How Half-Life and Carbon Dating Actually Work
The exponential decay formula behind radioactive half-life, explained with the carbon dating example.
Half-life describes one of the most reliable patterns in physics: no matter how much of a radioactive substance you start with, exactly half of it decays over one fixed time period, every time.
The formula
N(t) = N₀ × (1/2)^(t / half-life)
where N₀ is the starting quantity, t is elapsed time, and half-life is the substance's characteristic decay period.
Worked example: three half-lives
Start with 100g of a substance with a 10-year half-life. After 1 half-life (10 years): 50g remains. After 2 half-lives (20 years): 25g. After 3 half-lives (30 years): 12.5g. Notice it's not a straight-line decline — each successive half-life removes half of what's currently left, not half of the original amount, which is exactly why the quantity never mathematically reaches absolute zero, only ever approaching it.
Carbon dating: half-life in action
Carbon-14 has a half-life of approximately 5,730 years. If a sample contains 100g of carbon-14-bearing material when an organism dies, and researchers measure only 25g remaining today, that means exactly 2 half-lives have elapsed (since (1/2)² = 1/4): 25/100 = 0.25 = (1/2)^n solves to n=2, so elapsed time = 2 × 5,730 = 11,460 years. This is literally how radiocarbon dating estimates the age of organic archaeological finds — measure the remaining fraction, solve for elapsed half-lives, multiply by the known half-life period.
Why half-life is constant regardless of quantity
Unlike most decay processes people intuit from everyday experience (like a hot cup of coffee cooling faster when it's hotter), radioactive half-life doesn't depend on how much material you have — a single atom and a mountain of the same isotope both have identical odds of decay per unit time, statistically speaking. This constancy is what makes half-life dating mathematically reliable across such enormous timescales.
When half-life dating stops being useful
Carbon-14 dating becomes impractical for samples older than about 8-10 half-lives (roughly 50,000-60,000 years), since the remaining carbon-14 fraction becomes too small to measure reliably against background contamination. For older samples, scientists switch to isotopes with much longer half-lives (like potassium-40 or uranium-238), applying the exact same exponential formula, just with a vastly longer characteristic period.
Common mistakes to avoid
- Assuming decay is linear (losing a fixed amount each period) rather than exponential (losing a fixed fraction each period)
- Forgetting that "half-life" refers to a probabilistic average over large numbers of atoms — for a very small sample, the actual decay timing has real statistical variance
- Confusing elapsed half-lives with elapsed years — always multiply the number of half-lives by the specific isotope's half-life duration to get actual years
Model your own decay scenario with the half-life calculator.