₀ / N) | N₀, N, t |
The number of half-lives and remaining fraction
The most intuitive way to think about half-life is in terms of the number of half-lives elapsed, n = t / t½. After each integer number of half-lives, exactly half of the previous amount remains:
| Half-lives (n) | Fraction | % remaining |
|---|---|---|
| 0 | 1 | 100% |
| 1 | ½ | 50% |
| 2 | ¼ | 25% |
| 3 | ⅛ | 12.5% |
| 4 | 1/16 | 6.25% |
| 5 | 1/32 | 3.125% |
| 7 | 1/128 | 0.781% |
| 10 | 1/1024 | 0.0977% |
After 7 half-lives, less than 1% of the original remains. After 10 half-lives, just under 0.1% persists. This is why 10 half-lives is often used as a practical "fully decayed" threshold in pharmacology and radiochemistry.
Radioactive decay and nuclear physics
In nuclear physics, half-life describes how long it takes for half the nuclei in a radioactive sample to undergo spontaneous disintegration (α, β, or γ decay). The half-life is a fixed physical constant for each isotope, independent of temperature, pressure, chemical environment, or the amount of material present ; this is what makes radioactive decay uniquely useful for dating and measurement.
Half-lives span an extraordinary range across the periodic table:
| Isotope | Half-life | Application |
|---|---|---|
| Polonium-214 | 1.643 × 10⁻⁴ s | Nuclear physics |
| Radon-222 | 3.82 days | Radiation safety |
| Iodine-131 | 8.02 days | Cancer treatment |
| Carbon-14 | 5,730 years | Archaeological dating |
| Caesium-137 | 30.17 years | Industrial, chernobyl |
| Plutonium-239 | 24,110 years | Nuclear weapons/waste |
| Uranium-238 | 4.468 × 10⁹ years | Geological dating |
Radiocarbon dating: how it works
Carbon-14 (14C) is a radioactive isotope of carbon with a half-life of 5,730 years. It is continuously produced in the upper atmosphere by cosmic-ray neutrons colliding with nitrogen-14. While an organism is alive, it continuously exchanges carbon with its environment, maintaining a constant ratio of 14C to stable 12C. When the organism dies, exchange stops and the 14C begins to decay.
By measuring the current 14C/12C ratio and comparing it to the known atmospheric ratio, archaeologists can calculate elapsed time using:
A sample with 75% of its original 14C remaining has decayed for 5730 × log₂(1 / 0.75) ≈ 5730 × 0.415 ≈ 2,378 years. The technique is reliable back to ~50,000 years, beyond which the remaining 14C concentration becomes too small to measure accurately.
Drug pharmacokinetics: elimination half-life
In pharmacology, elimination half-life (t½) is the time for the concentration of a drug in the bloodstream to fall by 50%. Most drugs follow first-order kinetics — the same exponential decay law, where a constant fraction of the drug is eliminated per unit time, regardless of the current concentration.
Practical implications:
- Steady state: A drug reaches steady-state concentration after approximately 4–5 half-lives of regular dosing. A drug with t½ = 12 hours reaches steady state in ~2–3 days.
- Clearance: After 5 half-lives, ~97% of a drug has been eliminated. After 7 half-lives, ~99.2%. This guides washout periods before switching medications.
- Dosing interval: Drugs with short half-lives (ibuprofen: ~2h) require frequent dosing to maintain therapeutic levels. Drugs with long half-lives (fluoxetine: ~4 days) can be dosed weekly.
Other applications of exponential decay
- Capacitor discharge: A capacitor discharging through a resistor follows exponential decay with time constant τ = RC. The "half-life" equivalent is t½ = RC × ln(2).
- Newton's law of cooling: Temperature excess above ambient decays exponentially toward zero.
- Atmospheric pressure: Pressure decreases approximately exponentially with altitude (scale height model).
- Population dynamics: Populations undergoing constant fractional mortality follow exponential decay.
Disclaimer
Results are calculated using the standard exponential decay formula with the values you enter. For forensic, medical, or safety-critical applications, consult a qualified physicist, pharmacologist, or relevant specialist.