How Half-Life Works: Radioactive Decay, Carbon Dating, and Drug Clearance Explained

Half-life is one of those ideas that sounds simple until you try to do the arithmetic. "It halves every so often" — fine. But what is left after two and a half half-lives? How old is a bone that still has 18% of its carbon-14? How much of a 600 mg dose is in your bloodstream a day later? The shortcut answer of "just keep dividing by two" only works at whole half-lives, and real measurements almost never land there. This post walks through the actual math, why decay is exponential rather than linear, and how the same model shows up in archaeology, nuclear medicine, and pharmacology.

The one formula that runs everything

Every half-life problem comes down to a single equation:

N = N₀ · (1/2)^(t/T½)

Here N₀ is how much you started with, N is how much is left, t is the elapsed time, and T½ is the half-life. The exponent t/T½ is simply "how many half-lives have gone by." Plug in t = T½ and you get N = N₀ · (1/2)¹ = half the starting amount. Plug in t = 2·T½ and you get N₀ · (1/4). The formula is continuous, so it answers fractional cases too: at t = T½/2, the exponent is 0.5 and (1/2)^0.5 ≈ 0.707, meaning 70.7% remains — not the 75% you might guess by splitting the first half-life in two.

The same law can be written with the natural exponential, N = N₀ · e^(−λt), where λ is the decay constant. The two forms are identical; λ just packages the half-life as λ = ln2 / T½, with ln2 ≈ 0.6931. A long half-life gives a small λ (slow decay), and a short half-life gives a large one.

Why decay is exponential, not linear

The common mistake is to treat decay as a straight line: if 50% is gone after one half-life, surely it is all gone after two? It is not. Each half-life removes half of whatever is left, so you march through 50%, 25%, 12.5%, 6.25%, forever approaching zero without ever reaching it.

The reason is physical, not just mathematical. Each atom decays independently with a fixed probability per unit time, and it has no memory of how long it has already existed. When the rate of decay is proportional to how much remains (dN/dt = −λN), the only solution is an exponential curve. That is also why the decay constant matters: λ is the per-atom, per-unit-time probability of decay, and it is what ties the median (the half-life) to the average atom lifetime, the mean lifetime τ = T½ / ln2 ≈ 1.4427 · T½. The mean lifetime is longer than the half-life because a stubborn minority of atoms survive far past the median and drag the average up.

A worked example: three half-lives

Let me do one slowly. Suppose a drug has an elimination half-life of 8 hours and you take 600 mg. How much is left after 24 hours?

Twenty-four hours is exactly three half-lives (24 / 8 = 3), so the exponent t/T½ = 3:

• After 8 hours: 600 → 300 mg
• After 16 hours: 300 → 150 mg
• After 24 hours: 150 → 75 mg

That is N₀ · (1/2)³ = 600 · (1/8) = 75 mg. After three half-lives, exactly 1/8 of the original amount remains. This is worth memorizing as a sanity check: one half-life leaves 1/2, two leave 1/4, three leave 1/8, four leave 1/16. If a calculation hands you something wildly off these landmarks, recheck your units — pairing days with years is the single most common slip. When I'm grading my own back-of-envelope work, the 1/8 checkpoint is the first thing I reach for; it has caught more unit-mismatch errors than anything else I do.

The neat part is that biology and physics share this math. First-order drug clearance follows the exact same exponential law as a decaying nucleus. The body eliminates a fixed fraction of the remaining drug per unit time, just as a sample loses a fixed fraction of its remaining atoms.

Carbon dating: solving for time instead of amount

Radiocarbon dating flips the problem around. Living things absorb Carbon-14 at a roughly steady rate, so a fresh sample starts with a known ratio. Once the organism dies, the C-14 decays with a half-life of 5,730 years and nothing replaces it. Measure how much is left, and you can solve the formula for t.

Say a charcoal fragment retains 18% of its original C-14. Rearranging N = N₀ · (1/2)^(t/T½) for time gives t = T½ · log₂(N₀/N). With N₀/N = 100/18 ≈ 5.56, that works out to roughly 14,170 years — a value that sits between two and three half-lives, which is exactly why you cannot just round to the nearest half-life. The method stays reliable out to about 50,000 years; beyond that, too little C-14 survives to measure. To run this yourself, set the Half-Life Calculator to solve for elapsed time, load the Carbon-14 preset, and enter 100 and 18 as the initial and remaining amounts. Because the result is a fraction of the original, converting it to a clean percentage with a percentage calculator makes lab notes easier to read.

Reading the decay constant and mean lifetime

When you solve a half-life problem, three numbers fall out together, and confusing them is a classic exam error:

• Half-life T½ — the time for half the sample to decay (the median).
• Decay constant λ = ln2 / T½ — the probability per unit time of decay. For Carbon-14, λ ≈ 1.21 × 10⁻⁴ per year.
• Mean lifetime τ = T½ / ln2 = 1/λ — the average lifetime of a single atom.

For Cobalt-60 (half-life 5.27 years), λ comes out to about 0.1315 per year and τ to about 7.6 years. Students routinely swap λ and τ or drop the ln2 factor entirely. Seeing all three side by side makes the relationships concrete: λ and T½ are inversely related, and τ is always a touch larger than T½. If you are working a chain of conversions by hand, a scientific calculator keeps the natural logs honest.

Where this shows up

The half-life model is everywhere once you start looking. Nuclear-medicine techs use it to time isotope doses — a Technetium-99m sample (half-life 6.01 hours) prepared three hours before a scan retains about 71% of its activity, so they draw extra to hit the target window. Physics labs run it backwards to identify unknown isotopes from a decay curve: start at 5,000 counts per minute, measure 1,250 cpm after 40 minutes, and since the activity dropped to a quarter, exactly two half-lives passed, giving a 20-minute half-life. Pharmacologists use it to plan dosing intervals. Geologists stretch it across billions of years with Uranium-238.

The underlying equation never changes. Whether you are dating a bone, timing a scan, or tracking a medication, you are solving N = N₀ · (1/2)^(t/T½) for one of its four quantities. Get comfortable with that one formula and the rest is bookkeeping.

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Made by Toolora · Updated 2026-06-13