Select a nuclide and calculate radioactive decay in real time. Log-scale graphs of N(t) and A(t), decay chain visualization, and carbon-14 radiocarbon dating — all included.
Student 🙋: Why does the decay law show an exponential, not a linear, decrease?
Professor 🎓: Because radioactive decay is a purely statistical process — each nucleus decays independently with a constant probability λ per unit time, regardless of how long it has already existed or how many other nuclei surround it. If you have N nuclei, the rate of decay at any moment is dN/dt = −λN, which is a differential equation whose solution is N(t) = N₀e^(−λt). The more nuclei you have, the more decays happen per second; as the number drops, so does the rate — a self-reinforcing relationship that naturally produces an exponential curve. This is fundamentally different from, say, a radioactive source that runs out at a fixed rate regardless of how much remains.
What exactly is "half-life"? It sounds like the time it takes for something to become half as radioactive, but is that right?
🎓
Basically, you're close! The half-life is the time it takes for half of the radioactive atoms in a sample to decay. It's a constant for each specific nuclide. In practice, this means the activity, or the number of decays per second, also halves. Try selecting Carbon-14 in the simulator above. You'll see its half-life is about 5730 years. If you set the time slider to that value, the "Current Activity Ratio A/A₀" will drop to 0.5.
🙋
Wait, really? So if Carbon-14 halves every 5730 years, does it just disappear after two half-lives? That seems too fast for dating ancient artifacts.
🎓
Good catch! It doesn't disappear; it halves again. After one half-life, 50% remains. After two half-lives (11,460 years), you have 50% of that 50%, so 25% of the original remains. The decay is exponential, not linear. Move the "Time Range" slider further and watch the graph curve. A common case is that after about 10 half-lives, the activity has fallen to less than 0.1% of the original, which is often considered negligible for many applications.
🙋
What's the difference between the number of atoms N(t) and the activity A(t) that the formulas show? And what is this lambda (λ) parameter?
🎓
Great question. N(t) is the number of radioactive atoms left. A(t) is the rate at which they are decaying, measured in Becquerels (Bq = decays per second). They're connected by the decay constant, λ (lambda). Think of λ as the probability per second that any single atom will decay. So, A(t) = λ * N(t). In the simulator, when you change the "Initial Activity A₀," you're effectively setting λN₀. The relationship λ = ln(2) / Half-life is key—it's how we convert the intuitive half-life into the math of the exponential equation.
Physical Model & Key Equations
The core of radioactive decay is a first-order process: the decay rate is directly proportional to the number of remaining radioactive nuclei. This leads to the exponential decay law.
$$N(t) = N_0 e^{-\lambda t}$$
N(t): Number of radioactive nuclei at time t. N₀: Initial number of nuclei at t=0. λ (lambda): Decay constant [s⁻¹], the probability of decay per unit time. t: Elapsed time.
The activity A(t) is the measurable quantity—the number of decays per second. It follows the same exponential form. The half-life (T₁/₂) is the time for N or A to drop to half its initial value, providing a direct link between the microscopic constant λ and an observable timescale.
A(t): Activity at time t [Bq]. A₀: Initial activity. T₁/₂: Half-life. After time n × T₁/₂, activity is A₀ × (1/2)ⁿ.
Real-World Applications
Radiocarbon Dating: This simulator directly models the principle behind Carbon-14 dating. Living organisms absorb C-14. When they die, absorption stops and the C-14 decays. By measuring the remaining A(t) in an ancient sample and comparing it to the assumed initial A₀ (similar to modern levels), scientists can calculate the time t since death, dating artifacts and fossils.
Medical Radioisotope Therapy & Imaging: Nuclides like Technetium-99m (Tc-99m) have short half-lives (6 hours). Doctors calculate the initial activity A₀ needed so that A(t) delivers the correct diagnostic dose by the time it's used in a patient. The simulator can model this decay to ensure safe and effective medical procedures.
Nuclear Waste Management & Decommissioning: Before decommissioning a nuclear plant or storing waste, engineers must know the nuclide inventory's activity over millennia. They use these decay laws to predict when materials will decay to safe levels. For instance, they might calculate how long Cs-137 (T₁/₂ ~30 years) must be stored to reduce its hazard.
Environmental Radiation Monitoring: After events like nuclear accidents, monitoring for isotopes like Iodine-131 (T₁/₂ ~8 days) and Cesium-137 is crucial. The decay equation helps predict how quickly contamination will diminish naturally and informs decisions on land use and food safety.
Common Misconceptions and Points to Note
Here are some points where beginners often stumble when mastering this tool. First, the misconception that "radioactivity becomes zero after the half-life has passed". Let's consider C-14 with its 5730-year half-life. After 5730 years, it's 1/2, after the next 5730 years, it's 1/4, then 1/8... theoretically, it never reaches zero. In practice, after about 10 half-lives, the activity is roughly 1/1000 of the initial value, which is often judged as "effectively negligible." Conversely, this means that nuclides with long half-lives (e.g., U-238's ~4.5 billion years) hardly decay on a human timescale.
Next, confusion of units when inputting parameters. The tool asks for "Half-life (years)", but when dealing with a nuclide like I-131 with an 8-day half-life, you need to convert and input it as "8/365 ≈ 0.022 years". Getting this wrong will produce a completely different graph. Similarly, the unit for the decay constant λ is fundamentally [1/second]. If the half-life is in years, remember to convert it to seconds using λ=ln2 / (half-life[seconds]).
Finally, do not equate "activity" with "number of nuclei". The tool's graph typically shows the decay of "activity A(t)". Even with the same number of nuclei N(t), a nuclide with a larger decay constant λ (shorter half-life) will have higher activity. For example, for the same one million atoms, a nuclide with a 1-hour half-life will have a vastly greater "number of decays at this very moment" than one with a 1-year half-life.