Half-Life Calculator

What half-life means

A half-life is the time it takes for half of something to disappear through decay. Radioactive nuclei, first-order chemical reactions and many drugs in the body all lose a fixed fraction of what is left in each interval of time. That gives the exponential decay law:

N = N₀ × (1/2)^(t ÷ T½)

After one half-life, 50% remains. After two, 25%. After three, 12.5%. The half-life stays the same no matter how much you start with, which is why it is a useful fingerprint for an isotope or a reaction.

How to use the half-life calculator

Pick the quantity you don’t know:

1. Amount remaining: enter the initial amount, the time elapsed and the half-life.
2. Initial amount: enter what is left now, the time elapsed and the half-life to work backward.
3. Time elapsed: enter the starting and remaining amounts and the half-life. This is the radiocarbon dating case.
4. Half-life: enter two measurements and the time between them.

Amounts can be in any unit, from grams and atoms to becquerels, millicuries or percent, as long as both use the same one. Time and half-life each have their own unit selector, so a half-life in days and an elapsed time in years is fine. Every answer comes with the steps, the number of half-lives, the decay constant λ, the mean lifetime and a table of the amount left after each half-life.

How to calculate half-life problems by hand

Amount remaining

Cobalt-60 has a half-life of 5.2711 years. How much of a 100 mCi source is left after 20 years?

• Number of half-lives: 20 ÷ 5.2711 = 3.794
• Fraction left: (1/2)^3.794 = 0.0721
• Amount left: 100 × 0.0721 = 7.21 mCi

Time elapsed

How long does it take cesium-137 (half-life 30.018 years) to fall to 10% of its starting activity?

t = T½ × log₂(N₀ ÷ N) = 30.018 × log₂(10) = 30.018 × 3.3219 = 99.7 years

On a basic calculator, log₂(x) = ln(x) ÷ ln(2).

Half-life from data

A sample’s count rate falls from 800 to 50 counts per minute over 12 days. N₀ ÷ N = 16, and log₂(16) = 4 half-lives, so T½ = 12 ÷ 4 = 3 days.

Decay constant and mean lifetime

The decay constant λ = ln 2 ÷ T½ is the fraction decaying per unit time when the fraction is small. For cobalt-60, λ = 0.693 ÷ 5.2711 = 0.1315 per year. The mean lifetime, τ = 1 ÷ λ = 7.60 years, is the average time a nucleus survives, which is always about 1.44 times the half-life.

Fraction left after each half-life

Half-lives of common radioactive isotopes

Recommended values from the Decay Data Evaluation Project (DDEP), published by LNHB.

Many textbooks, including OpenStax Chemistry 2e, still give carbon-14 as 5,730 years. The newer evaluated value is 5,700 ± 30 years. Both are within the uncertainty for most homework problems, but say which one you used.

Where half-life calculations are used

• Nuclear medicine: working out how much activity a tracer has at the time of a scan, or when a patient’s dose has decayed.
• Radiocarbon dating: estimating the age of wood, bone or cloth from the carbon-14 left.
• Chemical kinetics: predicting how long a first-order reaction takes to reach a target conversion.
• Pharmacology: estimating how much of a single dose remains. A common rule of thumb is that a drug is largely eliminated after about 5 half-lives, when about 3% remains.
• Radiation safety: deciding how long to store waste before its activity is low enough to handle.

The halving pattern is exponential decay, the mirror image of exponential growth. The population growth calculator handles the growing case, and the exponent calculator evaluates powers such as (1/2)^3.794.

Common mistakes

• Dividing the time by the half-life and stopping there. t ÷ T½ is the number of half-lives, not the fraction left. You still need (1/2) to that power.
• Assuming straight-line decay. Half of the sample is gone after one half-life, but the rest does not vanish after a second one; a quarter remains.
• Mixing time units. A half-life in hours and an elapsed time in days must be converted to the same unit. The calculator does this for you.
• Using half-life for reactions that aren’t first order. For zero- and second-order reactions, the half-life changes as the concentration falls.
• Applying it to repeated doses. Medicines taken regularly build up toward a steady state, which a single-decay formula doesn’t describe.

Glossary

• Half-life (T½): time for the amount to fall to half its value.
• Decay constant (λ): ln 2 ÷ T½, the probability per unit time that a nucleus decays.
• Mean lifetime (τ): 1 ÷ λ, the average survival time.
• Activity: decays per second, measured in becquerels (Bq) or curies (Ci). It is proportional to the number of nuclei, so it follows the same half-life.
• First-order reaction: a reaction whose rate is proportional to the concentration of one reactant.