About the Half-Life calculator
The half-life calculator solves exponential decay problems. Given any three of the initial quantity, the remaining quantity, the elapsed time and the half-life, it finds the one that is missing. It also converts between half-life, mean lifetime and the decay constant.
Half-life appears in radioactive decay, radiocarbon dating, medicine (how quickly a drug leaves the body) and in any process where a quantity falls by the same fraction over equal intervals.
Who should use the Half-Life calculator
- Physics, chemistry and biology students.
- Anyone working with dating methods or decay.
- Learners who want to see exponential decay worked through.
How to use the Half-Life calculator: what each input means
- What to find
- Remaining quantity, time to decay, half-life, initial quantity, or a conversion between decay constants.
- Initial and remaining quantity
- In any consistent unit: grams, atoms, percent or activity.
- Elapsed time and half-life
- In the same unit of time, such as years or days.
How the Half-Life calculator works: method and formula
After each half-life, half of what remains has decayed. Mathematically N(t) = N₀ × (½)^(t ÷ t½). To find time or half-life, the equation is rearranged using natural logarithms. The decay constant λ equals ln 2 divided by the half-life, and the mean lifetime τ equals 1 ÷ λ.
- Remaining
- N(t) = N₀ × (1/2)^(t ÷ t½)
- Half-life
- t½ = t × ln 2 ÷ ln(N₀ ÷ N)
- Constants
- λ = ln 2 ÷ t½, τ = t½ ÷ ln 2
How to read your Half-Life calculator result
After 1 half-life, 50% remains; after 2, 25%; after 3, 12.5%. The result also reports how many half-lives have elapsed so you can sense the scale quickly.
Use the same time unit throughout. The time-unit label field is for display only.
Half-Life Calculator worked example
Carbon-14 has a half-life of 5,730 years. How much of 100 g remains after 11,460 years?
- 11,460 ÷ 5,730 = 2 half-lives.
- 100 × (1/2)² = 25.0 g.
- Working backwards, the time for 100 g to fall to 25 g is 11,460 years.
25.0 g remains after two half-lives.
Practical uses of the Half-Life calculator
- Radiocarbon dating examples.
- Estimating when a drug concentration falls below a threshold.
- Converting between half-life and decay constant for physics problems.
Assumptions and limitations of this calculator
- Assumes ideal first-order exponential decay with a constant half-life. Real systems, such as drug elimination, may not follow this exactly.
- Not a substitute for dating-laboratory calibration or clinical dosing decisions.