How do you calculate how much remains after a certain time?

Divide the elapsed time by the half-life to get the number of half-lives, then multiply the starting amount by one-half raised to that power. 80 g of iodine-131 (half-life 8.02 days) after 24.06 days: 3 half-lives, 80 × ⅛ = 10 g. Enter the three values and the page shows the power.

What is the decay constant, and how is it related to half-life?

λ = ln 2 ÷ t½ ≈ 0.693 ÷ t½: the fraction decaying per unit time, used in the exponential form N = N₀e^(−λt). Carbon-14's is 1.21 × 10⁻⁴ per year. The two describe the same curve; half-life is easier to picture, λ is easier to calculate with.

How long until only 10% (or 1%) is left?

t = t½ × log₂(N₀ ÷ N): 10% left is log₂(10) = 3.32 half-lives; 1% is 6.64. For carbon-14 that is 19,000 and 38,000 years. Enter elapsed times until the fraction line reads what you want, or use the formula directly.

Does the mass ever reach zero?

Mathematically no — each half-life halves what is left and never removes it all. Physically the last atom decays at some point, and long before that the sample is indistinguishable from background; ten half-lives (0.1% left) is a common practical 'gone'.

Does temperature change a half-life?

No. Radioactive decay is unaffected by temperature, pressure or chemical form, which is what makes it a dependable clock for dating.

Half-Life Calculator (Radioactive and Exponential Decay)

How much of a substance remains after a given time from its half-life, with the decay constant, mean lifetime and the working.

Inputs

Initial amount

0.0011,000,000,000,000,000

Half-life(years)

0.001 years1,000,000,000,000,000 years

Elapsed time(years)

0 years1,000,000,000,000,000 years

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Remaining

Initial amount

Remaining25(25%)
Decayed75(75%)

Working: 100 × (½)^(11460 ÷ 5730) = 100 × (½)^2 = 25
Fraction remaining: 25%
Decayed amount: 75
Half-lives elapsed: 2
Decay constant λ = ln 2 ÷ t½: 1.2097e-4 per unit time
Mean lifetime 1 ÷ λ: 8,266.6426

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What each Half-Life input means

Initial amount: The starting amount, in any unit — grams, atoms, becquerels or a percentage — since the answer comes back in the same unit.
Half-life: The isotope's half-life, in any time unit, as long as the elapsed time uses the same one.
Elapsed time: How much time has passed, in the same unit as the half-life.

Half-Life formula

N = N₀ × (½)^(t ÷ t½) · equivalently N = N₀ e^(−λt) with λ = ln 2 ÷ t½ · fraction left after n half-lives = 1 ÷ 2ⁿ · mean lifetime τ = 1 ÷ λ = 1.443 × t½

N = N_0 \left(\tfrac{1}{2}\right)^{t / t_{1/2}} = N_0 e^{-\lambda t}, \qquad \lambda = \frac{\ln 2}{t_{1/2}}

In radioactive decay — and in any first-order process, from drug clearance to a capacitor discharging — a fixed fraction of what is left disappears in each equal interval. The half-life is the interval in which half goes, so after one half-life 50% remains, after two 25%, after three 12.5%, and the amount left is the starting amount times one-half raised to the number of half-lives elapsed, which need not be a whole number. The same law written with the exponential gives the decay constant λ = ln 2 ÷ t½, the probability per unit time that any one nucleus decays, and the mean lifetime 1 ÷ λ, which is 44% longer than the half-life. The page shows the power of one-half, the fraction remaining, the amount gone, λ and the mean lifetime, in whatever time unit the half-life was entered in.

Half-Life worked example

Carbon-14 has a half-life of 5,730 years. After 11,460 years, that is 11,460 ÷ 5,730 = 2 half-lives, so 100 × (½)² = 25 remains — a quarter of the original — and 75 has decayed. The decay constant is 0.693 ÷ 5,730 = 1.21 × 10⁻⁴ per year and the mean lifetime 8,267 years. OpenStax's cobalt-60 example runs the same way: half-life 5.27 years, so 15.81 years is three half-lives and 12.5% is left.

More Half-Life examples

OpenStax's cobalt-60: after 15.81 years, exactly three half-lives of 5.27 years, a 1 g source has 0.125 g left — one-eighth.

A 10 mCi dose of iodine-131 (half-life 8.02 days) after 24 days: just under three half-lives, so 1.2565 mCi remains, 12.6% of the start.

Where the Half-Life calculator can give the wrong answer

Time and half-life must be in the same unit. A half-life in years and an elapsed time in days gives nonsense unless one is converted; the page does no conversion.
The law is statistical. It predicts the fraction of a large number of nuclei exactly and says nothing about when a particular nucleus decays; with a handful of atoms the actual count fluctuates around the prediction.
Half-lives are constants of the nuclide, unaffected by temperature, pressure or chemistry. Half-lives of drugs and other first-order processes are not — they change with dose, organ function and conditions.
Radiocarbon dating uses this arithmetic in reverse (find t from the fraction left), but real dates also need calibration for past changes in atmospheric ¹⁴C; the page gives the raw exponential only.
A decay chain — where the product is itself radioactive — does not follow this single-step formula for the intermediate products.

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Half-Life reference table

Half-lives of some well-known isotopes
Isotope	Half-life	Use or source
Technetium-99m	6.01 hours	Medical imaging
Iodine-131	8.02 days	Thyroid treatment
Cobalt-60	5.27 years	Radiotherapy, sterilising
Radium-226	1,600 years	Natural uranium decay chain
Carbon-14	5,730 years	Radiocarbon dating
Plutonium-239	24,110 years	Reactor fuel
Uranium-238	4.5 billion years	Dating rocks

Source: OpenStax Chemistry 2e, §21.3 'Radioactive Decay' and §21.6 for the half-lives

Half-Life FAQ

: Divide the elapsed time by the half-life to get the number of half-lives, then multiply the starting amount by one-half raised to that power. 80 g of iodine-131 (half-life 8.02 days) after 24.06 days: 3 half-lives, 80 × ⅛ = 10 g. Enter the three values and the page shows the power.
: λ = ln 2 ÷ t½ ≈ 0.693 ÷ t½: the fraction decaying per unit time, used in the exponential form N = N₀e^(−λt). Carbon-14's is 1.21 × 10⁻⁴ per year. The two describe the same curve; half-life is easier to picture, λ is easier to calculate with.
: t = t½ × log₂(N₀ ÷ N): 10% left is log₂(10) = 3.32 half-lives; 1% is 6.64. For carbon-14 that is 19,000 and 38,000 years. Enter elapsed times until the fraction line reads what you want, or use the formula directly.
: Mathematically no — each half-life halves what is left and never removes it all. Physically the last atom decays at some point, and long before that the sample is indistinguishable from background; ten half-lives (0.1% left) is a common practical 'gone'.
: No. Radioactive decay is unaffected by temperature, pressure or chemical form, which is what makes it a dependable clock for dating.