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Half-Life Calculator

Find the amount left, the time elapsed, the half-life or the starting amount of a decaying substance, plus its decay constant.

Level: Grades 10–12

  • Common Core standard HSF.LE.A.1.cRecognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
  • Common Core standard HSF.LE.A.4For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Printable worksheet

Half-Life Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Solve for

Any unit: grams, atoms, becquerels or percent. The remaining amount uses the same unit.

The time for half of the substance to decay, in the same unit as the elapsed time.

Result

Amount remaining (N)

12.565

Amount remaining is 12.565; 12.565% of the original amount remains after 2.9925 half-lives.

Decay constant (λ)
0.08642733 per d
Mean lifetime (τ = 1/λ)
11.5704 d
Fraction remaining
12.565%
Half-lives elapsed
2.9925

See the step-by-step solution ↓

Step-by-step solution

  1. Count the half-lives that have passed: n = t ÷ T½

    n=24 d8.02 d=2.992519

  2. Halve the starting amount once per half-life: N = N₀ × (1/2)ⁿ

    N=100×(12)2.992519=12.564989

  3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½

    λ=0.6931478.02 d=0.08642733 per d

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out how much of a decaying substance is left, how long it has been decaying, or what its half-life is. A half-life is the time it takes for half of a decaying quantity to disappear. After one half-life, 50% is left; after two, 25%; after three, 12.5%; and so on. Radioactive isotopes behave this way, and so do many first-order processes in chemistry and pharmacology, so the same equation turns up in many places.

This calculator works the decay law in all four directions. Give it any three of the starting amount, the amount remaining, the elapsed time and the half-life, and it finds the fourth. It also reports the decay constant λ, the mean lifetime, the percent remaining and the number of half-lives that have passed.

Where it's used

Why this math matters outside the classroom:

  • Nuclear waste. Engineers planning nuclear waste storage note that after 10 half-lives about 0.1% of the original material remains.
  • Medical isotopes. Hospitals order technetium-99m close to use because, with a half-life of about 6 hours, a quarter is left after 12 hours.
  • Amount remaining. Students who start with 80 g of a substance with a 5-day half-life find 80 × (1/2)³ = 10 g left after 15 days.
  • Finding a half-life. If a sample falls from 200 to 50 counts per minute in 8 hours, two half-lives have passed, so the half-life is 4 hours.

Grade level and standards: Grades 10–12; Common Core State Standards for Mathematics HSF.LE.A.1.c, HSF.LE.A.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose what you want to find: the amount remaining, the time elapsed, the half-life or the initial amount.
  2. Enter the other three values. Both amounts must use the same unit, and both times must use the same unit.
  3. Pick the time unit so the decay constant and mean lifetime are labeled correctly.
  4. Read the answer and the working. The steps show how many half-lives have passed and how the formula was rearranged.

What the inputs mean

Initial amount (N₀)
The amount at the start. It can be a mass, a number of atoms, an activity in becquerels or just 100 for percent.
Amount remaining (N)
The amount left at the end, in the same unit as the initial amount.
Half-life (T½)
The time for any amount of the substance to halve. It doesn't depend on how much you start with.
Time unit (for both times)
Only a label: the calculator never converts between units, so the elapsed time and half-life must already agree.

Formula and method

N = N₀ × (1/2)^(t / T½)
t = T½ × ln(N₀ / N) ÷ ln 2
T½ = t × ln 2 ÷ ln(N₀ / N)
N₀ = N × 2^(t / T½)
λ = ln 2 ÷ T½        τ = 1 ÷ λ

In the formulas, N₀ is the starting amount, N is the amount left, t is the time elapsed and T½ is the half-life. The ratio t / T½ counts how many half-lives have passed, and each one multiplies the amount by one half. The count doesn't have to be a whole number: 2.5 half-lives leaves (1/2)^2.5, about 17.7%.

To solve for time or half-life, the exponent has to come down, and that needs a logarithm. Here ln means the natural logarithm, a button on any scientific calculator. Dividing ln(N₀ / N) by ln 2 gives the number of halvings between the two amounts.

The decay constant λ (the Greek letter lambda) describes the same process as a rate. It is the fraction of the remaining substance that decays per unit time, at any instant. The equivalent exponential form is N = N₀e^(−λt), where e is the number about 2.718. The mean lifetime τ (tau) = 1/λ is the average time an individual atom survives, about 1.44 half-lives.

Worked example

A lab holds 100 g of iodine-131, which has a half-life of 8.02 days. How much is left after 24 days?

  1. Number of half-lives: n = 24 ÷ 8.02 = 2.992519.
  2. Remaining: N = 100 × (1/2)^2.992519 = 12.565 g.
  3. Decay constant: λ = 0.693147 ÷ 8.02 = 0.08642733 per day.

Result: About 12.565 g (12.565%) of the iodine-131 remains after 24 days, and the decay constant is 0.0864 per day.

Understanding the result

The percent remaining depends only on the number of half-lives elapsed, not on the substance. Any isotope at three half-lives has 12.5% left; whether that takes seconds or millennia is set by T½.

Decay never quite reaches zero in the formula, but for a real sample it effectively ends. After 10 half-lives, less than 0.1% is left. After 20, less than one part in a million remains.

Assumptions and limitations

  • Decay is first-order: a constant fraction decays per unit time, regardless of amount, temperature or pressure.
  • Nothing is added to the sample during the time interval, and the substance has a single decay path with one half-life.
  • Amounts are large enough for averages to apply; a handful of individual atoms decays randomly.
  • Decay chains, where a daughter product is itself radioactive, need separate equations for each member.
  • The calculator doesn't convert time units. Convert both times to the same unit before entering them.
  • Biological half-lives of medicines vary from person to person; this tool is not for dosing decisions.

Frequently asked questions

Is the half-life the same as the time for everything to decay?

No. After one half-life, half remains. The second half doesn't vanish in a second half-life, because each half-life removes half of what is left at that point. That is why decay is exponential, not linear.

How are half-life and the decay constant related?

They carry the same information. λ = ln 2 ÷ T½, so a half-life of 10 years means λ is about 0.0693 per year. A large decay constant means a short half-life.

Can I use percent instead of grams?

Yes. Enter 100 as the initial amount and the percent you have left as the remaining amount. Any unit works, as long as both amounts use it.

Why does solving for the half-life fail when the amounts are equal?

If nothing has decayed, there is no information about how fast decay happens, and the formula would divide by ln 1 = 0. The amount remaining must be smaller than the initial amount.

How this calculator is checked

Method: First-order (exponential) decay law N = N₀(1/2)^(t/T½) and its algebraic rearrangements; λ = ln 2 / T½; τ = 1/λ. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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