Half-life calculator

Pick the unknown and enter the other values. The calculator uses the decay law N = N₀ · (1/2)^(t/T½), shows every step and a table of what is left after 1 to 10 half-lives.

What do you want to find?

Half-life

Your own value or an example nuclide (data: IAEA/NNDC)

Half-life T½

calculated in “half-life” mode

Half-life unit

Initial amount N₀

mass, activity, number of atoms, concentration … – N and N₀ in the same unit

Remaining amount N

calculated in “remaining amount” mode

Elapsed time t

calculated in “time” mode

Time unit

Your inputs are saved in this browser only.

Result

Result: N = 25
Percent remaining: 25 %
Number of half-lives (t / T½): 2
Decay constant λ: 1.21605 · 10⁻⁴ per year (3.8535 · 10⁻¹² s⁻¹)
Mean lifetime τ: 8,223.36 years
Step-by-step solution: Formula: N = N₀ · (1/2)^(t / T½)
Number of half-lives: t / T½ = 11,400 / 5,700 = 2
Substitute: N = 100 · (1/2)^2 = 100 · 0.25 = 25
Decay constant: λ = ln 2 / T½ = 0.693147 / 5,700 = 1.21605 · 10⁻⁴ a⁻¹
Mean lifetime: τ = 1 / λ = T½ / ln 2 = 8,223.36 years
Fraction remaining: N / N₀ = 25 / 100 = 25 %
Amount left after k half-lives: Half-lives Time Amount left Percent
0 0 years 100 100 %
1 5,700 years 50 50 %
2 11,400 years 25 25 %
3 17,100 years 12.5 12.5 %
4 22,800 years 6.25 6.25 %
5 28,500 years 3.125 3.125 %
6 34,200 years 1.5625 1.5625 %
7 39,900 years 0.78125 0.78125 %
8 45,600 years 0.390625 0.390625 %
9 51,300 years 0.195313 0.195313 %
10 57,000 years 0.0976563 0.0976563 %; 1 year = 365.2422 days (tropical year, NIST SP 811) – the same convention nuclear data tables use. For medicines the model only applies to first-order kinetics and is an approximation. Not medical advice – always ask your doctor or pharmacist about doses.

Half-lives	Time	Amount left	Percent
0	0 years	100	100 %
1	5,700 years	50	50 %
2	11,400 years	25	25 %
3	17,100 years	12.5	12.5 %
4	22,800 years	6.25	6.25 %
5	28,500 years	3.125	3.125 %
6	34,200 years	1.5625	1.5625 %
7	39,900 years	0.78125	0.78125 %
8	45,600 years	0.390625	0.390625 %
9	51,300 years	0.195313	0.195313 %
10	57,000 years	0.0976563	0.0976563 %

How it is calculated

The half-life formula

The half-life T½ is the time it takes for half of a quantity to decay. After two half-lives a quarter is left, after three an eighth. That gives the exponential decay law:

N(t) = N₀ · (1/2)^(t / T½) = N₀ · e^(−λt)

N₀ is the initial amount and N the amount remaining after time t. Grams, becquerels, number of atoms or mg/L all work – N and N₀ just need the same unit.

Solving for each variable

Unknown	Formula
Remaining amount	N = N₀ · (1/2)^(t / T½)
Initial amount	N₀ = N · 2^(t / T½)
Time	t = T½ · ln(N₀ / N) / ln 2
Half-life	T½ = t · ln 2 / ln(N₀ / N)
Decay constant	λ = ln 2 / T½ ≈ 0.693147 / T½
Mean lifetime	τ = 1 / λ = T½ / ln 2 ≈ 1.4427 · T½

Worked example: radiocarbon dating

Carbon-14 has a half-life of 5,700 years. A wood sample still contains 25% of its original C-14. How old is it?

Ratio: N₀ / N = 100 / 25 = 4
t = 5,700 · ln 4 / ln 2 = 5,700 · 2 = 11,400 years
Decay constant: λ = 0.693147 / 5,700 ≈ 1.21605 × 10⁻⁴ per year
Mean lifetime: τ = 5,700 / 0.693147 ≈ 8,223 years

The other way round: how much of 80 mg of iodine-131 (T½ = 8.0252 days) is left after 30 days? t / T½ = 30 / 8.0252 ≈ 3.738, so N = 80 · 0.5^3.738 ≈ 5.99 mg.

Mean lifetime and decay constant

The decay constant λ is the fraction that decays per unit of time (for short intervals). The mean lifetime τ is the average time a single nucleus survives – about 44% longer than the half-life. After one mean lifetime, 1/e ≈ 36.8% is left.

Drug half-life

Many drugs are cleared from the blood by roughly first-order kinetics, so the same formula applies. A common rule of thumb is that a drug is largely eliminated after 4 to 5 half-lives (3.125% remains after 5). Some substances, such as alcohol, are broken down at a constant rate instead, and then this model does not fit. This calculator is not a substitute for advice from a doctor or pharmacist.

Useful for decay problems: powers such as (1/2)^n with the exponent calculator, very small or large values with the scientific notation converter, and element data in the interactive periodic table.

Frequently asked questions

How do you calculate half-life?

From the initial amount N₀, remaining amount N and elapsed time t: T½ = t · ln 2 / ln(N₀/N). Example: 100 g drop to 12.5 g in 6 hours → N₀/N = 8, ln 8 / ln 2 = 3, so T½ = 6 h / 3 = 2 h.

How much is left after 3 half-lives?

(1/2)³ = 1/8 = 12.5%. In general, the fraction left after k half-lives is 0.5^k: 3.125% after 5 and just under 0.1% after 10.

What is the difference between half-life and mean lifetime?

Half-life is the time until half is gone; mean lifetime τ is the average lifetime of a nucleus. τ = T½ / ln 2 ≈ 1.443 · T½, and after τ about 36.8% remains.

How are the decay constant and half-life related?

λ = ln 2 / T½. For carbon-14 (5,700 years), λ ≈ 1.216 × 10⁻⁴ per year. The shorter the half-life, the larger the decay constant and the higher the activity.

How does carbon-14 dating work?

Living things take in C-14; after death it decays with a 5,700-year half-life. The fraction left gives the age: t = 5,700 · ln(N₀/N) / ln 2. 50% left means 5,700 years, 25% means 11,400 years. Real dates are additionally calibrated.

Can I use this for medication?

For a rough estimate, yes, if the drug follows first-order kinetics: with the half-life from the package leaflet you can see how much is left in theory after a given time. This is not medical advice – ask your doctor or pharmacist about dosing.

Sources and legal basis

As of: 27 September 2026