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ATOMIC STRUCTURE · CHAPTER 7 OF 12

What does half-life tell us?

We cannot predict when one nucleus will decay. Yet a large sample follows a pattern: after each half-life, about half of the undecayed nuclei remain. How can both statements be true?

By the end, you’ll be able to…

  • define half-life using undecayed nuclei, activity or corrected count rate
  • explain how random decays produce a predictable overall pattern
  • find half-life from data and graphs, including background correction
  • calculate quantities remaining and elapsed time
  • calculate fractions, percentages and ratios remaining or lost.

Half of what remains

The half-life of a radioactive isotope is the time taken for the number of its undecayed nuclei in a sample to fall to half. It is also the time taken for its activity to halve.

A background-corrected count rate can be used to find half-life if the detector arrangement stays the same. The measured rate then follows the decreasing activity of the isotope.

Each half-life halves the amount still undecayed, rather than removing the same number every time. Half-life is a property of the isotope, not the sample’s starting size.

INTERACTIVE HALVING MODEL

What fraction of nuclei remains?

Choose a half-life to see the fraction of the original group still undecayed. Pink + circles represent nuclei. A yellow “D” marks a nucleus that has decayed; the product is still present.

64 undecayed · 0 decayedAll of the original group is undecayed.
32 undecayed · 32 decayedFraction remaining: 1/2 · 50%
16 undecayed · 48 decayedFraction remaining: 1/4 · 25%
8 undecayed · 56 decayedFraction remaining: 1/8 · 12.5%
4 undecayed · 60 decayedFraction remaining: 1/16 · 6.25%

* An idealised halving pattern, not a random simulation. A real sample of only 64 nuclei would not usually halve exactly at each step. Yellow marks track decay; nuclei do not vanish. The diagram does not show emitted radiation or nuclear composition.

Two half-lives do not mean that every nucleus has decayed. Half of half is one quarter remaining. Half-life is not a deadline for an individual nucleus.

Random individuals, predictable large samples

Each undecayed nucleus of the isotope has the same probability of decay per unit time. Waiting longer does not make a surviving nucleus “due” to decay.

With very many nuclei, the overall number remaining follows a smooth exponential pattern closely. Small samples and measured counts show random fluctuations around the expected pattern.

INTERACTIVE COMPARE SMALL SAMPLES

Must two samples halve exactly?

These illustrative results start with 64 nuclei of the same isotope. Each column represents another equal interval of one half-life.

SampleStart1 half-life23
Expected pattern6432168
Illustrative A6434159
Illustrative B6430177
Do the different small-sample results mean the isotope has two different half-lives?
Try again. A fluctuation does not mean the isotope’s decay probability has changed.
Correct. Half-life describes the statistical pattern; individual decays remain random.
Try again. Exact halving is an idealisation, especially for small numbers.

Read the time between two halvings

A smooth decay curve gets less steep as time passes: fewer undecayed nuclei remain to decay. It is not a straight-line decrease.

Choose an activity, halve it, and find the two corresponding times. The difference between those times is the half-life. You do not have to start at time zero.

INTERACTIVE GRAPH EXPLORER

One curve. The same half-life.

Estimate the half-life before showing the guides. Then check it using later pairs of values.

Activity (Bq) 0 200 400 600 800 0 2 4 6 8 Time (min) 100 Activity (Bq) 0 200 400 600 800 0 2 4 6 8 Time (min) 100
First half-life: 800 Bq at 0 min becomes 400 Bq at 2 min. Half-life = 2 − 0 = 2 min.
Second half-life: 400 Bq at 2 min becomes 200 Bq at 4 min. Half-life = 4 − 2 = 2 min.
Third half-life: 200 Bq at 4 min becomes 100 Bq at 6 min. Half-life = 6 − 4 = 2 min. The activity has fallen by a smaller amount, but the time taken to halve is unchanged.
INTERACTIVE COMPARE ISOTOPES

Why do some isotopes decay faster than others?

Both samples start with the same number of undecayed nuclei. Which isotope has the shorter half-life: A or B? Look at where each curve reaches 50%.

Undecayed nuclei (%) 100 75 50 25 0 0 2 4 6 8 Time (min)
Isotope A · solid Isotope B · dashed

Isotope A decays faster. Its half-life is 2 minutes, compared with 4 minutes for B.

But why are their half-lives different? Different isotopes have different arrangements of protons and neutrons and different nuclear energies. These affect how likely their unstable nuclei are to decay.

Each nucleus of isotope A has a greater probability of decaying in a given time than a nucleus of B. So, in a large sample, A loses half its undecayed nuclei sooner.

Count the half-lives first

If a half-life is 3 hours, then 9 hours contains three half-lives. Halve the starting quantity three times. Keep the time units consistent.

To find the number of half-lives:

Number of half-lives = elapsed time ÷ half-life

You can then halve the starting number of undecayed nuclei, activity or corrected count rate once for each whole half-life.

WATCH THE HALF-LIVES

How much time has passed?

Starting with 32 undecayed nuclei, connect elapsed time to the number of half-lives as half of the remaining pink nuclei drift away every 3 hours.
Time elapsed
0 hours 3 hours 6 hours 9 hours
Undecayed nuclei
32 16 8 4
Start: 32 undecayed nuclei. 3 hours later: 16 remain. 6 hours later: 8 remain. 9 hours later: 4 remain.
0 hours 32
3 hours 16
6 hours 8
9 hours 4
9 hours = 3 half-lives
32 → 16 → 8 → 4
After three half-lives, one-eighth of the original nuclei remain undecayed.

The circles that drift away represent nuclei that have decayed into other nuclei, not matter disappearing. Radioactive decay is random, so the numbers illustrate the expected pattern rather than the exact behaviour of a small sample.

Three worked examples

Calculate remaining activity:

An isotope starts at 640 Bq. Its half-life is 2 h. Find its activity after 6 h.

Number of half-lives = time ÷ half-life

6 ÷ 2 = 3; 640 → 320 → 160 → 80

Activity = 80 Bq

Calculate elapsed time:

A corrected count rate falls from 240 to 30 counts/s. The half-life is 5 min.

Time = number of half-lives × half-life

240 → 120 → 60 → 30; time = 3 × 5

Time = 15 min

Calculate half-life:

An activity falls from 800 to 100 Bq in 12 days.

Half-life = elapsed time ÷ number of half-lives

800 → 400 → 200 → 100; half-life = 12 ÷ 3

Half-life = 4 days

Correct background before halving a count rate

A detector measures radiation from the source and the surroundings. Subtract the background count rate before finding the source’s half-life.

Use the background-corrected rate to find half-life, and keep the detector arrangement fixed.

INTERACTIVE BACKGROUND CORRECTION

Why doesn’t this curve head towards zero?

Background is 20 counts/s. Follow the three stages to understand the graph and find the half-life of the source's signal.

Count rate (counts/s) 0 50 100 150 200 0 2 4 6 8 Time (min)
At the start, the total count rate is 180 counts/s. The dashed line shows the background count rate of 20 counts/s. As the source decays, background radiation is still detected, so the total count rate approaches 20 counts/s, not zero.
Count rate (counts/s) 0 50 100 150 200 0 2 4 6 8 Time (min)
Subtract 20 counts/s at every time. The dark curve is the corrected source signal; the pale curve is the original total reading. Initially: 180 − 20 = 160 counts/s.
Count rate (counts/s) 0 50 100 150 200 0 2 4 6 8 Time (min) 160 80
Starting corrected count rate 160 counts/s
Half the starting value 80 counts/s
Time taken 2 minutes
Half of 160 counts/s is 80 counts/s. The corrected curve reaches this value after 2 minutes. Therefore, half-life = 2 minutes.
What goes wrong if you halve 180 directly?

The 180 counts/s reading includes background radiation. Halving 180 gives 90 counts/s, but this is not half of the corrected source count rate.

Subtract the background first: 180 − 20 = 160 counts/s. Then halve 160 to get 80 counts/s.

At this point, the detector's total reading is actually 100 counts/s, because it still includes the 20 counts/s background.

HIGHER TIER FOCUS · FRACTIONS AND RATIOS

Remaining is different from lost

After three half-lives, ⅛ remains and ⅞ has decayed. These fractions add to 1.

A compact way to express repeated halving:

Fraction remaining = (½)n
Quantity remaining = starting quantity × (½)n

Here n is the number of half-lives. Repeated halving gives the same answer without using the power formula.

INTERACTIVE RATIO EXPLORER

Which way round is the ratio?

Choose the number of half-lives. The pink section is the fraction remaining; the white sections represent the fraction lost from the original undecayed group.

Remaining: 1/2 = 50%.

Lost: 1/2.

Remaining : original = 1 : 2
Original : remaining = 2 : 1
Lost : original = 1 : 2

Remaining: 1/4 = 25%.

Lost: 3/4.

Remaining : original = 1 : 4
Original : remaining = 4 : 1
Lost : original = 3 : 4

Remaining: 1/8 = 12.5%.

Lost: 7/8.

Remaining : original = 1 : 8
Original : remaining = 8 : 1
Lost : original = 7 : 8

Remaining: 1/16 = 6.25%.

Lost: 15/16.

Remaining : original = 1 : 16
Original : remaining = 16 : 1
Lost : original = 15 : 16

* These ratios describe the original isotope or its emission rate under fixed conditions. “Lost” means no longer in the original undecayed group, not that matter has disappeared.

Always label the order of a ratio. After three half-lives, remaining : original is 1 : 8, while lost : original is 7 : 8. The net decrease is 87.5%, not 12.5%.

Can you use any half-life graph?

Check what the vertical axis represents. It may show undecayed nuclei, activity or count rate. Subtract background if needed, then compare a value with half that value.

The clean halving rule applies to the isotope being studied. A mixture of radioactive isotopes, or radiation from radioactive daughter products, can produce a more complicated measured pattern.

INTERACTIVE QUICK CHECK

Check your understanding

Choose one answer for each question. Feedback appears underneath.

1. What fraction of the original undecayed nuclei is expected to remain after two half-lives?
Try again. Each interval halves what remains, rather than removing the other original half.
Correct. 1 → ½ → ¼.
Try again. One half remains after one half-life.
2. A curve shows 300 Bq at 4 min and 150 Bq at 7 min. What is the half-life?
Correct. The time for the activity to halve is 7 − 4 = 3 min.
Try again. 7 min is a time coordinate, not the interval between the two readings.
Try again. Subtract the two times rather than adding them.
3. Total count rate is initially 90 counts/s and background is 10 counts/s. Which total reading corresponds to one half-life later?
Try again. The background does not halve along with the source signal.
Try again. 40 counts/s is the corrected rate, not the total.
Correct. Source rate starts at 80, halves to 40, then background adds 10: total = 50 counts/s.
INTERACTIVE CHALLENGE

The total count rate falls from 100 to 30 counts/s in 18 minutes. Background stays at 20 counts/s. Find the half-life and the fraction of the source’s initial signal lost.

Correct both readings before counting the halvings. Give a clearly labelled ratio as well.

Reveal the worked answer

Initial corrected rate = 100 − 20 = 80 counts/s. Final corrected rate = 30 − 20 = 10 counts/s.

80 → 40 → 20 → 10: three half-lives occur in 18 minutes.

Half-life = 18 ÷ 3 = 6 min.

Fraction remaining = ⅛. Fraction lost = ⅞ = 87.5%. Lost : original = 7 : 8; remaining : original = 1 : 8.

CHAPTER 7

Chapter summary

  • Half-life is the time for undecayed nuclei or activity to halve.
  • Every half-life halves what remains, not a fixed number of nuclei.
  • Individual decay times are random; large samples follow a predictable overall pattern.
  • Find half-life from the time difference between any value and half that value on a suitable curve.
  • Subtract background before using detector rates; keep the measurement arrangement fixed.
  • Count the half-lives, then halve repeatedly or use the fraction (½)n.
  • Distinguish fractions remaining from fractions lost, and label ratio order.