Half-Life Calculator
Calculate the remaining amount of a substance after radioactive or exponential decay.
Try an example
78.5106
About this calculator
What Is Half-Life?
Half-life is the time it takes for a quantity to fall to half of its starting value. The idea is most familiar from radioactive decay, where it describes how long it takes half the atoms in a sample to decay, but it applies to any process that decreases exponentially — the amount of a drug left in your blood, a cooling object's temperature difference, or the charge on a discharging capacitor. This half-life calculator lets you enter any three of the four quantities in the decay equation and solves for the missing one, and it converts between half-life, mean lifetime and the decay constant.
The Half-Life Formulas
N(t) = N₀ × (1/2)^(t ÷ t½), where N₀ is the initial amount, N(t) is the amount remaining after time t, and t½ is the half-life. The same law can be written with the exponential function: N(t) = N₀ × e^(−λt), where λ is the decay constant. The three decay measures are linked by:
- t½ = ln 2 ÷ λ ≈ 0.693 ÷ λ
- Mean lifetime τ = 1 ÷ λ = t½ ÷ ln 2 ≈ 1.443 × t½
After one half-life half remains; after two, a quarter; after three, an eighth, and so on. So 80 g of a substance with a 5-day half-life leaves 40 g after 5 days, 20 g after 10 and 10 g after 15.
Worked Example
100 units of carbon-14 (half-life 5,730 years), after 2,000 years, decays to about 78.51 units remaining — 78.5% of the original amount.
Where Half-Life Is Used
- Carbon-14 dating: carbon-14 has a half-life of about 5,730 years, and the fraction left in an organic sample reveals its age.
- Medicine: doctors use a drug's half-life to set dosing schedules, and nuclear medicine uses isotopes whose half-lives are short enough to clear the body quickly.
- Nuclear safety and waste: half-lives tell us how long materials stay hazardous.
- Chemistry and physics: first-order reactions and many natural processes follow the same pattern.
Why It Works Like This
In decay, each atom (or molecule) has the same chance of decaying in any time interval, so the amount that decays is proportional to the amount present. That proportionality produces an exponential curve — fast loss when there is a lot, slower loss as less remains, never quite reaching zero. Solving for time or amount involves logarithms, which you can explore with the Log Calculator, and the powers involved are covered by the Exponent Calculator.
Worked example
100 units of carbon-14 (half-life 5,730 years), after 2,000 years, decays to about 78.51 units remaining — 78.5% of the original amount.
Frequently asked questions
What is a half-life?
The time it takes for exactly half of a decaying substance to break down or be eliminated. It's a constant for a given substance — it doesn't speed up or slow down as the amount decreases.
Why does the amount never reach exactly zero?
Exponential decay approaches zero but never technically reaches it in the formula — each half-life removes half of whatever remains, so a tiny fraction always remains mathematically, even if it becomes practically undetectable.
Does this apply to things other than radioactive decay?
Yes — the same math describes any exponential decay process, including how the body eliminates certain medications, or how a drug's concentration decreases over time.
