Exponential Decay Calculator

Exponential decay: N(t) = N₀·e^(−kt)

Whenever a quantity shrinks at a rate proportional to how much of it is left, you get exponential decay: dN/dt = −k·N, whose solution is N(t) = N₀·e^(−kt). The decay constant k (in 1/time) controls how fast things fall, and the half-life follows from it: T = ln(2)/k. Take N₀ = 100 and k = 0.1/h. After t = 5 h you land on N ≈ 100·e^(−0.5) ≈ 60.65. Radioactivity is the textbook case, but the same equation shows up in RC capacitor discharge (V(t) = V₀·e^(−t/RC)), caffeine elimination (half-life around 5 h), drug pharmacokinetics (first-order PK) and Newton's law of cooling, where a temperature gap closes in on ambient exponentially.

Applications

It turns up in pharmacokinetics (drug clearance, dosing by half-life), in electronics (RC and RL transients), in mechanics where viscous damping bleeds energy out of an oscillator, in population ecology when mortality stays constant, in atmospheric chemistry for pollutant breakdown, and even in finance through continuous depreciation models.

FAQ

k vs half-life T? Same information, two ways of stating it. T = ln(2)/k ≈ 0.693/k, so a bigger k means quicker decay and a shorter half-life.

What's the time constant τ? τ = 1/k is when the quantity falls to 1/e ≈ 36.8 % of its initial value. After 5τ, less than 1 % remains — a common engineering settling criterion.

When does pure exponential decay break down? When the assumption of first-order kinetics fails — saturable enzymes (Michaelis-Menten), nonlinear damping, or when the population is so small that stochastic fluctuations dominate.