2. Half-Life

Half-Life

Half-life is the time required for the number of undecayed radioactive nuclei in a sample to decrease to half of its previous value.

Half-life can also describe the time required for a radioactive sample's:

• activity
• corrected count rate

to decrease to half of its previous value.

The symbol for half-life is:

t₁/₂

Radioactive decay is random for individual nuclei, but when a sample contains a very large number of nuclei, its overall decay follows a predictable pattern.

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What Happens During One Half-Life?

Suppose a sample begins with:

1000 radioactive nuclei

After one half-life, approximately half remain undecayed:

1000 → 500

After another half-life:

500 → 250

Then:

250 → 125

The sample does not lose the same number of nuclei during each half-life.

Instead, it loses the same fraction:

1/2 of the radioactive nuclei present at the beginning of that interval.

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Successive Half-Lives

A useful pattern is:

Each half-life multiplies the remaining amount by:

1/2

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Half-Life Does Not Mean Half the Sample Disappears

When radioactive nuclei decay, they transform into other nuclei and emit radiation.

Therefore, saying:

"half the sample disappears"

is usually inaccurate.

Instead:

half of the original radioactive parent nuclei have decayed.

The daughter nuclei produced by the decay may still remain physically present in the sample.

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Half-Life and Random Decay

Half-life does not tell us exactly when a particular nucleus will decay.

Suppose an isotope has a half-life of:

20 years

This does not mean every nucleus survives exactly 20 years and then decays.

Some nuclei may decay:

• almost immediately
• after 5 years
• after 19 years
• after 40 years
• after much longer

However, in a very large population, approximately half of the original radioactive nuclei are expected to remain after one half-life.

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Half-Life Describes a Population

Consider one radioactive nucleus.

We cannot predict when it will decay.

Now consider:

10²⁰ radioactive nuclei

We still cannot predict which individual nuclei will decay.

However, we can predict very accurately how the population will change.

This gives us the important distinction:

individual decay → random

large population → statistically predictable

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Calculating the Fraction Remaining

After n half-lives:

fraction remaining = (1/2)ⁿ

where:

n = number of half-lives

For example, after 3 half-lives:

fraction remaining = (1/2)³

= 1/8

Therefore:

12.5% remains

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Calculating the Number Remaining

The expected number of radioactive nuclei remaining can be calculated using:

N = N₀(1/2)ⁿ

where:

• N = number remaining
• N₀ = initial number
• n = number of half-lives

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Example: Number of Nuclei

A radioactive sample initially contains:

12,800 radioactive nuclei

After one half-life:

12,800 → 6400

After two:

6400 → 3200

After three:

3200 → 1600

Therefore:

1600 nuclei remain after three half-lives.

Using the equation:

N = 12,800(1/2)³

N = 12,800/8

N = 1600

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Example: Four Half-Lives

A sample initially contains:

9600 radioactive nuclei.

After four half-lives:

N = 9600(1/2)⁴

N = 9600/16

N = 600 nuclei

Therefore:

6.25% remains.

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Calculating Mass Remaining

The same method can be used when a question gives the mass of a pure radioactive isotope.

Suppose a sample initially contains:

80 g

of a radioactive isotope.

After one half-life:

80 g → 40 g

After two:

40 g → 20 g

After three:

20 g → 10 g

After four:

10 g → 5 g

Therefore, after four half-lives:

5 g of the original radioactive isotope remains.

The total physical mass of all decay products is a separate issue; half-life calculations normally track the amount of the radioactive parent isotope.

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Example: Mass Calculation

A radioactive isotope has:

initial mass = 240 mg

After three half-lives:

240 → 120 → 60 → 30

Therefore:

30 mg remains

Alternatively:

m = 240(1/2)³

m = 240/8

m = 30 mg

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Activity

The activity of a radioactive sample is the number of nuclear decays occurring per second.

The SI unit is the:

becquerel (Bq)

1 Bq means:

1 decay per second

Activity also decreases according to the isotope's half-life.

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Activity After Several Half-Lives

Suppose:

initial activity = 6400 Bq

After one half-life:

6400 → 3200 Bq

After two:

3200 → 1600 Bq

After three:

1600 → 800 Bq

After four:

800 → 400 Bq

The same halving pattern applies.

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Activity Equation

After n half-lives:

A = A₀(1/2)ⁿ

where:

• A = activity remaining
• A₀ = initial activity
• n = number of half-lives

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Example: Activity

A radioactive source has an initial activity of:

4800 Bq

After five half-lives:

A = 4800(1/2)⁵

A = 4800/32

A = 150 Bq

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Count Rate and Half-Life

A radiation detector may record count rate rather than activity directly.

If the experimental setup remains unchanged:

corrected count rate ∝ activity

Therefore, corrected count rate also decreases with the same half-life.

For example:

800 counts/min

→ 400 counts/min

→ 200 counts/min

→ 100 counts/min

This represents three half-lives.

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Remember Background Radiation

Detector readings usually include:

source radiation + background radiation

Therefore:

corrected count rate = measured count rate − background count rate

Half-life should normally be determined from the corrected count rate, not from an uncorrected detector reading.

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Example with Background Radiation

Suppose:

background = 30 counts/min

Initial measured rate:

830 counts/min

Corrected initial rate:

830 − 30 = 800 counts/min

Later, measured rate:

430 counts/min

Corrected:

430 − 30 = 400 counts/min

The source count rate has halved.

Therefore, the time between these readings represents:

one half-life

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Half-Life Graphs

Radioactive decay produces an exponential decay curve.

The graph usually has:

x-axis → time

and:

y-axis → number of radioactive nuclei, activity, or corrected count rate

The curve:

• decreases rapidly at first
• becomes progressively less steep
• approaches zero
• does not suddenly reach zero after a particular number of half-lives

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Why the Decay Graph Is Curved

Suppose the sample begins with:

8000 nuclei

During the first half-life:

8000 → 4000

Number decayed:

4000

During the second:

4000 → 2000

Number decayed:

2000

During the third:

2000 → 1000

Number decayed:

1000

The same fraction decays each time:

1/2

but the actual number decaying becomes smaller.

Therefore, the graph is curved rather than a straight line.

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Constant Fraction, Not Constant Amount

This is one of the most important ideas in half-life calculations.

Radioactive decay does not normally mean:

−100 nuclei every hour

or:

−20 Bq every day

Instead, it means a constant fractional decrease over each half-life.

For example:

800 → 400 → 200 → 100 → 50

The decreases are:

−400

−200

−100

−50

The amount lost changes, but the fraction lost remains:

1/2

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Finding Half-Life from a Graph

Suppose a graph begins at:

1000 Bq

First find half:

1000/2 = 500 Bq

Find where the decay curve reaches 500 Bq.

Suppose this occurs at:

8 hours

Therefore:

t₁/₂ = 8 hours

You can check by finding when the activity reaches:

250 Bq.

This should occur approximately another 8 hours later:

16 hours

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A Reliable Graph Method

To determine half-life from a graph:

1. Choose a convenient y-value on the curve.
2. Divide that value by 2.
3. Find both values on the graph.
4. Read their corresponding times.
5. Subtract the times.

For example:

600 Bq occurs at 4 h.

300 Bq occurs at 10 h.

Therefore:

t₁/₂ = 10 − 4

t₁/₂ = 6 h

You do not have to start at t = 0.

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Example: Interpreting a Graph

Suppose a graph gives:

The activity halves every:

5 days

Therefore:

half-life = 5 days

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Finding the Number of Half-Lives

Use:

number of half-lives = elapsed time / half-life

or:

n = t/t₁/₂

Suppose:

elapsed time = 24 hours

half-life = 6 hours

Then:

n = 24/6

n = 4 half-lives

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Example: Combining Time and Decay

An isotope has:

half-life = 4 days

initial mass = 160 mg

How much remains after:

12 days?

First find the number of half-lives:

n = 12/4

n = 3

Then:

160 → 80 → 40 → 20

Therefore:

20 mg remains

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Example: Activity After a Given Time

A source has:

initial activity = 12,000 Bq

half-life = 3 hours

Find its activity after:

9 hours.

Number of half-lives:

n = 9/3

n = 3

Activity:

12,000 → 6000 → 3000 → 1500

Therefore:

A = 1500 Bq

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Finding Elapsed Time

A radioactive sample decreases from:

1600 Bq

to:

100 Bq

The sequence is:

1600 → 800 → 400 → 200 → 100

This is:

4 half-lives

If the half-life is:

6 hours

then:

time = 4 × 6

time = 24 hours

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Finding the Original Quantity

Sometimes the final amount is given.

Suppose:

25 mg remains after 3 half-lives.

Work backwards by doubling:

25 → 50 → 100 → 200

Therefore:

initial mass = 200 mg

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Percentage Remaining

The percentage remaining can be calculated:

percentage remaining = (1/2)ⁿ × 100%

For example, after 3 half-lives:

percentage = (1/2)³ × 100%

= 1/8 × 100%

= 12.5%

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Percentage Decayed

If 12.5% remains, then:

percentage decayed:

= 100% − 12.5%

= 87.5%

Be careful to distinguish between:

percentage remaining

and:

percentage decayed

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Example: Percentage Decayed

After four half-lives:

fraction remaining = 1/16

percentage remaining:

= 6.25%

Therefore:

percentage decayed:

= 100 − 6.25

= 93.75%

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The Exponential Decay Equation

For times that are not exact whole numbers of half-lives, radioactive decay can be described using:

N = N₀e^(−λt)

where:

• N = number remaining
• N₀ = initial number
• λ = decay constant
• t = time

The equivalent activity equation is:

A = A₀e^(−λt)

These equations describe the smooth exponential decay curve.

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Half-Life and Decay Constant

Half-life and decay constant are related:

t₁/₂ = 0.693/λ

or:

λ = 0.693/t₁/₂

where λ is the decay constant.

A large λ corresponds to:

short half-life

A small λ corresponds to:

long half-life

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Example: Decay Constant

An isotope has:

t₁/₂ = 8 hours

Then:

λ = 0.693/8

λ ≈ 0.0866 h⁻¹

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Comparing Isotopes

Different radioactive isotopes can have dramatically different half-lives.

Some decay extremely rapidly.

Others decay over:

• days
• years
• thousands of years
• millions or billions of years

Half-life is a characteristic property of a particular radioactive isotope.

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Short Half-Life

An isotope with a short half-life has a relatively high probability of decay per unit time.

Therefore, a sample typically:

• loses activity rapidly
• produces many decays per unit time relative to the number of nuclei present
• becomes much less active relatively quickly

For the same number of radioactive nuclei, a shorter half-life generally corresponds to greater activity.

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Long Half-Life

An isotope with a long half-life has a lower decay probability per unit time.

Therefore:

• individual nuclei are statistically more likely to survive longer
• activity decreases slowly
• radioactive material may persist for long periods

A long half-life does not mean the radiation itself is more penetrating.

Penetrating power depends mainly on the type and energy of radiation, not simply the half-life.

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Comparing Two Isotopes

Suppose:

Isotope A: t₁/₂ = 2 hours

Isotope B: t₁/₂ = 20 hours

After 20 hours:

Isotope A has experienced:

20/2 = 10 half-lives

Fraction remaining:

1/1024

Isotope B has experienced:

20/20 = 1 half-life

Fraction remaining:

1/2

Therefore, isotope A decreases much more rapidly.

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Comparing Decay Curves

On a graph:

Short half-life

The curve:

• falls steeply
• reaches half its initial value quickly

Long half-life

The curve:

• decreases more gradually
• takes longer to reach half its initial value

If two samples start with the same number of radioactive nuclei, the steeper decay curve represents the isotope with the:

shorter half-life

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Half-Life and Activity

Activity is:

A = λN

where:

• A = activity
• λ = decay constant
• N = number of undecayed nuclei

Suppose two samples contain the same number of radioactive nuclei.

The isotope with the larger decay constant has:

• greater activity
• shorter half-life

The isotope with the smaller decay constant has:

• lower activity
• longer half-life

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Half-Life Does Not Determine Radiation Type

An isotope's half-life does not tell us whether it emits:

• alpha
• beta
• gamma

For example, two isotopes may have similar half-lives but emit different forms or energies of radiation.

Half-life describes:

how rapidly a radioactive population decays

Radiation type describes:

what is emitted during the decay

These are different properties.

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Half-Life Does Not Determine Penetration

A common misconception is:

long half-life = more penetrating radiation

This is incorrect.

Penetrating ability depends on radiation type and energy.

For example:

• alpha has low penetration
• beta has moderate penetration
• gamma has high penetration

regardless of whether the particular isotope has a relatively short or long half-life.

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Why Different Half-Lives Are Useful

Different applications require different half-lives.

An isotope used as a medical tracer may need to remain radioactive long enough for:

• preparation
• administration
• imaging

but ideally not remain unnecessarily radioactive for a very long time.

By contrast, dating ancient materials requires isotopes with half-lives appropriate to the timescale being measured.

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Carbon-14

Carbon-14 undergoes beta-minus decay.

Its half-life is approximately:

5730 years

This makes it useful for estimating the age of certain once-living materials over archaeological timescales.

The remaining carbon-14 decreases predictably over time.

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Carbon-14 Example

Suppose a sample originally had a certain carbon-14 activity.

After:

5730 years

approximately:

1/2 remains

After:

11,460 years

approximately:

1/4 remains

After:

17,190 years

approximately:

1/8 remains

This predictable decrease can be used in radiometric dating.

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Long-Lived Isotopes and Geological Dating

Very long-lived radioactive isotopes can be useful for investigating geological timescales.

For example, uranium isotopes can have half-lives measured in hundreds of millions or billions of years.

Such long half-lives allow radioactive decay to function as a natural clock over Earth's geological history.

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Half-Life in Medicine

Radioactive isotopes used in medicine are selected partly according to half-life.

An appropriate isotope should often:

• remain active long enough to perform its function
• decay sufficiently quickly to limit prolonged unnecessary exposure
• emit radiation suitable for the intended procedure

Therefore, selecting an isotope involves balancing:

half-life + radiation type + radiation energy + chemical behaviour

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Half-Life and Nuclear Waste

Half-life is also important when considering radioactive waste.

Short-lived isotopes may have relatively high activity initially but decrease substantially over shorter timescales.

Long-lived isotopes may remain radioactive for much longer.

Waste management therefore considers:

• half-life
• activity
• radiation type
• radiation energy
• chemical properties
• biological effects

Half-life alone does not determine overall hazard.

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Half-Life Is Not the Same as "Safe Life"

After one half-life, the material has not become non-radioactive.

It still has approximately:

50% of its original radioactive nuclei

After five half-lives:

3.125% remains

Even after many half-lives, some radioactive nuclei can remain.

Therefore, there is no universal rule that a source becomes automatically "safe" after one or two half-lives.

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Radioactive Material Does Not Reach Zero After a Fixed Number of Half-Lives

Mathematically, exponential decay approaches zero.

For example:

100%

→ 50%

→ 25%

→ 12.5%

→ 6.25%

→ 3.125%

→ 1.5625%

The expected fraction becomes progressively smaller.

For actual samples containing a finite number of nuclei, eventually the remaining radioactive nuclei will individually decay, but the ideal exponential model does not suddenly become zero after a fixed number of half-lives.

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Problem-Solving Strategy

When solving half-life problems:

Identify the initial quantity

This could be:

• mass of radioactive isotope
• number of nuclei
• activity
• corrected count rate

Identify the half-life

For example:

t₁/₂ = 4 days

Find the elapsed time

For example:

t = 12 days

Calculate the number of half-lives

n = t/t₁/₂

Halve once for each half-life

or use:

remaining = initial × (1/2)ⁿ

Check your answer

The remaining amount should:

• decrease with time
• halve during each half-life
• never become negative

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Worked Example

A radioactive isotope has:

initial activity = 9600 Bq

half-life = 6 hours

Find the activity after:

24 hours.

Number of half-lives:

n = 24/6

n = 4

Now halve four times:

9600 → 4800 → 2400 → 1200 → 600

Therefore:

A = 600 Bq

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Worked Example: Finding Time

A sample decreases from:

320 mg

to:

20 mg

Sequence:

320 → 160 → 80 → 40 → 20

This represents:

4 half-lives

If:

t₁/₂ = 7 days

then:

t = 4 × 7

t = 28 days

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Worked Example: Finding Half-Life

A source has an initial corrected count rate of:

1600 counts/min

After 18 hours:

200 counts/min

Determine the half-life.

Halving sequence:

1600 → 800 → 400 → 200

This is:

3 half-lives

Therefore:

t₁/₂ = 18/3

t₁/₂ = 6 hours

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Worked Example: Comparing Isotopes

Isotope X has:

t₁/₂ = 4 days

Isotope Y has:

t₁/₂ = 12 days

Both begin with 8000 nuclei.

After 12 days:

Isotope X

Number of half-lives:

12/4 = 3

Remaining:

8000(1/2)³

= 1000 nuclei

Isotope Y

Number of half-lives:

12/12 = 1

Remaining:

8000(1/2)

= 4000 nuclei

Therefore, isotope X decays more rapidly because it has the shorter half-life.

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Common Mistakes

Mistake: Saying half-life is the time for all nuclei to decay.
Half-life is the time for the expected number of undecayed nuclei to decrease by half.

Mistake: Subtracting the same amount during every half-life.
The quantity is multiplied by 1/2, not reduced by a fixed number.

Mistake: Halving the original quantity every time.
Each new half-life halves the remaining quantity.

Mistake: Saying an individual nucleus lives for exactly one half-life.
Individual decay is random.

Mistake: Forgetting to calculate how many half-lives have passed.
Use n = t/t₁/₂.

Mistake: Using uncorrected detector data.
Background radiation should normally be subtracted before determining half-life from count-rate data.

Mistake: Saying a short half-life means low activity.
For equal numbers of radioactive nuclei, a shorter half-life corresponds to a larger decay constant and greater activity.

Mistake: Saying a long half-life means more penetrating radiation.
Half-life and penetrating ability are different properties.

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Key Terms

Half-life (t₁/₂) — time required for the expected number of undecayed nuclei or activity of a radioactive sample to fall to half its previous value.

Radioactive decay — spontaneous transformation of an unstable nucleus.

Parent nucleus — original unstable radioactive nucleus.

Daughter nucleus — nucleus produced by radioactive decay.

Activity — number of nuclear decays occurring each second.

Becquerel (Bq) — SI unit of activity; 1 Bq = 1 decay/s.

Count rate — number of radiation events detected per unit time.

Corrected count rate — detector count rate after background radiation has been subtracted.

Decay constant (λ) — measure of the probability of decay per unit time.

Exponential decay — decrease in which the rate of change depends on the amount remaining.

Background radiation — radiation present in the environment independently of the source being investigated.

Radiometric dating — determination of age using predictable radioactive decay.

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Key Takeaways

• Half-life is the time required for the expected amount of radioactive parent material to decrease to half its previous value.
• The symbol for half-life is t₁/₂.
• Half-life can be applied to number of nuclei, activity, radioactive-isotope mass and corrected count rate.
• After one half-life, 50% remains.
• After two, 25% remains.
• After three, 12.5% remains.
• After four, 6.25% remains.
• Each half-life halves the quantity that is currently remaining.
• The same fraction decays during each half-life, not the same number.
• Individual nuclear decay is random.
• Half-life describes predictable statistical behaviour of large populations.
• Radioactive decay produces an exponential curve.
• The number of half-lives is n = t/t₁/₂.
• The expected remaining quantity can be calculated using N = N₀(1/2)ⁿ.
• Background radiation should be removed when interpreting detector count-rate data.
• Different isotopes have different characteristic half-lives.
• A shorter half-life means more rapid decay.
• A longer half-life means slower decay.
• Half-life does not determine whether radiation is alpha, beta or gamma.
• Half-life does not directly determine penetrating ability.
• The usefulness of a radioactive isotope depends partly on whether its half-life suits the intended application.

Check Your Understanding

1. Define half-life.
2. What percentage of a radioactive sample remains after one half-life?
3. What fraction remains after three half-lives?
4. What percentage remains after four half-lives?
5. Explain why the same number of nuclei does not decay during every half-life.
6. A sample begins with 6400 radioactive nuclei. How many are expected to remain after three half-lives?
7. A radioactive isotope has an initial mass of 160 mg. How much of the parent isotope remains after five half-lives?
8. A source begins with an activity of 8000 Bq. What is its expected activity after four half-lives?
9. An isotope has a half-life of 6 hours. How many half-lives pass in 30 hours?
10. A 320 mg sample has a half-life of 5 days. How much parent isotope remains after 20 days?
11. A source's activity decreases from 2400 Bq to 300 Bq. How many half-lives have passed?
12. If the half-life in Question 11 is 4 hours, how much time has passed?
13. Explain how half-life can be determined from a decay graph.
14. Why is a radioactive-decay graph curved rather than a straight line?
15. Explain why background radiation should be subtracted when finding half-life from detector readings.
16. Isotope A has a half-life of 2 hours and isotope B has a half-life of 20 hours. Which decays more rapidly? Explain.
17. Explain why a longer half-life does not necessarily mean more penetrating radiation.
18. A sample has a half-life of 10 years. Does this mean every nucleus decays after exactly 10 years? Explain.
19. After six half-lives, what fraction and percentage of the original radioactive nuclei are expected to remain?
20. A source has an initial corrected count rate of 6400 counts/min. After 15 hours it has fallen to 800 counts/min. Determine its half-life.