Half-Life Explained: Exponential Decay in One Formula

Carbon-14 keeps half of its atoms for 5,730 years. Caffeine keeps half of its molecules in your bloodstream for about 5 hours. Radon manages 3.8 days; polonium-214 manages a fraction of a millisecond. Different substances, wildly different clocks — and yet one mathematics describes every one of them. Half-life is the great equaliser of decay.

The formula

A decay process removes entities at a rate proportional to how many remain: twice as many atoms, twice as many decays per second. That single assumption forces the curve to be exponential, and the half-life h — the time for any amount to halve — pins it down completely:

N(t) = N₀ × 2^(−t/h)          N(t) = N₀ × e^(−λt),  λ = ln 2 / h

N₀ is the starting amount, t the elapsed time, and h and t must share their units — hours with hours, years with years. The base-2 form makes the halving logic visible; the base-e form is what calculus and textbooks prefer, with λ (lambda), the decay constant, swapping in through the identity λ = ln 2 / h. The two forms are the same curve wearing different coordinates.

The halving ladder

Because each half-life halves whatever is left, the sequence of remaining percentages is worth memorising once and for all:

100 → 50 → 25 → 12.5 → 6.25 → 3.125
    → 1.5625 → 0.78 → 0.39 → 0.20 → 0.10 %

Ten half-lives leaves one part in a thousand — the working convention for "effectively gone", and the reason laboratories treat seven to ten half-lives as the point where a sample's activity disappears into the background. Notice what the ladder implies: the substance never reaches exactly zero. Exponential decay approaches zero asymptotically, which is not a quirk of measurement but a theorem — there is always a next half.

λ and τ: the other two constants

Half-life is the friendly face of a family of three constants. The decay constant λ is the per-unit-time decay probability — for carbon-14, λ = ln 2 / 5730 ≈ 0.000121 per year, meaning each atom has about a 1-in-8,260 chance of decaying in any given year. The mean lifetime τ = 1/λ = h / ln 2 ≈ 1.44 × h is how long an average atom lasts before decaying: carbon-14's is about 8,267 years. All three describe the same physics, and each exists because a different profession got tired of multiplying: radiometric dating speaks in half-lives, reactor physics in λ, and particle physics in τ. The exponential that ties them together is unpacked gently in how exponents and logarithms work.

Running the curve backwards

Turn the formula around and elapsed time falls out of any measured remainder:

t = h × log₂(N₀ / N)

This is radiocarbon dating in one line. A charcoal sample whose carbon-14 sits at 25% of the atmospheric level is two half-lives old: t = 5730 × log₂(4) ≈ 11,460 years. At 12.5%, three half-lives, about 17,190 years. The same inversion drives medicine — when will this drug fall below its effective threshold? — and salvage accounting — how much of a reserve is left after n halvings? The only target with no answer is zero, which is why dating statements are always ranges and why "gone" is always an engineering judgement, not a measurement.

Where the model applies

Radioactive decay is the textbook case: per-atom, memoryless, and precise to many decimal places. Pharmacokinetics borrows the model with honest caveats — the body eliminates many drugs in proportion to their concentration, so plasma levels fall on roughly exponential curves, but hydration, metabolism and dosing intervals bend the bookkeeping. The everyday example is caffeine: its 3–7 hour population-average half-life is why an afternoon espresso lingers into the evening, a story told in full — with a dose- and time-aware tool — in caffeine half-life explained. The general rule of thumb: wherever removal is proportional to remainder, half-life thinking works; wherever it is not (flushing, filtering, first-order-plus-threshold models), the ladder only approximates.

Try it yourself

Open the Half-Life Calculator and enter 100 with a half-life of 5730 and an elapsed time of 11,460 years: 25 remains, and the decay constant reads 0.000121 per year. Add a target of 12.5 to flip the question around — the calculator returns 17,190 years, three half-lives exactly. Then try a drug-style case, 200 mg with a 5-hour half-life after 10 hours: 50 mg left, four half-lives in between. For writing the very small λ values and very large particle counts that decay problems produce, the Scientific Notation Converter keeps the zeros under control.