Half-Life Decay Simulator

Choose an isotope, starting atom count, and seed.

Watch individual atoms decay beside the smooth exponential curve, pause at any time, and print a matching investigation sheet and answer key.

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Why a smooth decay curve comes from random atoms

Start 1,000 identical unstable atoms at the same moment. They do not carry timers set to 5,730 years or 8.0252 days. Each nucleus has the same chance to decay during the next short interval, and the ones that survive face the same chance again. One atom may decay almost immediately while another lasts for several half-lives.

The surprise is what happens when those independent events are counted as a group. The expected amount follows N = N0 × (1/2)t/T, where T is the half-life. At t = T, the exponent is 1 and one half remains. At t = 2T, the exponent is 2 and one quarter remains. The solid curve in the simulator draws those expected values without randomness.

The dashed line asks every surviving atom to make a random decay decision 24 times per half-life. The chance on each check is about 2.847%. That may look too small, but surviving a whole half-life means avoiding the event 24 times. The probability of doing that is exactly 50%. The model is therefore not forcing half the atoms to disappear at the boundary; it is using small independent chances that produce one-half survival on average.

With only 50 atoms, a difference of five atoms is 10 percentage points, so the dashed line can look rough. With 5,000 atoms, the same five-atom difference is only 0.1 percentage point. Large groups reveal the smooth curve because early and late decays balance one another more closely.

Predict the graph before pressing Play

A half-life graph becomes easier to read when you mark the whole half-lives first. Suppose the run begins with 1,600 atoms. The smooth checkpoints are fixed before the simulation starts:

  • 0 half-lives: 1,600 atoms, or 100%.
  • 1 half-life: 800 atoms, or 50%.
  • 2 half-lives: 400 atoms, or 25%.
  • 3 half-lives: 200 atoms, or 12.5%.
  • 4 half-lives: 100 atoms, or 6.25%.

Notice that the same amount is not lost each time. The first interval loses 800 atoms, while the fourth loses only 100. What stays constant is the fraction lost: half of whatever was present at the start of that interval. That is why radioactive decay makes a curve instead of a straight descending line.

The time labels depend on the isotope. Three half-lives mean 24.0756 days for iodine-131, 17,190 years for carbon-14, and 13.404 billion years for uranium-238. The percentages are still 100%, 50%, 25%, and 12.5% because half-life scaling works the same way at every timescale.

What carbon dating mode can and cannot tell you

Carbon dating mode reverses the decay equation. Instead of asking how much C-14 remains after a known time, it asks how much time would produce a measured fraction. If 25% remains, two half-lives have passed because 100% becomes 50% and then 25%. Two times 5,730 gives an idealized age of 11,460 years.

For a percentage that is not an exact halving, the tool uses age = 5,730 × log2(100 ÷ remaining percentage). Entering 10% gives about 19,034 years, or 3.322 half-lives. That is the right answer for the simple exponential model taught in algebra, chemistry, physics, and earth science.

A real radiocarbon report needs more work. Atmospheric C-14 has varied over time, so laboratories compare conventional radiocarbon ages with calibration curves built from dated records such as tree rings. They also evaluate contamination, instrument background, and reservoir effects that can make an organism begin with a different apparent C-14 level. The calibrated result is often a range of calendar years rather than one exact year.

Use the mode to check the mathematics, explore how percentages map to half-lives, and prepare classroom examples. Do not use it to assign a professional date to a real sample or artifact.

A 15-minute classroom investigation

Set the isotope to carbon-14, keep the seed at classroom-14, and start with 50 atoms. Before pressing Play, have the class predict the smooth amounts after one, two, and three half-lives: 25, 12.5, and 6.25. The random run must show whole atoms, so it cannot equal every decimal checkpoint.

  1. Run the simulation and record the dashed-line count at each whole half-life.
  2. Change only the starting count to 500, keeping the same seed. Compare the difference from the curve in atoms and in percentage points.
  3. Repeat with 5,000 atoms. The atom difference may still be visible, but the percentage difference should usually be smaller.
  4. Press Different run once. Ask which features change and which stay fixed. The dashed path changes; the smooth curve and every N0 divided by 2 to the power n checkpoint do not.
  5. Print the student sheet for calculations and the matching answer key for discussion.

This sequence separates three ideas that students often mix together: the half-life is a fixed property of the isotope, the smooth curve is an expected group pattern, and the decay time of one atom is random. Keeping the seed visible also turns a fleeting animation into evidence the class can reproduce exactly.

Frequently Asked Questions

Common questions about the Half-Life Decay Simulator

The smooth model leaves one half after one half-life, one quarter after two, and one eighth after three. In symbols, the amount after n half-lives is N0 divided by 2 to the power n. Starting with 1,000 gives exact model values of 500, 250, and 125.

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