Half-life calculator

A half-life calculator works out how a radioactive sample, or anything else that halves at a steady rate, shrinks over time. The half-life t½ is the time for half of it to decay. Our Half-Life Calculator finds the amount left after any time, the half-life from your measurements, or the time to decay to a given amount, and shows how many half-lives pass at each step.

It opens on the three half-life modes, Amount left, Half-life from data and Time to decay. Any time unit works, seconds, days or years, as long as every time uses the same one.

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Half Life Calculator

Calculate radioactive decay and half-life

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How to use the half-life calculator

1. Under Half-life, choose Amount left, Half-life from data or Time to decay.
2. Type the values. The starting amount N₀ and the amount left N can be a number of nuclei, a mass, a percentage or an activity, as long as both use the same unit.
3. Use the same time unit for the half-life and the time.
4. Read the answer and the working: how many half-lives pass, then halving that many times.

A decay curve shows the amount falling over time. A second group of modes, under Decay constant (λ), converts between the half-life and the decay constant and uses N = N₀e^(−λt). For practice, Practice and help sets half-life questions with worked solutions.

The half-life formulas

The key number is t ÷ t½, the number of half-lives that pass. After 1 half-life, half is left; after 2, a quarter; after 3, an eighth. The form N = N₀e^(−λt) gives the same answers, because λ = ln 2 ÷ t½.

Worked examples you can check in the calculator

Example 1: a whole number of half-lives

A sample has 200 g of a substance with a half-life of 8 days. How much is left after 24 days? 24 ÷ 8 = 3 half-lives, so halve three times: 200, 100, 50, 25.

In Amount left, type 200, 8 and 24. The calculator shows 3 half-lives and 25.

Here x is the time in days and y is the mass left in grams. The curve halves every 8 days and never quite reaches zero.

Example 2: not a whole number of half-lives

Carbon-14 has a half-life of about 5,730 years, the value most textbooks use. What percentage of the carbon-14 in a sample is left after 10,000 years? 10,000 ÷ 5,730 = 1.745 half-lives:

Type 100 as the starting amount to work in percentages. The calculator gives 29.83.

Example 3: the half-life from data

A sample's count rate falls from 640 to 40 counts per minute in 12 hours. What is its half-life? 640 ÷ 40 = 16, which is 2⁴, so it has halved 4 times in 12 hours:

In Half-life from data, type 640, 40 and 12. The calculator works out log₂(16) = 4 and shares the 12 hours between the halvings. If the data does not give a whole number of halvings, it uses t½ = t ln 2 ÷ ln(N₀/N): from 1,000 to 300 in 20 minutes gives 11.5 minutes.

Example 4: the time to decay

How long until only 12.5% of a carbon-14 sample is left? 100 ÷ 12.5 = 8 = 2³, so it takes 3 half-lives: 3 × 5,730 = 17,190 years. In Time to decay, type 100, 12.5 and 5730. For 10% left with an 8-day half-life, log₂(10) = 3.32, so it takes 26.6 days.

Try a half-life sum in the course, then press Mark:

Half-life and the decay constant

The decay constant λ is the probability per unit time that a nucleus decays. It links to the half-life by λ = ln 2 ÷ t½, so a short half-life means a large decay constant. For carbon-14, λ = 0.693 ÷ 5,730 = 1.21 × 10⁻⁴ per year. Use the Decay constant (λ) modes to convert, and N = N₀e^(−λt) for the amount left. A-level and university courses use this form; it gives the same numbers as the half-life form.

Other things with a half-life

Anything that loses the same fraction of itself in each equal time interval decays exponentially and has a half-life. That includes many medicines leaving the body and the charge on a capacitor through a resistor, as well as radioactive nuclei. The same formulas work for all of them, but real drug half-lives vary from person to person, so treat any medical figure as general information rather than advice.

Common mistakes

• Mixing time units. A half-life in days and a time in hours gives the wrong number of half-lives. Convert first.
• Subtracting half the original each time. Each half-life halves what is left, not the starting amount: 200, 100, 50, 25, not 200, 100, 0.
• Thinking it reaches zero. In theory the amount keeps halving and never quite reaches zero.
• Using the time instead of the number of half-lives as the power. The power is t ÷ t½.
• Confusing the amount decayed with the amount left. After 2 half-lives, 25% is left and 75% has decayed.

Practise half-life in a short timed paper

Half-life comes up in High School Physics and AP Physics 2 practice, and in GCSE Physics Higher in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics. The Nuclear Energy course covers radioactive decay from alpha, beta and gamma to half-life, and our alpha, beta and gamma guide covers the radiation itself.

Read next

Physics · Atoms and particlesHalf-life: definition, graphs and calculationsWhat half-life means, how to find it from a decay graph, GCSE half-life calculations and net decline, and the A-level decay constant, with worked examples.