๐งช Half Life Calculator: Decay, Time and Half-Life
By ToolNimba Editorial Team ยท Updated 2026-06-20
Tip: keep time and half-life in the same unit (both years, both hours, etc.). The amount unit is up to you.
Enter the known values to compute the unknown one.
Half-life is the time it takes for a quantity to fall to half of its starting value. This half life calculator uses the standard decay relationship N = N0 times 0.5 raised to the power (t divided by T), and it lets you solve for whichever piece you are missing: the remaining amount, the elapsed time, or the half-life itself. Alongside the answer you also see how many half-lives have passed and what percent of the original amount is left.
What is the Half Life Calculator?
Half-life describes any process where a quantity shrinks by a fixed fraction in equal time steps. After one half-life, half of the original amount remains. After two half-lives, a quarter remains. After three, an eighth. The defining feature is that the same proportion disappears in every equal interval, which is what makes the decay exponential rather than linear. This pattern shows up in radioactive isotopes, drug concentrations in the bloodstream, the discharge of a capacitor, and the cooling of an object toward room temperature.
The working formula is N = N0 times 0.5^(t/T), where N0 is the initial amount, N is the amount remaining after time t, and T is the half-life. The exponent t/T is simply the number of half-lives that have elapsed. If t/T equals 1 you have one half-life and N is half of N0; if t/T equals 3.5 you have three and a half half-lives and N is 0.5^3.5, or about 8.8 percent, of N0. Because the base is one half, every whole step in the exponent halves the amount again.
The same equation rearranges to answer the reverse questions. To find the elapsed time from a known fraction remaining, use t = T times log base 2 of (N0/N). To find the half-life from a measured decay, use T = t divided by log base 2 of (N0/N). These forms come straight from taking a base-2 logarithm of the original equation, and they are exactly what this calculator applies when you switch the solve-for selector. The amount units cancel out, so only the ratio N0/N matters, not whether you measure in grams, milligrams, counts per minute, or percent.
Half-life is closely related to the decay constant and to mean lifetime, but they are not the same number. The decay constant k equals the natural log of 2 divided by T (about 0.693/T), and the mean lifetime tau equals 1/k, which is longer than the half-life. If a source quotes a decay constant or a time constant instead of a half-life, convert first so every value in the calculator refers to the same quantity. Mixing a half-life with a decay constant is one of the most common sources of wrong answers.
When to use it
- Working out how much of a radioactive sample is left after a given number of years for a physics or chemistry problem.
- Estimating how long a medication stays meaningfully active in the body based on its elimination half-life.
- Finding an unknown half-life from a measured before-and-after amount and the time between the two readings.
- Checking carbon-14 dating style calculations, where the fraction of the original isotope reveals the age of a sample.
How to use the Half Life Calculator
- Choose what to solve for: the remaining amount, the elapsed time, or the half-life.
- Enter the initial amount (N0) and the other two known values in the same time unit.
- Read the headline answer, which updates instantly as you type.
- Check the half-lives elapsed and percent remaining panels to sanity-check the result.
Formula & method
Worked examples
A 100 mg sample of carbon-14 (half-life 5730 years) decays for 11460 years. How much remains?
- Find the number of half-lives: t/T = 11460 / 5730 = 2.
- Apply the formula: N = 100 times 0.5^2 = 100 times 0.25.
- So N = 25 mg, which is 25 percent of the original.
Result: 25 mg remains (2 half-lives, 25 percent left).
A drug starts at 200 mg and only 50 mg is left after some time. Its half-life is 8 hours. How long has passed?
- Find the ratio N0/N = 200 / 50 = 4.
- Number of half-lives = log base 2 of 4 = 2.
- Elapsed time t = T times 2 = 8 hours times 2 = 16 hours.
Result: About 16 hours have passed (2 half-lives elapsed).
Fraction remaining after each half-life
| Half-lives elapsed (t/T) | Fraction remaining | Percent 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 | 3.125% |
| 10 | 1/1024 | about 0.098% |
Half-life of some well-known isotopes
| Isotope | Half-life | Common use |
|---|---|---|
| Carbon-14 | 5730 years | Radiocarbon dating of organic material |
| Iodine-131 | 8.02 days | Thyroid imaging and treatment |
| Cobalt-60 | 5.27 years | Cancer radiotherapy and sterilisation |
| Uranium-238 | 4.47 billion years | Dating very old rocks |
| Technetium-99m | 6.01 hours | Medical diagnostic imaging |
Common mistakes to avoid
- Mixing time units between t and T. The elapsed time and the half-life must use the same unit, because only the ratio t/T matters. If the half-life is in days, the elapsed time must also be in days. Plugging in 16 hours against an 8-day half-life gives a wildly wrong answer.
- Confusing half-life with the decay constant. The half-life T and the decay constant k are different numbers, linked by k = 0.693/T. If a problem gives you k or a time constant, convert it to a half-life first, or you will be off by a factor of about 0.693.
- Treating decay as linear. Decay does not subtract a fixed amount each step; it multiplies by one half each half-life. After two half-lives you have a quarter left, not zero. Assuming it reaches zero after two half-lives is a frequent error.
- Entering a remaining amount larger than the initial. In a decay process the remaining amount N can never exceed the initial amount N0. If N is bigger than N0 the formula has no valid solution, so double-check that you have not swapped the two values.
Glossary
- Half-life (T)
- The time required for a quantity to decay to half of its current value. It is constant for a given decay process.
- Initial amount (N0)
- The starting quantity before any decay has taken place, measured at time t = 0.
- Remaining amount (N)
- The quantity left after a time t has elapsed, given by N = N0 times 0.5^(t/T).
- Exponential decay
- A process where a quantity decreases by the same proportion in each equal time interval, producing a curve that flattens toward zero.
- Decay constant (k)
- The rate parameter of decay, equal to 0.693 divided by the half-life. It sets how fast the quantity falls.
- Number of half-lives
- The value t/T, telling you how many halving steps have occurred. A value of 3 means the amount has halved three times.
Frequently asked questions
What is the half-life formula?
The half-life formula is N = N0 times 0.5^(t/T), where N0 is the initial amount, N is the remaining amount, t is the elapsed time and T is the half-life. The exponent t/T is the number of half-lives that have passed. The same equation rearranges to t = T times log base 2 of (N0/N) when you need the time.
How do I calculate the remaining amount after a number of half-lives?
Multiply the initial amount by 0.5 raised to the number of half-lives. After 1 half-life you have half left, after 2 a quarter, after 3 an eighth. For 2.5 half-lives you would compute N0 times 0.5^2.5, which is about 17.7 percent of the original.
How do I find an unknown half-life?
Use T = t divided by log base 2 of (N0/N). Take the initial amount and the remaining amount to get the ratio N0/N, take the base-2 logarithm to get the number of half-lives, then divide the elapsed time by that number. This calculator does it automatically when you set it to solve for half-life.
Is half-life the same as the decay constant?
No. They describe the same decay but are different numbers. The decay constant k equals 0.693 divided by the half-life T. If your source gives a decay constant or a time constant, convert it to a half-life before using this calculator, otherwise your answer will be off.
Does the unit of the amount matter?
No. Only the ratio N0/N affects the time and half-life calculations, so grams, milligrams, counts per minute, or percent all work as long as you are consistent. The remaining-amount mode returns its answer in whatever unit you used for the initial amount.
Can half-life calculations apply outside radioactivity?
Yes. Any process with constant proportional decay follows the same math, including drug elimination in the body, a discharging capacitor, and a cooling object approaching room temperature. Wherever a quantity loses the same fraction in equal time steps, the half-life formula applies.