SCIENCE · INVESTIGATE AND EXPLAIN

Predict an ensemble without assigning a lifetime to one nucleus

Use exponential survival and distinguish an expected ensemble count from a deterministic decay schedule.

Learning outcome: Use exponential survival and distinguish an expected ensemble count from a deterministic decay schedule.

Starting recap

A half-life gives a 50% survival probability over that interval for each independent parent in this model.

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Try

Start with an original ensemble of 1600 parents and compare one, two and four half-lives.

Open the experiment

Predict

After two half-lives, are all original parents gone?

Your prediction and reason

Experiment

Select 4,8 and 16 quarter-half-lives at a 20 s half-life. Calculate survivors before reading results. Then select a fractional half-life.

Compare survivor number and activity

Worked teaching model · synthetic values · Original ensemble of 1600 independent identical parents, no replenishment or daughter feedback. Smooth values are expectations, not integer observations or a clock for one nucleus. Activity is decays per second, not dose.

Elapsed quarter-half-lives: 4 quarters Half-life: 20 s

Reset keeps your prediction, captures and practice history.

Quantity	Calculated value	Unit
Expected parents remaining	800.0000	expected nuclei
Expected parents transformed	800.0000	expected nuclei

800.000 expected survivors after 1 half-lives.

Elapsed time 20 s; activity 27.7259 Bq. Changing half-life at fixed half-life count preserves survivors but changes time and activity. No dose or safety limit follows.

Model boundary: Original ensemble of 1600 independent identical parents, no replenishment or daughter feedback. Smooth values are expectations, not integer observations or a clock for one nucleus. Activity is decays per second, not dose.

All investigations are virtual. Do not obtain radioactive sources, expose eyes to bright beams, or build high-voltage apparatus. Notebook captures preserve selected recipes, not physical measurements.

800.000 expected survivors after 1 half-lives.

Elapsed time 20 s; activity 27.7259 Bq. Changing half-life at fixed half-life count preserves survivors but changes time and activity. No dose or safety limit follows.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Repeated fractions form an exponential

Begin with 1600 identical parents in the synthetic ensemble. After one half-life the expected count is 800, after two it is 400, and after three it is 200. It is half of the remaining population each time, not another fixed 800 parents. The equation N=N₀2^(−t/T) captures that repeated fractional change, where T is the half-life.

In the browser, the first control counts quarter-half-lives. A setting of 4 means t/T=1, not 4 s. With T=20 s, it corresponds to 20 s elapsed. A setting of 8 gives 40 s and 400 expected survivors. The second control changes T; at fixed t/T it changes elapsed time while preserving the expected surviving fraction.

Interpret a fractional result

At half a half-life, expected survivors are 1600/√2≈1131.37. A real observed sample cannot contain 0.37 of a parent nucleus. The decimal is an ensemble expectation, analogous to an expected number of heads over many repeated coin experiments. Individual observed counts fluctuate around it, and this deterministic display does not simulate those fluctuations.

For this memoryless exponential model, a nucleus that has survived one half-life still has a 50% chance of surviving the next. The half-life is not a deadline by which every nucleus disappears. A large ensemble can retain a small population after many half-lives; absolute guarantees about a single nucleus do not follow from the smooth equation.

Infer an interval from a ratio

An original comparison sample falls from 2400 expected parents to 300. The ratio is 1/8, so three half-lives have passed. If T=12 days, the interval is 36 days. This inference assumes a closed parent inventory, known initial count and no replenishment. Mixing samples or losing parents by another process breaks that simple interpretation.

Transfer beyond whole half-lives

If N/N₀=0.30, then t/T=−ln(0.30)/ln 2≈1.737. It is between one and two half-lives. Rounding it to two before further calculation would discard useful information.

Limits of a decay curve

The model contains no daughter decay chain, transport, background detector counts or dose calculation. It predicts parents under specific assumptions, not the safety of an unknown sample. All values are virtual; no radiation source is needed or permitted by this lesson.

Source and authorship

Reference for the physical relationship. Source checked 30 October 2026. The explanations, numerical cases, questions and diagrams here are original; reference material is not copied. Exact h,c,e follow NIST SI definitions.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Explain why the smooth count can be noninteger and why an old surviving nucleus is not forced to decay in the next interval.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Conserve nucleon number and charge in nuclear change Next: Distinguish activity from remaining inventory and dose

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