Radioactive Decay (N = N₀ e^(−kt))

Quantity remaining after time t given decay constant k or half-life.

Inputs

Result

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General calculation reads

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How to use this calculator

  • Enter N₀ and decay parameter (t½ or k).
  • Enter elapsed time.
  • Read remaining N(t).

About this calculator

First-order radioactive decay: N(t) = N₀ × e^(−kt). After n half-lives, N/N₀ = (½)ⁿ. Each radioactive nucleus has constant decay probability per unit time — independent of past. Bulk decay is exponential because of large numbers. Same math applies to many other first-order processes: drug elimination, RC capacitor discharge, exponential cooling.

Frequently asked

Why exponential?+
dN/dt = −kN (rate proportional to amount). Solution: N = N₀e^(−kt).
Activity vs. quantity?+
Activity = decay events/second = kN. Half-life governs both. Activity also decays exponentially.
Decay chain?+
Some isotopes decay into other radioactive species (U-238 → 13 stages → Pb-206). For chains, use Bateman equations.
Carbon dating sensitivity?+
Useful 100-50,000 years. Beyond ~10 half-lives: too little ¹⁴C left to measure reliably.
Why first-order?+
Each nucleus decays independently with constant probability per unit time. Doesn't depend on neighbors or how long it's been there.

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