MotionLab

Half-Life Calculator

How much of something remains after repeated halvings — radioactive decay, drug elimination, and more.

Initial amount

Half-life (time units)

Elapsed time (same units)

Remaining amount

27.041

N = N₀ × (0.5)^(t / half-life)

What you entered

Initial amount: 100Half-life (time units): 5.3Elapsed time (same units): 10

1
Number of half-lives elapsed
10 ÷ 5.3= 1.8868
2
N = N₀ × (0.5)^(t/half-life)
100 × 0.5^1.8868= 27.0408

Result

Remaining amount: 27.041

After 10 time units, 27.041 of the original 100 remains.

Radioactive decay: how much remains

Each half-life halves the remaining quantity. Rows one to three use carbon-14 at 5,730 years.

Initial amount	Half-life	Time elapsed	Remaining	Half-lives
100	5730	5730	50.0000	1
100	5730	11460	25.0000	2
100	5730	17190	12.5000	3
100	12.3	12.3	50.0000	1
1000	8	24	125.0000	3

Decay is exponential, not linear: after three half-lives an eighth remains, not nothing, and in principle the quantity never reaches exactly zero. That is why radiocarbon dating works out to about ten half-lives - roughly 50,000 years - beyond which too little carbon-14 survives to measure. Half-life is a statistical property of huge numbers of atoms, so it says nothing about when any individual nucleus will decay, and it is unaffected by temperature, pressure or chemical state.

Half-life never reaches exactly zero

Exponential decay approaches zero but mathematically never quite reaches it — after 10 half-lives, less than 0.1% of the original amount remains, which is why '10 half-lives' is often used as a practical 'essentially gone' benchmark.

Beyond radioactivity

The same math describes drug elimination from the bloodstream, capacitor discharge, and even some population decline models — any process where the rate of decrease is proportional to the current amount follows this pattern.

Frequently asked questions

Carbon-14 has a half-life of 5,730 years. An artifact has 25% of its original C-14. How old is it?

25% = (½)² — that's 2 half-lives. Age = 2 × 5,730 = 11,460 years. This is the basis of radiocarbon dating, usable for organic materials up to about 50,000 years old (after ~9 half-lives, too little C-14 remains to measure).

A patient takes a drug with a 6-hour half-life. How much remains after 24 hours?

24 hours = 4 half-lives. Remaining fraction = (½)⁴ = 1/16 = 6.25%. From a 200 mg dose, only 12.5 mg remains after 24 hours. Doctors use the rule '5 half-lives to clear' (3.1% remaining) for drug elimination.

Iodine-131 (half-life 8 days) is used to treat thyroid cancer. How much remains after a month?

30 days ÷ 8 days/half-life = 3.75 half-lives. Remaining = (½)^3.75 = 7.4%. From an initial 100 mCi dose, about 7.4 mCi remains after a month — low enough that radiation precautions can be relaxed.

What's the difference between half-life and mean lifetime?

Mean lifetime (τ) = half-life / ln(2) ≈ half-life × 1.443. Half-life is when 50% has decayed; mean lifetime is the average time any individual atom survives. For C-14: half-life = 5,730 years, mean lifetime ≈ 8,267 years.

Can half-life be used for non-radioactive processes?

Yes — any exponential decay has a half-life. Drug elimination from the body, capacitor discharge, beer foam collapse, and even the cooling of hot coffee follow the same math. The half-life formula N = N₀ × (½)^(t/t½) works for all of them.