Exponential Decay Calculator
Continuous exponential decay follows N = N₀ · e^(−k·t), where N₀ is the starting amount, k is the decay rate (per unit time) and t is the elapsed time.
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How to use this calculator
- Fill in any three of the four fields — initial amount N₀, remaining amount N, decay rate k, and elapsed time t — and leave exactly one blank.
- Press Calculate: the tool solves for the blank field and also reports the half-life, mean lifetime and percent remaining.
- Keep the decay rate k and the time t in the same time units (a k in “per year” needs t in years).
How it works
Continuous exponential decay follows N = N₀ · e^(−k·t), where N₀ is the starting amount, k is the decay rate (per unit time) and t is the elapsed time. Every unit of time multiplies the amount by e^(−k), so the quantity falls by the same fraction each period rather than by a fixed amount.
Rearranging the same equation solves for any variable: N₀ = N · e^(k·t), k = ln(N₀ ÷ N) ÷ t, and t = ln(N₀ ÷ N) ÷ k. The decay rate ties directly to the familiar half-life and mean lifetime: T½ = ln 2 ÷ k ≈ 0.6931 ÷ k, and τ = 1 ÷ k (the average time before an item decays).
Worked example
100 units, decay rate k = 0.2 per hour, after 10 hours. With N = N₀ · e^(−k·t): N = 100 × e^(−0.2 × 10) = 100 × e^(−2) = 13.533528 units remaining, i.e. 13.5335% left. The decay constant gives a half-life T½ = ln 2 ÷ 0.2 = 3.465736 hours and a mean lifetime τ = 1 ÷ 0.2 = 5 hours.
Common mistakes
- Mixing time units — if k is “per hour”, then t must be in hours. A k in per-day with t in hours gives a wildly wrong answer.
- Confusing the decay rate k with the half-life. They are reciprocal-like: a larger k means faster decay and a shorter half-life (T½ = ln 2 ÷ k), not a longer one.
- Treating decay as linear. It is exponential — after two “rate periods” you do not lose twice as much; each period removes the same fraction of what is left.
Frequently asked questions
What is the exponential decay formula?
N = N₀ · e^(−k·t), where N₀ is the initial amount, N is the amount left after time t, and k is the decay rate (per unit time). This calculator can also invert it to solve for N₀, k or t when you know the other three values.
How is the decay rate k related to the half-life?
The half-life is T½ = ln 2 ÷ k ≈ 0.6931 ÷ k. So a decay rate of k = 0.2 per hour gives a half-life of about 3.47 hours. The mean lifetime is τ = 1 ÷ k, always longer than the half-life.
What is the difference between this and a half-life calculator?
They describe the same physics with different parameters. This tool works from the decay rate (or constant) k in N = N₀ · e^(−k·t), while a half-life calculator is parameterised by the half-life T½ directly. Use whichever number you already have — the tool reports both.
Does exponential decay only apply to radioactivity?
No. The same maths models drug concentration clearing the body, a capacitor discharging, Newton's cooling, atmospheric pressure with altitude and any process that loses a constant fraction per unit time.
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