Half-life and Radioactive Decay

Because radioactive decay is random, we cannot say when any single nucleus will decay. But when a sample contains a huge number of unstable nuclei, their average behaviour is very predictable. We describe this average using half-life.

Half-life is the time taken for the number of unstable nuclei in a sample to fall to half its original value.

Since fewer nuclei means fewer decays each second, the half-life can be measured in a second, equivalent way that is easier to detect in the lab.

Half-life is equally the time taken for the count-rate (or activity) of a sample to fall to half its original value.

Both definitions give the same time, because the count-rate is proportional to the number of undecayed nuclei present. The half-life of a given isotope is always the same, whether you have a large lump or a tiny speck.

At first it seems odd that a random process gives such a reliable, repeatable time. The link is probability.

Each unstable nucleus has a fixed chance of decaying in any given time. With billions of nuclei present, the fraction that decays in one half-life is always very close to one half, even though you cannot say which particular nuclei will go.

This is why the amount always halves in the same time, never falling by the same fixed number each time. Consider a sample starting with 1000 nuclei:

Half-lives passed	Nuclei remaining
0	1000
1	500
2	250
3	125

Each step removes half of what is currently there, not a fixed amount. So the number falls quickly at first and then more slowly, which is the shape of every decay curve.

A very common exam task is to read a half-life from a table or graph of count-rate against time. The method is to find how long the count-rate takes to fall to half its value.

Worked example — a sample has a count-rate of 800 counts per second. The count-rate is recorded over time:

Time (minutes)	0	5	10	15
Count-rate (counts/second)	800	400	200	100

Work through the halvings:

• 800 falls to 400 (half) in 5 minutes.
• 400 falls to 200 (half again) in the next 5 minutes.
• 200 falls to 100 in the next 5 minutes.

The count-rate halves every 5 minutes, so the half-life is 5 minutes. Checking two or three halvings, rather than just one, guards against reading the wrong value.

Sometimes you are given only a start and end value plus the time between them, and asked for the half-life. Count the number of halvings needed to get from the start value to the end value.

Worked example — the activity of a source falls from 240 Bq to 60 Bq over 10 years. Find the half-life.

Count the halvings from 240 to 60:

• $240\to 120$ (one half-life)
• $120\to 60$ (two half-lives)

So 2 half-lives take 10 years. One half-life is therefore:

$\text{half-life}=\frac{10}{2}=5\text{years}$

The key step is counting how many times you must halve to reach the final value, then dividing the total time by that number of half-lives.

Each half-life multiplies the remaining amount by one half, so after n half-lives the fraction of the original nuclei still present follows a simple rule.

After n half-lives, the fraction of the original nuclei remaining is ${\left(\frac{1}{2}\right)}^n$.

Half-lives (n)	Fraction remaining	As a fraction
1	${\left(\frac{1}{2}\right)}^1$	1/2
2	${\left(\frac{1}{2}\right)}^2$	1/4
3	${\left(\frac{1}{2}\right)}^3$	1/8
4	${\left(\frac{1}{2}\right)}^4$	1/16

Worked example — a sample starts with 6.4 × 10⁶ undecayed nuclei. How many remain after 3 half-lives?

After 3 half-lives the fraction remaining is ${\left(\frac{1}{2}\right)}^3=\frac{1}{8}$.

$\text{nuclei remaining}=6.4\times 10^6\times \frac{1}{8}=8.0\times 10^5$

So 800 000 nuclei remain undecayed after 3 half-lives.

(Higher Tier only) This slide covers expressing the net decline as a ratio, which is a Higher Tier requirement.

The net decline is the overall drop in the number of radioactive nuclei. It is often expressed as a ratio comparing the number that have decayed to the number that remain.

If the fraction remaining after n half-lives is ${\left(\frac{1}{2}\right)}^n$, then the fraction decayed is whatever is left over from the whole.

Worked example — a source is left for 2 half-lives. Express the net decline as a ratio of decayed nuclei to remaining nuclei.

After 2 half-lives the fraction remaining is ${\left(\frac{1}{2}\right)}^2=\frac{1}{4}$.

• Remaining = $\frac{1}{4}$ of the original
• Decayed = $1-\frac{1}{4}=\frac{3}{4}$ of the original

The ratio of decayed to remaining is $\frac{3}{4}:\frac{1}{4}$, which simplifies to 3 : 1.

Worked example — express the net decline after 4 half-lives as a ratio of decayed to remaining.

After 4 half-lives the fraction remaining is ${\left(\frac{1}{2}\right)}^4=\frac{1}{16}$, so 15/16 has decayed. The ratio of decayed to remaining is 15 : 1. A neat check: with 1 part remaining out of $2^4=16$ parts, the decayed part must be $16-1=15$.

1. Thinking the same number decays each half-life

Each half-life removes half of what remains, not a fixed number. From 800 the drops are 400, then 200, then 100. The step size shrinks each time.

2. Reading only one halving from a graph

Confirm the count-rate halves over the same time interval at least twice. A single reading can be misjudged; a consistent interval proves the half-life.

3. Confusing the fraction remaining with the fraction decayed

After 3 half-lives, 1/8 remains but 7/8 has decayed. Read the question carefully to see which one it wants.

4. Forgetting half-life can be defined by count-rate as well as by number of nuclei

Half-life is the time for the number of nuclei to halve, or equally the time for the count-rate or activity to halve. Both give the same value.

5. Muddling the net-decline ratio (Higher Tier)

Express it clearly as decayed to remaining. After 2 half-lives that is 3 : 1 (three parts decayed to one part left), not 1 : 3. State which quantity each part of the ratio refers to.