Half-Life and Exponential Decay: Medications, Isotopes, and Radiocarbon

Decay Table

After n half-lives, the fraction remaining is (1/2)^n. After about 4 to 5 half-lives, more than 94% has gone, which is why clinicians say a drug is essentially eliminated after about 5 half-lives. The amount never reaches exactly zero, but it becomes negligible.

For non-integer numbers of half-lives, use the formula directly: after 9 hours with T = 6, the exponent is 1.5 and the fraction remaining is 0.5^1.5 = 0.354, or 70.7 mg of 200.

Time to Reach a Target Level

To find when the amount falls to a fraction f of the original, solve t = T × log(f) ÷ log(0.5). To fall to 5% with a 6-hour half-life: t = 6 × ln(0.05) ÷ ln(0.5) = 6 × 4.32 = 25.9 hours. For carbon-14 (half-life 5,730 years), a sample with 25% of the original carbon-14 is two half-lives old, or 11,460 years. For iodine-131 (half-life 8 days), after 40 days (5 half-lives), 3.1% remains.

Repeated Dosing and Accumulation

When doses are repeated before the previous one has gone, the drug accumulates until the amount eliminated between doses equals the dose. If you take 100 mg every 6 hours with a half-life of 6 hours, the peak just after a dose approaches 200 mg at steady state. The accumulation factor is 1 ÷ (1 − 0.5^(interval/T)), which equals 2 when the interval equals the half-life, so the peak reaches 2 × 100 = 200 mg and the trough just before the next dose is 100 mg. Steady state is reached after about 4 to 5 half-lives. Dosing intervals are chosen with this in mind.

Related Rates

The decay constant is λ = ln 2 ÷ T = 0.693 ÷ T, so for T = 6 hours, λ = 0.1155 per hour, and N = N₀ × e^(−λt). The same math, with a growth rate in place of the decay rate, describes doubling times. In medicine, the half-life depends on the drug, the person's kidney and liver function, and age, so these calculations are illustrations; follow your prescriber's instructions.