Half-Life Calculator
Model exponential decay for isotopes, pharmacology, and scientific research.
Half Life Calculator
The half-life of a substance is the time it takes for half of it to decay or break down. Used in nuclear physics, pharmacology, chemistry, and archaeology, this calculator solves any variable in the half-life equation.
How to Use This Calculator
- Enter the initial quantity (N₀).
- Enter the half-life (t½) in the appropriate time unit.
- Enter the time elapsed.
- Click Calculate to find the remaining quantity.
- Or enter initial + remaining quantity + half-life → find time elapsed.
Half-Life Formula
N(t) = N₀ × (1/2)^(t/t½)
Where: N(t) = remaining quantity | N₀ = initial quantity | t = elapsed time | t½ = half-life
Equivalent form: N(t) = N₀ × e^(−λt) where λ = ln(2)/t½ is the decay constant.
Example Calculation
A radioactive sample starts with 100 mg. The half-life is 5 years. How much remains after 15 years?
- Number of half-lives: 15 ÷ 5 = 3
- Remaining: 100 × (1/2)³ = 100 × 1/8 = 12.5 mg
Real-World Applications
- Carbon dating: Carbon-14 has a half-life of ~5,730 years. Used to date organic material up to ~50,000 years old.
- Medical imaging: Technetium-99m (t½ = 6 hours) is widely used in diagnostic scans.
- Pharmacology: Drug half-life determines dosing frequency. Ibuprofen t½ ≈ 2 hours; Diazepam t½ ≈ 36–100 hours.
- Nuclear waste: Plutonium-239 has a half-life of 24,100 years — explaining why nuclear waste storage is such a long-term challenge.
Common Mistakes to Avoid
- Confusing half-life with decay constant — λ = ln(2)/t½ ≈ 0.693/t½. They're related but distinct. Use the formula appropriate for your context.
- Mixing time units — If t½ is in days and t is in hours, convert to consistent units before calculating.
- Assuming half-life is constant under all conditions — Radioactive decay is truly constant. Chemical half-life can vary with temperature, pH, and concentration.
Frequently Asked Questions
Can something fully decay?
Theoretically never — each half-life reduces the amount by half, approaching but never reaching zero. Practically, after 10 half-lives, less than 0.1% of the original remains (effectively undetectable).
What is the mean life vs. half-life?
Mean life (τ) is the average time a nucleus exists before decay. τ = t½ ÷ ln(2) ≈ 1.443 × t½. The mean life is always longer than the half-life.
How is half-life used in drug dosing?
After 4–5 half-lives, a drug reaches steady-state concentration. Drugs with short half-lives need more frequent dosing (ibuprofen q4–6h); long half-life drugs can be dosed once daily or weekly (fluoxetine weekly).
Conclusion
Half-life calculations are essential across physics, chemistry, medicine, and archaeology. Use this calculator for any radioactive decay or exponential decline problem — it handles all three possible unknowns (remaining quantity, time, or initial quantity).
Related: Exponent Calculator | Log Calculator | Percent Calculator
Science Tip
After 10 half-lives, less than 0.1% of the initial substance remains. This is often used as a rule of thumb for when a radioactive source or drug is considered "gone" from a system.
