6.23-6.27 - Activity, half-life and decay calculations

Activity and random radioactive decay

Consider a source containing a very large number of unstable nuclei of one radioactive isotope. Each decay is a nuclear event. The activity of the source is the rate at which its unstable nuclei decay.

An activity of 600 Bq therefore means a decay rate of about 600 decays each second. It is a rate, not the number of undecayed nuclei in the source. Because individual decays are random, the number occurring in one particular second can fluctuate around this value.

For the simple single-isotope sources considered here, no new nuclei of the isotope are being made. Once one of its nuclei decays, that nucleus is no longer an undecayed nucleus of the original isotope. As time passes, fewer undecayed nuclei remain, so there are fewer possible decays per second and the source's activity decreases.

Radioactive decay is random. It is not possible to predict when one named nucleus will decay, or which nucleus will decay next. This does not prevent useful predictions for a very large population: although the individual events are unpredictable, the proportion of nuclei likely to remain after a given time follows a stable pattern. A smooth decay curve represents this large-population prediction; short measurements can scatter around the trend.

What half-life means

The half-life is a time interval. It can be described in two equivalent ways:

These descriptions agree because, for one isotope, activity depends on how many undecayed nuclei remain. If the number of undecayed nuclei halves, the activity also halves.

The half-life is characteristic of the isotope and does not depend on the starting activity. A 1000 Bq source and a 200 Bq source made from the same isotope have the same half-life; each loses half of its own activity in that time.

Successive half-lives produce repeated halving:

The crucial phrase is half of what remains. If a source begins at 800 Bq, equal half-life intervals give 800 → 400 → 200 → 100 Bq. The decreases are 400 Bq, then 200 Bq, then 100 Bq: the same fraction is lost each time, not the same amount. This multiplicative change produces a curved decay graph rather than a straight line.

After two half-lives, 1/4 of the original nuclei remain and 3/4 have decayed. Half-life is not the time for all the nuclei to decay, and it is not a countdown attached to one particular nucleus.

Calculations with repeated halving

Build each simple calculation from the meaning of half-life. Repeated halving is enough; the compact power expression below is optional shorthand:

1. Put the elapsed time and half-life in the same unit.
2. Find how many half-lives have passed: number of half-lives = elapsed time / half-life.
3. Halve the activity or number of undecayed nuclei once for each half-life.
4. Check whether the question asks for the amount remaining or the amount that has decayed.

The repeated steps can be recorded compactly as:

This power form does not describe a different method; it is shorthand for halving the value the required number of times.

Worked example 1: calculate the activity remaining

A source has an initial activity of 960 Bq and a half-life of 3.0 hours. Calculate its predicted activity after 12 hours.

First count the half-lives:

number of half-lives = 12 hours / 3.0 hours = 4

Now halve the value four times, keeping the time intervals visible:

0 h: 960 Bq → 3 h: 480 Bq → 6 h: 240 Bq → 9 h: 120 Bq → 12 h: 60 Bq

So the predicted activity after 12 hours is 60 Bq. The answer is plausible because four half-lives leave 1/16 of the original activity, and 960 / 16 = 60.

Do not subtract 480 Bq four times. Only the first halving removes 480 Bq; each later halving acts on the smaller value that remains.