About this calculator
The half-life calculator tells you how much of a decaying substance is left after a given time, using the initial amount, the half-life, and the elapsed time. It works for any first-order exponential decay, from radioactive isotopes to drugs clearing from the bloodstream.
Half-life is the time required for a quantity to fall to half its value. The idea was introduced by Ernest Rutherford in 1907 while studying radioactivity, and it is powerful because the halving time is constant: it does not matter whether you start with a kilogram or a microgram, the fraction remaining after one half-life is always 50 percent. That property underpins radiocarbon dating in archaeology, dosing schedules in medicine, and the management of nuclear waste. This calculator also reports how many half-lives have elapsed, the percentage remaining, and the time needed to fall to 1 percent.
How it works
The remaining amount follows a simple exponential law. The exponent is just the number of half-lives that have passed, so the maths reduces to repeated halving.
N = N0 x (1/2)^(t / T) decay form used by this tool N0 = initial amount t = time elapsed T = half-life t/T = number of half-lives passed Equivalent: N = N0 x e^(-lambda x t), lambda = ln(2) / T = 0.693 / T
- Units must match: t and T have to be in the same time unit (both days, both years, and so on).
- Decay constant lambda = ln(2) / T relates the half-life to the continuous exponential rate.
- Each half-life halves the amount: 100, 50, 25, 12.5, 6.25 percent, and so on.
- Fraction remaining after n half-lives is (1/2) to the power n, independent of the starting quantity.
Worked example
You start with 100 mg of a substance whose half-life is 10 hours. How much remains after 30 hours?
- Count half-lives: t / T = 30 / 10 = 3 half-lives.
- Apply the formula: N = 100 x (1/2)^3 = 100 x (1/8).
- Compute: 100 / 8 = 12.5 mg remaining.
- Percent decayed: 100 - 12.5 = 87.5 mg gone, or 87.5 percent.
- Time to about 1 percent: roughly 7 half-lives (1/2 to the 7th is 0.78 percent), so about 70 hours.
Decay and half-life reference
The left table shows how quickly any substance falls with each half-life; the right column lists real-world half-lives that span billions of years to a few hours.
| Half-lives elapsed | Percent remaining | Example real half-life |
|---|---|---|
| 1 | 50 percent | Caffeine in adults: ~5 hours |
| 2 | 25 percent | Iodine-131: ~8 days |
| 3 | 12.5 percent | Cobalt-60: ~5.3 years |
| 5 | 3.1 percent | Carbon-14: ~5,730 years |
| 7 | 0.78 percent | Plutonium-239: ~24,100 years |
| 10 | 0.098 percent | Uranium-238: ~4.5 billion years |
Common pitfalls
- Mismatched units. The most frequent error is mixing units, such as a half-life in days with an elapsed time in hours. Convert both to the same unit first.
- Assuming linear decay. Decay is exponential, not straight-line. After two half-lives you have 25 percent left, not zero; the substance never reaches exactly zero, only ever-smaller fractions.
- Confusing half-life with full lifetime. A substance is not gone after one half-life; it is merely halved. Practical "fully gone" is conventionally taken as about 5 half-lives.
- Mixing up half-life and mean lifetime. The mean lifetime is 1/lambda = T / ln(2), about 1.44 times the half-life, not the same number.
- Applying nuclear constancy to biology. Radioactive half-lives are fixed, but a drug's biological half-life varies with liver and kidney function, age, and interactions, so quoted values are averages.
- Rounding the constant. Using 0.69 instead of ln(2) = 0.6931 for lambda introduces small errors that compound over many half-lives.
Related tools
Frequently asked questions
What is half-life?
Half-life is the time it takes for a quantity that decays exponentially to fall to half of its starting value. It is a fixed property of the substance, so it does not depend on how much you start with: a sample halves in one half-life, halves again to a quarter after two, to an eighth after three, and so on. The concept applies to radioactive isotopes, drug clearance in the body, and any first-order exponential decay.
What is the half-life decay formula?
The amount remaining is N = N0 x (1/2)^(t/T), where N0 is the initial amount, t is the time elapsed, and T is the half-life. The exponent t/T is simply the number of half-lives that have passed. An equivalent continuous form is N = N0 x e^(-lambda x t), where the decay constant lambda equals ln(2) divided by T, which is about 0.693 / T.
Why are drugs considered cleared after about five half-lives?
After five half-lives only (1/2) to the fifth power remains, which is 1/32, or about 3.1 percent of the original dose. Pharmacologists treat that as effectively cleared because the leftover amount is usually too small to have a clinical effect. The same five-half-lives rule of thumb is used in reverse to estimate when a drug taken repeatedly reaches steady-state concentration.
What are some real half-life values?
They span an enormous range. Carbon-14, used for radiocarbon dating, has a half-life of about 5,730 years; Iodine-131, used in medicine, is about 8 days; Uranium-238 is around 4.5 billion years. In pharmacology, caffeine has a half-life of roughly 5 hours in a healthy adult, ibuprofen about 2 hours, and some antidepressants several days. The wide spread is why half-life is always quoted with its unit.
Does half-life depend on temperature or how much you start with?
For radioactive decay, no. The half-life is a fixed nuclear property, unaffected by temperature, pressure, chemical state, or the amount of material present, which is what makes radiometric dating reliable. Biological half-lives in the body can vary between people because they depend on liver and kidney function, age, and other drugs, so a drug's quoted half-life is a population average rather than an exact personal value.
Sources
- Rutherford, Ernest (1907) work on radioactive decay - origin of the half-life concept.
- IUPAC Compendium of Chemical Terminology - definitions of half-life and decay constant.
- Goodman and Gilman, The Pharmacological Basis of Therapeutics - the five-half-lives clearance and steady-state rules.