Calculate remaining amount using half-life periods and exponential decay equations.
Overview
Half-life is the time required for half of a quantity to decay in exponential processes: radioisotopes, drug metabolism, and certain chemical reactions. Each equal time interval multiplies the remaining amount by one-half.
The model N(t) = N₀ × (1/2)^(t/t½) counts how many half-lives fit in elapsed time t. Three half-lives reduce material to one-eighth of the original, not zero, because decay is continuous in the underlying exponential form N(t) = N₀e^(−λt).
Logarithms solve for time when you know remaining fraction: t = t½ × log₂(N/N₀). Radiocarbon dating and pharmacokinetics rely on these relationships with domain-specific half-life constants.
Exponential decay assumes a constant half-life, which may not hold for complex biological clearance phases. Still, single half-life models are the standard teaching baseline and quick estimation tool.
Steps
Provide starting amount N₀.
Time for 50% decay with consistent units.
How long decay has proceeded.
Apply exponential decay formula.
Rearrange to find time to a target fraction.
Keep half-life and elapsed time in the same unit.
Math
N(t) = N₀ × (1/2)^(t/t_half) = N₀ × e^(−λt)
Each half-life multiplies remaining fraction by 1/2. Decay constant λ relates to half-life via λ = ln(2)/t_half.
Worked cases
200 × (1/2)^(15/5) = 200 × (1/8)
25 g remaining.
Multiply by 1/2 once
500 remaining.
Three half-lives = 9 hours
9 hours to decay to 12.5%.
Why use this
Questions