Law of Radioactive Decay Explained Simply with Easy Examples
Radioactive decay is random for a single atom — nobody can say when one nucleus will break apart. But for a large collection of atoms, the pattern is perfectly predictable: in each equal time interval, a fixed fraction of the remaining nuclei decays. This is the law of radioactive decay.
It matters because it lets us calculate how long a radioactive sample stays dangerous, how old ancient objects are, and how much radiation a medical dose delivers.
Key Concepts
- Decay constant (λ): probability per unit time that a nucleus decays, in s⁻¹. Larger λ means faster decay.
- N₀: number of undecayed nuclei at time zero.
- N: number remaining after time t.
Formulas
Decay law: N = N₀ e^(−λt), where e ≈ 2.718, λ in s⁻¹, t in seconds. Rate form: dN/dt = −λN. These assume a large sample so statistics are smooth, and that decayed nuclei are not replaced.
Worked Example
A sample starts with 8.0 × 10²⁰ nuclei and λ = 0.05 per day. How many remain after 10 days?
- N = N₀ e^(−λt) = 8.0 × 10²⁰ × e^(−0.05 × 10).
- Exponent: −0.5. e^(−0.5) = 0.6065.
- N = 8.0 × 10²⁰ × 0.6065 = 4.85 × 10²⁰ nuclei.
Common Misconceptions
- Students think decay is linear, losing the same number each day. It is exponential — the same fraction, so the number lost shrinks over time.
- Many believe heating or chemical reactions change decay rate. For almost all decays, λ is unaffected by temperature, pressure, or chemistry.
- Some think after two half-lives everything is gone. After two half-lives, one quarter remains; decay never quite reaches zero.
Real-World Applications
The decay law underpins carbon dating, planning radiotherapy doses, predicting nuclear waste storage times, and smoke-detector source lifetimes.
We can see this with dice as a rough picture: each undecayed nucleus is like a die that has a fixed chance of showing a six on each roll. You cannot predict one die, yet out of thousands you can predict closely how many sixes appear. Similarly, the decay law works because samples contain enormous numbers of nuclei, often 10²⁰ or more. The equation can also be written with half-life: N = N₀(1/2)^(t/T½), which is the same physics in a form that is easier to use with everyday time units.
Remember: N = N₀e^(−λt) — the fraction decaying per unit time is constant, individual atoms are random, but big samples follow this curve exactly.