Real-time visualization of exponential decay for C-14, I-131, Cs-137, U-238, and custom isotopes. Supports decay chains A→B→C with Bateman equations. Explore carbon dating, medical isotopes, and nuclear waste management.
Parameters
Initial Atoms N₀
Isotope
Half-Life T½
Time Unit
Time Span (× half-lives)
Decay Chain
T½_B / T½_A ratio
Presets
Check time (× T½)
Results
1000
Current N
—
Activity (Bq)
100%
% Remaining
0.0
Half-lives elapsed
Decay
Atom-dot visualization (blue = undecayed, red = decayed) - snapshot at the selected time
What exactly is a "half-life"? I see it's a time, but how does it work for something random like an atom decaying?
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Basically, it's the time it takes for half of a large group of identical, unstable atoms to decay. For a single atom, you can't predict when it'll decay—it's random. But for a huge number, like the thousands of dots in this simulator, the fraction that decays over a specific time becomes predictable. Try setting the "Initial Atoms" slider to a high number and watch the decay curve smooth out.
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Wait, really? So if I start with 1000 atoms and the half-life is 1 minute, I'll have exactly 500 left after a minute?
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In practice, yes—statistically, it'll be very close to 500. The key is that it's a constant fractional decay. After another minute, half of those 500 decay, leaving about 250. You can see this "halving" pattern perfectly by setting the "Time Span" control to something like "3 × half-lives" and watching the graph. Each time you pass a multiple of the half-life, the number of atoms is roughly cut in half.
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That makes sense for one isotope. But what's the "T½_B / T½_A ratio" slider for? Is that for when one decaying atom turns into another unstable one?
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Exactly! That's for modeling a decay chain, like Uranium-238 decaying to Thorium-234 and so on. Atom "A" decays into atom "B," which is also radioactive. That ratio controls how much faster or slower the second isotope decays compared to the first. If B decays much faster (a small ratio), it barely builds up. If it decays much slower (a large ratio), it accumulates. Try adjusting that ratio and watch the population of the red "B" atoms change on the graph.
Physical Model & Key Equations
The fundamental law of radioactive decay states that the rate of decay is proportional to the current number of undecayed atoms. This leads to an exponential decay function.
$$N(t) = N_0 \, e^{-\lambda t}$$
Here, $N(t)$ is the number of atoms remaining at time $t$, $N_0$ is the initial number of atoms (which you set with the slider), and $\lambda$ is the decay constant, which defines the probability of decay per unit time.
The decay constant $\lambda$ is intrinsically linked to the half-life $T_{1/2}$, the time for half the sample to decay. By setting $N(t) = N_0/2$, we can derive their relationship.
$$\lambda = \frac{\ln 2}{T_{1/2}}$$
This is why the simulator's "Half-Life" parameter directly controls the steepness of the decay curve. A shorter half-life means a larger $\lambda$ and faster decay. The activity $A = \lambda N$ (measured in Becquerels, Bq) is the decays per second, which is what radiation detectors measure.
Frequently Asked Questions
Yes, it can. This tool supports the Bateman equation and can plot the time-dependent changes of each nuclide in a decay chain (A→B→C) in real time. Simply input the initial number of atoms and half-lives, and it will automatically calculate the concentration changes from the parent nuclide to the daughter nuclide.
The horizontal axis represents time (with selectable units such as years, days, or seconds), and the vertical axis represents the number of remaining atoms or radioactivity (in Bq). You can freely adjust the time axis range using the slider or numerical input at the bottom of the graph, and you can zoom in and out using mouse operations.
By setting the half-life of C-14 (approximately 5730 years) and matching the initial value to its atmospheric ratio, the remaining ratio over time will be plotted as a graph. By comparing the measured value of a sample with the graph, you can intuitively learn the back-calculation process of dating.
By inputting the half-life of each isotope (Cs-137: about 30 years, I-131: about 8 days), the decay curve of radioactivity over time will be displayed. For example, while I-131 decreases to 1/1000 in about 80 days, Cs-137 remains for a long period, allowing you to visually compare the differences in management duration.
Real-World Applications
Archaeological & Geological Dating (C-14, U-238): Carbon-14 (half-life ~5,730 years) is used to date organic materials up to about 50,000 years old. By comparing the remaining C-14 to stable C-12 in a sample, archaeologists can estimate its age. Uranium-238's long decay chain (half-life ~4.5 billion years) is used in radiometric dating of rocks.
Nuclear Medicine (I-131, Tc-99m): Iodine-131 (half-life ~8 days) is a workhorse in medicine. It's used for both imaging and therapy for thyroid conditions because the thyroid gland naturally concentrates iodine. Its moderately short half-life delivers a therapeutic radiation dose while limiting long-term exposure.
Food Safety & Tracer Studies (Cs-137): Cesium-137 (half-life ~30 years) is a common fission product. Its gamma emissions make it detectable in minute quantities, so it's used as a tracer to study soil erosion and, unfortunately, to monitor contamination in food following nuclear accidents.
Engineering & CAE Simulation: In computer-aided engineering, simulating decay heat from fission products (with many linked half-lives, like in our decay chain model) is critical for designing nuclear reactor cooling systems after shutdown. Accurate decay chain modeling ensures safety analyses are robust.
Common Misconceptions and Points to Note
First, it is a major misconception to think that "radioactivity becomes zero after the half-life has passed." Since the half-life is the time it takes to "reduce by half," it decreases to 1/2 after one half-life, 1/4 after two, 1/8 after three, and so on. For example, simulating 1 million initial atoms of Co-60 (half-life 5.27 years) shows that even after 10 half-lives (about 53 years), approximately 1000 atoms remain. In practice, when considering the management period for radioactive waste, it is crucial to recognize the length of time required to reach this "practically negligible level."
Next, note that "radioactivity (Bq) is proportional to the number of atoms, but also depends on the half-life." Try experimenting with the tool. If you fix the initial number of atoms at 1 million and compare I-131 (half-life 8 days) with C-14 (half-life 5730 years), the initial radioactivity of I-131 is overwhelmingly higher. Conversely, the number of atoms needed to produce the same radioactivity of 1 MBq (megabecquerel) is enormous for C-14, with its long half-life. When handling radiation sources, remember that this "radioactivity" value is the direct indicator for safety standards.
Finally, the pitfall of the "daughter nuclide half-life" setting in the Bateman equations. If the daughter's half-life is extremely long compared to the parent's (e.g., setting the parent's half-life to 1 day and the daughter's to 100 years), the daughter nuclide accumulates with almost no decay, becoming a long-term management concern. This is precisely the issue with long-lived radioactive nuclides produced in nuclear reactors. When adjusting simulation parameters, get into the habit of considering them in the context of real nuclide data.
Set initial atom count (N₀) using the slider or numeric input—values from 1×10⁶ to 1×10¹² atoms typical for lab samples
Select half-life duration: choose preset isotopes (C-14: 5,730 years; I-131: 8.02 days; U-238: 4.468 billion years) or enter custom values in seconds
Define time span (tSpan) in appropriate units matching your half-life scale—use years for C-14, days for I-131, seconds for rapid decay studies
Enable decay chain simulation (chainCheck) to track daughter products if modeling multi-step nuclear decay
Click simulate to visualize atom count decrease and activity decay in Becquerels
Worked Example
Medical isotope I-131 treatment: initial activity 370 MBq (N₀ = 1×10¹³ atoms), half-life 8.02 days. After 24.06 days (3 half-lives), remaining atoms = 1×10¹³ ÷ 2³ = 1.25×10¹² atoms, activity drops to 46.25 MBq (12.5% original). In 5 days, approximately 1 half-life elapses with ~50% decay, useful for thyroid treatment dosimetry calculations.
Practical Notes
Carbon-14 dating requires large sample sizes (milligrams) and accounts for atmospheric variation—set N₀ around 1×10¹¹ atoms for archaeological specimens
Iodine-131 environmental monitoring uses decay chains: I-131 → Xe-131 (stable); enable chainCheck to model daughter accumulation in soil/water systems
Uranium-238 cascades through 14 intermediate isotopes to Pb-206; simulate individual steps separately for precise geochronology
Activity conversion: 1 Bq = 1 decay/second; scale your N₀ by decay constant λ = ln(2)/t₁/₂ to validate output against expected Becquerels