The Half-Life Formula, Explained with Examples
By ToolNimba Editorial Team June 20, 2026 8 min read
Quick answer
The half-life formula is remaining amount = initial amount times (1/2) raised to the power (t / half-life). After one half-life, half of the original amount remains; after two half-lives, a quarter remains; after three, an eighth, and so on.
A half-life is the time it takes for half of something to disappear, decay, or break down. It shows up in radioactive decay, in how long a medicine stays in your body, and in carbon dating. The idea is simple once you see the pattern: every half-life, whatever is left gets cut in half again. This guide walks through the formula, a decay chart, and several worked examples you can follow step by step.
What is a half-life?
A half-life is the fixed amount of time required for a quantity to fall to half of its starting value. The key word is fixed: no matter how much you start with, it always takes the same length of time to lose half. If a substance has a half-life of 10 years, then 10 years from now you will have half of what you have today, and 10 years after that you will have a quarter.
This kind of behaviour is called exponential decay. The amount does not drop by a fixed number each year; it drops by a fixed fraction. Because the leftover shrinks each round, the quantity gets smaller and smaller but, in theory, never quite reaches zero. If you have met exponential growth, half-life is its mirror image: instead of repeatedly multiplying by a number bigger than one, you repeatedly multiply by one half.
The half-life formula
The most useful form of the formula tells you how much is left after any amount of time has passed:
Half-life decay formula
N = N0 times (1/2) raised to the power (t / T), where N is the amount remaining, N0 is the initial amount, t is the time elapsed, and T is the half-life.
The exponent t / T counts how many half-lives have gone by. If t equals T, the exponent is 1, so you multiply by one half once and end up with half. If t equals 2T, the exponent is 2, so you multiply by one half twice and end up with a quarter. You do not need a whole number of half-lives: the formula works for any time t, because raising one half to a fractional power gives a value between the two whole-step results.
Notice that the starting amount N0 can be measured in anything: grams, milligrams, number of atoms, becquerels of activity, or a percentage. As long as N and N0 use the same units, the ratio takes care of itself. To turn a leftover fraction into a percentage you can use a percentage calculator, which is handy when a problem asks what percent remains.
Half-life decay chart
The fastest way to build intuition is to watch the fraction halve, half-life after half-life. This chart shows what is left of a starting amount as whole half-lives pass.
Fraction and percentage remaining after each half-life
| Half-lives passed | 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% |
A common rule of thumb taken from this chart: after about 10 half-lives, less than one tenth of one percent remains, which is why scientists often treat a substance as effectively gone by that point. Converting these tidy fractions into percentages is the same skill as in how to calculate percentage.
Worked example: counting whole half-lives
A sample starts with 80 grams of a substance whose half-life is 5 years. How much remains after 15 years?
- Find how many half-lives have passed: 15 years divided by 5 years equals 3 half-lives.
- Write the formula: N = N0 times (1/2) raised to the power (t / T).
- Substitute the values: N = 80 times (1/2) raised to the power 3.
- Evaluate the power: (1/2) raised to the power 3 equals 1/8.
- Multiply: 80 times 1/8 equals 10.
- State the answer: 10 grams remain after 15 years.
You can sanity-check this by halving step by step: 80 becomes 40 after 5 years, 40 becomes 20 after 10 years, and 20 becomes 10 after 15 years. Both routes agree.
Worked example: a time that is not a whole half-life
A 200 mg dose of a drug has a half-life of 6 hours. How much is left after 9 hours?
- Find the exponent: t / T = 9 / 6 = 1.5 half-lives.
- Write the formula: N = 200 times (1/2) raised to the power 1.5.
- Evaluate the power: (1/2) raised to the power 1.5 is about 0.3536.
- Multiply: 200 times 0.3536 is about 70.7.
- State the answer: roughly 70.7 mg remain after 9 hours.
As expected, the answer sits between the values for one half-life (100 mg) and two half-lives (50 mg), which is exactly what a fractional number of half-lives should give.
Where half-life shows up in real life
- Radioactive decay. Unstable isotopes lose half their atoms each half-life, from fractions of a second to billions of years depending on the element.
- Carbon dating. Carbon-14 has a half-life of about 5,730 years, so measuring how much remains in old material estimates its age.
- Medicine. A drug half-life tells doctors how quickly a dose clears, which sets how often you take a pill.
- Medical imaging. Tracers used in scans are chosen to have short half-lives so they fade quickly and limit exposure.
- Caffeine and the body. Caffeine has a half-life of roughly 5 hours, which is why an afternoon coffee can still affect sleep at night.
How to find the half-life itself
Sometimes the half-life is the unknown. You might know the starting amount, how much is left, and how long it took, and need to work backward to find T. This is the reverse of the problems above, and it needs a logarithm to pull the exponent down. Start from N = N0 times (1/2) raised to the power (t / T), divide both sides by N0, then take the logarithm of both sides so the exponent comes down in front.
Solving for the half-life
T = t times ln(2) / ln(N0 / N), or equivalently T = t / log base 2 of (N0 / N). Here ln is the natural logarithm, and N0 / N is the ratio of the starting amount to the amount that is left.
If you instead know the decay constant k from the natural-exponential form, the half-life is even quicker: T = ln(2) / k, which is about 0.693 / k. The two are tied together because halving and continuous decay describe the same curve from different angles. The same reverse logic powers the math behind exponential growth, where you solve for a doubling time instead of a halving time.
Worked example: solving for an unknown half-life
A 100 gram sample decays to 30 grams in 8 hours. What is its half-life?
- Write the reverse formula: T = t times ln(2) / ln(N0 / N).
- Find the ratio N0 / N: 100 divided by 30 equals about 3.333.
- Take the natural logs: ln(2) is about 0.6931, and ln(3.333) is about 1.2040.
- Substitute: T = 8 times 0.6931 / 1.2040.
- Evaluate: 8 times 0.6931 equals about 5.545, and 5.545 / 1.2040 equals about 4.61.
- State the answer: the half-life is about 4.6 hours.
Quick sanity check: 4.6 hours is a little more than half of 8 hours, so a bit fewer than two half-lives have passed in 8 hours. Two half-lives would leave 25 grams, and we have slightly more at 30 grams, so the answer is in the right range.
Half-life of common substances
Half-lives span an astonishing range, from tiny fractions of a second to billions of years. The table below collects values that come up often in chemistry, biology, and everyday life so you can compare them at a glance.
Approximate half-lives of well-known substances and isotopes
| Substance | Approximate half-life | Why it matters |
|---|---|---|
| Carbon-14 | 5,730 years | Radiocarbon dating of organic material |
| Uranium-238 | 4.5 billion years | Dating rocks and the age of the Earth |
| Plutonium-239 | 24,100 years | Nuclear fuel and long-term waste |
| Iodine-131 | 8 days | Thyroid treatment and imaging |
| Technetium-99m | 6 hours | Most common medical imaging tracer |
| Caffeine (in adults) | about 5 hours | Why late coffee disrupts sleep |
| Ibuprofen | about 2 hours | How often to take a dose |
Drug half-lives in particular guide dosing schedules: a short half-life means more frequent doses, while a long one means a drug lingers. Working out how a percentage of a dose clears each hour is the same arithmetic covered in how to calculate percentage.
Common mistakes to avoid
- Subtracting instead of halving. Each half-life removes half of what is currently left, not a fixed amount. Going from 80 to 40 removes 40, but the next step removes only 20.
- Mismatched time units. The elapsed time t and the half-life T must use the same unit. Convert one before dividing, the same care you take with any unit conversion.
- Thinking it ever reaches zero. Halving repeatedly gets very close to zero but never exactly there, so the amount remaining is always a small positive number.
- Forgetting the exponent counts half-lives. The power in the formula is t / T, not t alone. Plugging in raw time without dividing by the half-life gives a wildly wrong answer.
- Doubling a half-life to find the full lifetime. Two half-lives leaves a quarter, not nothing. A substance is never fully gone after a fixed number of steps.
Good to know
There is a second common way to write the same formula using the natural exponential, N = N0 times e raised to the power (negative k times t), where k is the decay constant. The two forms are linked by k = ln(2) / T, so a longer half-life means a smaller decay constant and slower decay. Both describe identical curves; the (1/2) version is friendlier for hand calculation, while the e version is preferred in calculus. Either way, the headline rule stays the same: one half-life halves it, two half-lives quarter it, and the pattern continues without end.
Frequently asked questions
What is the half-life formula?
The half-life formula is N = N0 times (1/2) raised to the power (t / T), where N0 is the starting amount, t is the time elapsed, and T is the half-life. The exponent t / T counts how many half-lives have passed, and each one cuts the remaining amount in half.
How much is left after two half-lives?
After two half-lives, a quarter of the original amount remains. The first half-life leaves one half, and halving that half again leaves one quarter, or 25 percent. After three half-lives an eighth remains, and after four half-lives a sixteenth remains.
How do you calculate half-life decay?
Divide the elapsed time by the half-life to find how many half-lives have passed, then multiply the starting amount by one half raised to that power. For example, 80 grams with a 5-year half-life after 15 years gives 80 times (1/2) cubed, which equals 10 grams.
Does a substance ever fully decay to zero?
Not exactly. Each half-life removes half of what remains, so the amount keeps shrinking but never reaches absolute zero in theory. In practice, after about 10 half-lives less than 0.1 percent is left, so scientists often treat the substance as effectively gone.
What is the half-life of carbon-14?
Carbon-14 has a half-life of about 5,730 years. Living things absorb carbon-14, and once they die it decays at a steady rate. Measuring how much remains lets scientists estimate the age of ancient organic material, a technique known as radiocarbon dating.
Can you have a fractional number of half-lives?
Yes. Time does not have to land on a whole half-life. If 9 hours pass for a substance with a 6-hour half-life, that is 1.5 half-lives, and you raise one half to the power 1.5. The result falls between the one and two half-life values, as expected.
How do you find the half-life if you only know the decay rate?
Take the natural logarithm of both sides of the decay equation to bring the exponent down. The shortcut is T = t times ln(2) / ln(N0 / N), where N0 is the starting amount, N is what is left, and t is the elapsed time. If you know the decay constant k instead, use T = ln(2) / k, which is about 0.693 / k.
What is the difference between half-life and decay constant?
The decay constant k measures how fast a substance decays at every instant in the natural-exponential form N = N0 times e raised to negative k times t. The half-life T is the time for half to vanish. They describe the same curve and are linked by T = ln(2) / k, so a larger decay constant always means a shorter half-life.
What is the half-life of caffeine?
Caffeine has a half-life of roughly 5 hours in a typical adult, though it varies with age, liver function, and pregnancy. After one half-life about half the caffeine remains, so a 200 mg afternoon coffee can still leave around 50 mg in your system at bedtime, which is why late caffeine often disrupts sleep.