Modern Physics
Radioactive Decay Calculator
Calculate remaining quantity after radioactive decay using N(t) = N₀e^(−λt). Enter half-life and time.
Remaining Quantity N(t)
N(t) = N₀ e^(−λt)
Decay Constant λ
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% Remaining
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Half-lives Elapsed
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Step-by-Step Solution
What is Radioactive Decay?
Radioactive decay is the spontaneous disintegration of an unstable atomic nucleus, emitting radiation in the form of alpha particles, beta particles or gamma rays. The rate of decay is proportional to the number of undecayed nuclei present, leading to exponential decay — the same mathematical form as bacterial growth, cooling and many other natural processes.
N(t) = N₀ e^(−λt) | λ = ln(2)/t_½
N₀ = initial quantity | λ = decay constant (s⁻¹) | t_½ = half-life | t = time elapsed
💡 After each half-life, exactly half the remaining nuclei have decayed. After 10 half-lives, only 1/1024 (about 0.1%) of the original remains. After 20 half-lives, only 1/1,048,576 remains.
Real-World Applications
Medical Imaging
Radioactive tracers with short half-lives (hours) are injected for PET/SPECT scans — they decay away quickly.
Carbon-14 Dating
C-14 (half-life 5,730 years) decays at a known rate, allowing archaeologists to date organic materials up to ~50,000 years old.
Nuclear Power
Uranium-235 and Plutonium-239 decay (by fission) release energy that drives nuclear power stations.
Geological Dating
U-238 (half-life 4.5 billion years) is used to date rocks and determine the age of the Earth.
Frequently Asked Questions
Why is radioactive decay exponential?
Each nucleus decays independently with a fixed probability per unit time. This means the rate of decay at any moment is proportional to the number of undecayed nuclei: dN/dt = -λN. Solving this differential equation gives the exponential N(t) = N₀e^(-λt).
What is the half-life?
The half-life (t½) is the time for exactly half the nuclei to decay. It is related to the decay constant by t½ = ln(2)/λ = 0.693/λ. Half-lives range from microseconds (unstable isotopes) to billions of years (stable isotopes).
Can you speed up radioactive decay?
In normal conditions, no. The decay constant λ is determined by nuclear physics and is not affected by temperature, pressure, chemical bonding or electromagnetic fields. This is why nuclear waste remains dangerous for thousands of years.