Radioactive Decay Simulator
Simulate radioactive decay of a sample. Adjust half-life and initial number of nuclei and watch the exponential decay and activity curve.
Radioactive Decay
Radioactive decay is a spontaneous, random process in which an unstable atomic nucleus emits radiation and transforms into a different nucleus (or a lower energy state). Individual decay events are unpredictable, but for large numbers of nuclei the average behaviour follows a precise exponential law.
Half-Life
The half-life t½ is the time after which, on average, half the nuclei have decayed. After one half-life: N = N₀/2. After two half-lives: N = N₀/4. After ten half-lives: N < 0.1% of original. Half-lives range from nanoseconds (Po-213: 4 μs) to billions of years (U-238: 4.47 × 10⁹ yr).
Decay Constant and Mean Lifetime
The decay constant λ = ln(2)/t½ is the probability per unit time that a given nucleus decays. The mean lifetime τ = 1/λ = t½/ln(2) ≈ 1.443 t½ is the average time a nucleus survives. Note τ > t½ because the exponential tail extends to infinity.
Activity
Activity A is the decay rate — the number of decays per second. A = λN(t) = λN₀ e^(−λt). It is measured in becquerel (Bq = 1 decay/s) or curie (Ci = 3.7 × 10¹⁰ Bq). Activity decreases exponentially at the same rate as N(t).
| Real isotope | Half-life | Decay type |
|---|---|---|
| Carbon-14 (¹⁴C) | 5730 years | β⁻ |
| Iodine-131 (¹³¹I) | 8.0 days | β⁻, γ |
| Radium-226 (²²⁶Ra) | 1600 years | α |
| Uranium-238 (²³⁸U) | 4.47 × 10⁹ yr | α |
| Polonium-214 (²¹⁴Po) | 164 μs | α |
Radioactive Decay Formulas
Exponential Decay Law
Decay Constant
Mean Lifetime
Activity
| Symbol | Name | Unit |
|---|---|---|
| N₀ | Initial number of nuclei | nuclei |
| N(t) | Number remaining at time t | nuclei |
| λ | Decay constant | s⁻¹ |
| t½ | Half-life | s (or any time unit) |
| τ | Mean lifetime | s |
| A | Activity | Bq (decays/s) |
Frequently Asked Questions
Why is radioactive decay exponential?
Each nucleus decays independently with a constant probability λ per unit time. The rate of decrease −dN/dt is proportional to how many nuclei remain: dN/dt = −λN. This first-order differential equation has the exponential solution N(t) = N₀ e^(−λt). The memoryless (Markov) property of quantum decay is the key physical reason.
What is the difference between half-life and mean lifetime?
The half-life t½ is when 50% of nuclei have decayed. The mean lifetime τ is the average survival time of individual nuclei. Because the exponential distribution is skewed (some nuclei survive very long), τ = t½/ln(2) ≈ 1.443 t½. If you wait τ, about 36.8% of nuclei remain (e⁻¹ ≈ 0.368).
How is carbon-14 dating possible?
Living organisms maintain a constant ratio of ¹⁴C/¹²C by exchanging carbon with the atmosphere. After death, no new ¹⁴C is absorbed and existing ¹⁴C decays with t½ = 5730 years. Measuring the remaining ¹⁴C fraction and solving N/N₀ = e^(−λt) gives the age. It works up to about 50,000 years.
Does a single nucleus decay exactly at t = t½?
No — individual nuclear decay is completely random. The half-life is a statistical property of a large ensemble. A single nucleus has a 50% chance of decaying before t½ and 50% after. It might decay in the first microsecond or last a million times longer — there is no way to predict.
What is the becquerel and how does it differ from the curie?
One becquerel (Bq) = one nuclear decay per second. One curie (Ci) = 3.7 × 10¹⁰ Bq, originally defined as the activity of 1 gram of radium-226. The curie is much larger than the becquerel. Background radiation in a typical room is ~0.1 Bq/m³ of air from radon. Radiotherapy sources are typically GBq (gigabecquerel) to TBq (terabecquerel).