Half-Life Calculator

Half-Life is the time required for a quantity to reduce to half its initial value. This concept applies to radioactive decay, chemical reactions, drug metabolism, and many other processes.

The exponential decay formula:

N(t) = N₀ × (1/2)^(t / t½)

Where:

• N(t) = remaining quantity after time t
• N₀ = initial quantity
• t = elapsed time
• t½ = half-life period

This calculator solves for three different unknowns:

1. Remaining quantity: Given initial amount, half-life, and elapsed time, calculate what remains
2. Elapsed time: Given initial and remaining amounts plus half-life, find how much time has passed
3. Half-life period: Given initial and remaining amounts plus elapsed time, determine the half-life

Rearranged formulas:

• Time elapsed: t = t½ × log(N₀/N) / log(2)
• Half-life: t½ = t × log(2) / log(N₀/N)

Real-world examples:

• Carbon-14 has a half-life of 5,730 years (used in archaeological dating)
• Iodine-131 has a half-life of 8.02 days (used in medical treatments)
• Caffeine in the human body has a half-life of about 5 hours
• Uranium-238 has a half-life of 4.47 billion years

After N half-lives, the fraction remaining is:

• 1 half-life: 50% remains
• 2 half-lives: 25% remains
• 3 half-lives: 12.5% remains
• 5 half-lives: 3.125% remains
• 10 half-lives: 0.098% remains (essentially gone)

Five half-lives is the rule of thumb for “gone” in medicine and pharmacology, because 3% of a dose is usually below the threshold where it does anything. In radiation safety the working figure is ten, at which point you are down to a thousandth.

Two numbers that go with every half-life

The decay constant λ = ln(2) / t½ is the probability per unit time that any one atom decays. It is what appears in the exponential form N(t) = N₀e^(−λt), and it is the number physicists actually work with, because it adds cleanly when several decay paths compete.

The mean lifetime τ = 1/λ = t½ / ln(2) is the average time an individual atom survives, and it is always longer than the half-life by a factor of 1.4427. That surprises people, but it makes sense: half the atoms are gone by t½, and the survivors keep going for an unbounded time, so the mean is dragged upward. Carbon-14’s half-life is 5,730 years and its mean lifetime is 8,267.

Why the half-life never changes

Radioactive decay is memoryless. An atom that has sat in a rock for a billion years is exactly as likely to decay in the next second as one created a moment ago. There is no ageing and no accumulated damage. That is the property that makes radiometric dating work at all, and it is also why heating, freezing, crushing or chemically bonding a radioactive sample does essentially nothing to its half-life.

The one real exception is electron capture, where the decay depends slightly on the electron density around the nucleus, so extreme chemical or pressure environments can shift the rate by a fraction of a percent. It is a genuine effect and it is far too small to matter outside a laboratory.