Starter track
Step 5 of 50 / 5 completeModern Physics
Earlier steps still set up Radioactivity and Half-Life.
1. Photoelectric Effect2. Atomic Spectra3. de Broglie Matter Waves4. Bohr Model+1 more steps
Previous step: Bohr Model.
Concept module
Radioactivity and Half-Life
Use one compact decay bench to see why each nucleus decays unpredictably, why large samples still follow a regular half-life curve, and how to read remaining-count graphs honestly.
Interactive lab
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Stable links
Why it behaves this way
Explanation
Radioactivity is the bounded decay case where single events are genuinely probabilistic, but large samples still produce a steady and predictable curve. A nucleus does not carry a countdown clock to a personal half-life. Instead, each nucleus has the same chance to survive each interval, and the sample-level pattern becomes regular only when many independent yes-or-no events are averaged together.
This page keeps one compact bench with a bounded sample tray, two linked time graphs, and one readout card. Sample size, half-life, graph hover previews, compare mode, worked examples, quick tests, overlays, and challenge checks all stay tied to that same honest decay state instead of splitting into a separate probability page and a separate curve page.
Key ideas
01Half-life means the expected sample count is multiplied by one half over each equal half-life interval. It does not mean each nucleus survives exactly half that long.
02For one nucleus, decay is a probabilistic yes-or-no event. For many nuclei, those independent events average into a smoother exponential trend.
03The decay curve should step downward for an actual sample and stay smooth only for the expectation. Both views belong on the same bench.
04An honest decay graph approaches zero without becoming negative, and the sample can wander above or below the expectation while still following the same law.
Frozen walkthrough
Step through the frozen example
Frozen walkthrough
Use the current sample and half-life directly from the live bench. The same tray, graphs, and readout card drive each worked result.
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View plansFrozen valuesUsing frozen parameters
For a sample of 64 nuclei with half-life 2.4 s, what expected count belongs at time 0 s?
Starting sample
64 nuclei
Half-life
2.4 s s
Elapsed time
0 s s
Expected remaining
64 nuclei
1. Count equal half-life intervals
The live time is 0 s, which is 0 half-lives of 2.4 s each.
2. Apply the half-life law to the expectation
That means the expected surviving fraction is about 100%, so the sample expectation is 64 nuclei.
3. Compare expectation with the live sample
The tray currently shows 64 nuclei still present, so the sample is being compared with an expectation instead of being forced to equal it exactly.
Expected benchmark
\(N_{\mathrm{exp}} \approx 64\) nuclei at \(t = 0 s\)
The expected count comes from repeating the same fractional halving law across equal half-life intervals rather than subtracting a fixed number each second.
Chance-versus-curve checkpoint
A graph shows a smooth curve halving from 80 to 40 to 20 while the sample tray drops in uneven steps. Which description is the most honest?
Make a prediction before you reveal the next step.
Answer from the difference between expectation and individual decays.
Check your reasoning against the live bench.
The smooth curve is the expected sample trend, while the tray shows the actual integer nuclei that happened to survive. Both are correct views of the same probabilistic decay law.
If the tray matched a perfectly smooth curve, the page would be hiding the fact that nuclei decay one by one. If the page showed only random dots with no expectation, it would hide the regular large-sample law.
Common misconception
A half-life of 2 seconds means every nucleus lives for 2 seconds and then exactly half of them disappear on schedule.
Half-life is a sample-level expectation, not a personal timer for each nucleus. One nucleus may decay early, late, or not yet at all.
The smooth exponential curve is the expected trend for many independent decays. The live sample count stays as integer nuclei and should fall in steps.
Quick test
Reasoning
Question 1 of 4
Answer from the live decay bench, not from a fake countdown picture.
What is the most honest meaning of a 2-second half-life on this page?
Use the live bench to test the result before moving on.
Accessibility
The simulation shows a bounded sample tray of nuclei on the left and an actual-versus-expected summary on the right. Each nucleus is either still present or marked as decayed, and optional cues can highlight recently decayed nuclei, equal half-life checkpoints, and one-nucleus survival language.
The readout card summarizes sample size, half-life, elapsed time, expected remaining count, actual remaining count, remaining fraction, deviation from the expectation, and the single-nucleus survival probability at the current time.
Graph summary
The remaining-count graph plots time against both the stepped actual count and the smooth expected count. The remaining-fraction graph shows the same story normalized by the starting sample so different sample sizes can be compared without changing the underlying half-life law.
Carry the modern-physics branch forward
Keep this idea moving
Open the next concept, route, or track only when you want the current model to widen into a larger branch.
Quantized-structure comparisonModern Physics
Bohr Model
Use a compact hydrogen bench to connect quantized energy levels, allowed transitions, and named spectral-line series while staying clear that Bohr is a useful historical model rather than the final quantum description.
Discrete-evidence comparisonModern Physics
Atomic Spectra
Link discrete emission and absorption lines to allowed energy-level gaps with one compact ladder-and-spectrum bench that keeps transitions, wavelengths, and mode changes tied together.
Another probability-versus-threshold clueModern Physics
Photoelectric Effect
Use one compact lamp-to-metal bench to see why light frequency sets electron emission, why intensity alone fails below threshold, and how stopping potential reads the electron energy honestly.