Half-Life Calculator

Using the half-life calculator

The Solve for buttons decide the unknown. The other fields change to match:

• Half-life: enter the initial quantity, the remaining quantity and the time that passed between them. The answer is given in the unit you chose for the time.
• Time elapsed: enter the initial and remaining quantities and the half-life. The answer comes back in the unit of the half-life.
• Initial quantity or Remaining quantity: enter the other quantity, the elapsed time and the half-life. The time and the half-life may use different units; the calculator converts them before dividing.

When the half-life is an input you only need to fill in one of Half-Life, Decay Constant or Mean Lifetime; they all describe the same rate and are tied by fixed relationships. Typing a value in one of them fills in the other two automatically, and choosing a time unit for one sets the same unit on all three. If more than one is filled, the half-life is used first, then the decay constant. The decay constant is a rate per unit of the time menu next to it: 0.0864 with Days selected means 0.0864 per day.

Quantities can be in any unit (grams, atoms, becquerels, percent), as long as the initial and remaining amounts use the same one. Time units run from nanoseconds to years. The calculator counts a month as 30.5 days and a year as 365.2421 days. Results are rounded to twelve decimal places.

Radioactive decay and half-life

An unstable nucleus can change into a different nucleus at any moment, and no one can say when a particular atom will go. For a large collection of identical atoms, however, the fraction that decays in a given time is very predictable. That makes the half-life a useful number: it is the time after which half of the original atoms are left. After a second half-life half of those have gone, leaving a quarter, and so on.

The same pattern appears anywhere a quantity falls by a fixed percentage in each equal time step. This is exponential decay, and it describes radioactive isotopes, the elimination of many substances from the body, a discharging capacitor and the cooling of a hot object toward room temperature.

Half-life formulas

N(t) = N₀ × (½)^(t ÷ T½)N₀ is the initial quantity, N(t) the amount left after time t, T½ the half-life

N(t) = N₀ × e^(−λ × t)the same law written with the decay constant λ

λ = ln 2 ÷ T½ τ = 1 ÷ λ = T½ ÷ ln 2decay constant λ and mean lifetime τ; ln 2 ≈ 0.693147

Solving the first equation for each unknown gives the formulas behind the four modes:

t = T½ × log₂(N₀ ÷ N)

T½ = t ÷ log₂(N₀ ÷ N) = t × ln 2 ÷ ln(N₀ ÷ N)

N₀ = N × 2^(t ÷ T½) N = N₀ × 2^(−t ÷ T½)

The mean lifetime τ is the average time an atom survives before decaying. It is about 1.44 times longer than the half-life, because a few atoms live much longer than the typical one. After exactly one mean lifetime, 1/e, or about 36.8%, of the sample remains.

Worked examples

Radiocarbon dating. Carbon-14 has a half-life of 5,730 years. A wood sample retains 60% of the carbon-14 found in living wood. Then t = 5,730 × log₂(100 ÷ 60) = 5,730 × 0.73697 ≈ 4,222.8 years. Use Time elapsed with 100 as the initial and 60 as the remaining quantity.

Finding a half-life. A sample of 100 g falls to 25 g in 10 seconds. That is two halvings (100 → 50 → 25), so the half-life is 10 ÷ 2 = 5 seconds. The calculator gets the same result through T½ = 10 ÷ log₂(4).

Remaining amount. Iodine-131 has a half-life of about 8.02 days. From 20 mg, after 30 days the amount left is 20 × 2^(−30 ÷ 8.02) = 20 × 2^(−3.741) ≈ 1.50 mg.

Decay constant and mean lifetime. For carbon-14, λ = 0.693147 ÷ 5,730 ≈ 1.2097 × 10^-4 per year, and τ = 5,730 ÷ 0.693147 ≈ 8,266.6 years.

Where half-life is used

• Dating. Carbon-14 works for organic material up to about 50,000 years old, roughly 8.7 half-lives, after which too little is left to measure. Longer spans use isotopes such as potassium-40 or uranium-238.
• Medicine and industry. Short-lived isotopes are chosen for imaging so they fade quickly; long-lived ones are used in sealed sources and in waste planning.
• Pharmacology. The time for the level of a substance in the body to halve is also called a half-life, and the same maths applies when elimination follows first-order kinetics.
• Electronics. The voltage across a discharging capacitor halves every R × C × ln 2. See the capacitor energy and charge calculator for the energy side.

To check a result by hand, the exponent calculator handles the power of ½ and the logarithm calculator handles the log₂ step.

Assumptions and limits

• Decay is random for individual atoms. The formulas describe the expected value for a large sample; a handful of atoms will scatter around it.
• The law applies to a single decaying species. If a decay produces another radioactive isotope, the sample as a whole no longer follows one exponential.
• When the remaining amount is larger than the initial amount, the solved half-life comes out negative, which means the quantity is growing, not decaying.
• The initial and remaining amounts cannot be zero or equal when solving for the half-life, because the logarithm of the ratio would be undefined or zero.

This calculator is a maths tool. Do not use it as the only basis for decisions about radiation safety, medication timing or the disposal of radioactive material. See the disclaimer.