Linking half-life to activity

Definition

Half-life

The half-life of a radioactive isotope is the time taken for half the undecayed nuclei in a sample to decay, or for the activity of a source to fall by half.

Definition

Activity

The activity of a radioactive source is the number of nuclei that decay each second.

The half-life of an isotope tells you what fraction of the undecayed nuclei in a sample decay in a given time, and it is fixed for that isotope.
Take two samples containing the same number of undecayed nuclei: the one with the shorter half-life has a larger fraction of its nuclei decaying each second, so it has the greater activity.
A greater activity means more ionising particles are emitted each second, so more of them pass through anyone standing nearby and more ionisation happens in their tissue.
Activity halves every half-life, so after $n$ half-lives the activity is $A=A0(12)nA=A_{0}\left(\dfrac{1}{2}\right)^{n}$ , where $A_{0}$ is the starting activity.
The number of half-lives that have passed in a time $t$ is $n=tT1/2n=\dfrac{t}{T_{1/2}}$ , where $T_{1/2}$ is the half-life.
Half-life therefore controls two different things at once: how intense the hazard is at the moment, and how long the hazard lasts.

Key Idea

A short half-life tends to mean an intense hazard for a short time.
A long half-life tends to mean a weaker hazard that persists for years or centuries.

Sources with a short half-life

A short half-life source emits a large number of ionising particles each second while it is fresh, so anyone close to it receives a large exposure in a short time.
The immediate hazard is therefore high, and a few minutes of careless handling can matter more than a whole day near a weaker source.
The activity then falls quickly, because each half-life removes half of the nuclei that are left.
This means the hazard has a short lifetime, so the material can simply be locked away and left to decay until its activity is close to background.
A short half-life is an advantage wherever the source has to stop emitting soon after it has done its job, and it is a disadvantage wherever a steady output is needed for years.
It also means the source has to be made shortly before it is used, because a great deal of it decays away while it is being transported and stored.

Sources with a long half-life

A long half-life source has only a small fraction of its nuclei decaying each second, so for a given number of nuclei its activity, and the immediate hazard, is lower.
That lower activity is spread out over a very long time, so the material is still emitting ionising radiation decades, centuries or thousands of years later.
The hazard cannot be waited out, so it has to be contained instead.
If long half-life material escapes into soil, water or the food chain, that land or water stays hazardous long after the accident that released it.
The responsibility for that containment also outlasts the people who created the material, so records and warnings have to be kept for future generations.
A long half-life is an advantage where a source must give an almost unchanging output for years, such as an industrial gauge that would otherwise need constant recalibration.

Example

Comparing two sources

Source A has a half-life of $hours6\ \text{hours}$ and an activity of $Bq4800\ \text{Bq}$ when it is delivered, and source B has a half-life of $years30\ \text{years}$ and the same starting activity.
For source A after one day, the number of half-lives is $n=tT1/2=246=4n=\dfrac{t}{T_{1/2}}=\dfrac{24}{6}=4$ .
Substituting into $A=A0(12)nA=A_{0}\left(\dfrac{1}{2}\right)^{n}$ gives $A=4800×(12)4A=4800\times\left(\dfrac{1}{2}\right)^{4}$ .
This works out as $BqA=4800\times\dfrac{1}{16}=300\ \text{Bq}$ .
Source B loses almost nothing in a day, and even after $years30\ \text{years}$ it is still emitting at $Bq2400\ \text{Bq}$ .
Both sources are equally hazardous on the day they arrive, but source A can be locked in a store and left to decay, whereas source B needs sealed containment and monitoring for the rest of the century.

Matching precautions to half-life

Because half-life sets both the intensity and the duration of the hazard, the precautions that matter most are different for the two kinds of source.
For a short half-life source, the priority is protecting people during the few hours or days when the activity is high.
1. Plan the work in advance so the source is out of its container for the shortest possible time.
2. Handle it remotely with tongs and keep the shielding in place while it is fresh.
3. Store it in a shielded container and measure its activity before deciding it is safe to dispose of as ordinary waste.
For a long half-life source, the priority is stopping the material escaping at any point over a very long period.
1. Seal it inside a corrosion-resistant container that will not fail as it ages.
2. Label the container clearly and keep written records of what it holds and where it is.
3. Store it in a secure site with restricted access, often deep underground in stable rock.
4. Monitor the site and the groundwater around it for as long as the material remains active.
Shielding is chosen by the type of radiation rather than by the half-life, so a long-lived gamma emitter still needs lead or concrete and a short-lived beta emitter still needs only a few millimetres of aluminium.

Exam technique

Answering half-life hazard questions

Decide first whether the question is about how dangerous the source is now or about how long it stays dangerous, because half-life answers those two questions in opposite directions.
Write the chain in full: short half-life, so a large fraction of nuclei decay each second, so the activity is high, so the exposure in a given time is large.
In a calculation, work out $n=tT1/2n=\dfrac{t}{T_{1/2}}$ first and show it, because the number of half-lives is usually a separate mark from the final activity.
When two sources are compared, say something about both of them; an answer that only describes the short-lived one cannot reach full marks.
Finish a choice question with a judgement that names the property, for example that the isotope with the shorter half-life is chosen because its activity falls to a safe level within days.

Common Mistake

Do not say a source is safe after one half-life; half of the original nuclei are still there and the activity is still half of what it was.
Do not say the activity reaches zero, because each half-life removes only half of what remains.
Do not claim a long half-life is always more dangerous; it lasts longer, but a short half-life source of the same size gives the larger exposure right now.
Do not treat half-life as the time for the whole sample to decay, or as a time that changes when the sample gets older or warmer.
Do not choose shielding from the half-life; the type of radiation emitted decides whether aluminium or lead is needed.

Self review

Explain why two samples with the same number of undecayed nuclei have different activities if their half-lives differ.
A source of half-life $hours6\ \text{hours}$ has an activity of $Bq4800\ \text{Bq}$ ; calculate its activity one day later.
Describe the main hazard created by a source with a very long half-life.
State two precautions that matter most for a short half-life source and explain why.
State two precautions that matter most for a long half-life source and explain why.
Explain why half-life does not decide which shielding material is used.

Linking half-life to activity

Definition

Half-life

The half-life of a radioactive isotope is the time taken for half the undecayed nuclei in a sample to decay, or for the activity of a source to fall by half.

Definition

Activity

The activity of a radioactive source is the number of nuclei that decay each second.

The half-life of an isotope tells you what fraction of the undecayed nuclei in a sample decay in a given time, and it is fixed for that isotope.
Take two samples containing the same number of undecayed nuclei: the one with the shorter half-life has a larger fraction of its nuclei decaying each second, so it has the greater activity.
A greater activity means more ionising particles are emitted each second, so more of them pass through anyone standing nearby and more ionisation happens in their tissue.
Activity halves every half-life, so after $n$ half-lives the activity is $A=A0(12)nA=A_{0}\left(\dfrac{1}{2}\right)^{n}$ , where $A_{0}$ is the starting activity.
The number of half-lives that have passed in a time $t$ is $n=tT1/2n=\dfrac{t}{T_{1/2}}$ , where $T_{1/2}$ is the half-life.
Half-life therefore controls two different things at once: how intense the hazard is at the moment, and how long the hazard lasts.

Key Idea

A short half-life tends to mean an intense hazard for a short time.
A long half-life tends to mean a weaker hazard that persists for years or centuries.

Sources with a short half-life

A short half-life source emits a large number of ionising particles each second while it is fresh, so anyone close to it receives a large exposure in a short time.
The immediate hazard is therefore high, and a few minutes of careless handling can matter more than a whole day near a weaker source.
The activity then falls quickly, because each half-life removes half of the nuclei that are left.
This means the hazard has a short lifetime, so the material can simply be locked away and left to decay until its activity is close to background.
A short half-life is an advantage wherever the source has to stop emitting soon after it has done its job, and it is a disadvantage wherever a steady output is needed for years.
It also means the source has to be made shortly before it is used, because a great deal of it decays away while it is being transported and stored.

Sources with a long half-life

A long half-life source has only a small fraction of its nuclei decaying each second, so for a given number of nuclei its activity, and the immediate hazard, is lower.
That lower activity is spread out over a very long time, so the material is still emitting ionising radiation decades, centuries or thousands of years later.
The hazard cannot be waited out, so it has to be contained instead.
If long half-life material escapes into soil, water or the food chain, that land or water stays hazardous long after the accident that released it.
The responsibility for that containment also outlasts the people who created the material, so records and warnings have to be kept for future generations.
A long half-life is an advantage where a source must give an almost unchanging output for years, such as an industrial gauge that would otherwise need constant recalibration.

Example

Comparing two sources

Source A has a half-life of $hours6\ \text{hours}$ and an activity of $Bq4800\ \text{Bq}$ when it is delivered, and source B has a half-life of $years30\ \text{years}$ and the same starting activity.
For source A after one day, the number of half-lives is $n=tT1/2=246=4n=\dfrac{t}{T_{1/2}}=\dfrac{24}{6}=4$ .
Substituting into $A=A0(12)nA=A_{0}\left(\dfrac{1}{2}\right)^{n}$ gives $A=4800×(12)4A=4800\times\left(\dfrac{1}{2}\right)^{4}$ .
This works out as $BqA=4800\times\dfrac{1}{16}=300\ \text{Bq}$ .
Source B loses almost nothing in a day, and even after $years30\ \text{years}$ it is still emitting at $Bq2400\ \text{Bq}$ .
Both sources are equally hazardous on the day they arrive, but source A can be locked in a store and left to decay, whereas source B needs sealed containment and monitoring for the rest of the century.

Matching precautions to half-life

Because half-life sets both the intensity and the duration of the hazard, the precautions that matter most are different for the two kinds of source.
For a short half-life source, the priority is protecting people during the few hours or days when the activity is high.
1. Plan the work in advance so the source is out of its container for the shortest possible time.
2. Handle it remotely with tongs and keep the shielding in place while it is fresh.
3. Store it in a shielded container and measure its activity before deciding it is safe to dispose of as ordinary waste.
For a long half-life source, the priority is stopping the material escaping at any point over a very long period.
1. Seal it inside a corrosion-resistant container that will not fail as it ages.
2. Label the container clearly and keep written records of what it holds and where it is.
3. Store it in a secure site with restricted access, often deep underground in stable rock.
4. Monitor the site and the groundwater around it for as long as the material remains active.
Shielding is chosen by the type of radiation rather than by the half-life, so a long-lived gamma emitter still needs lead or concrete and a short-lived beta emitter still needs only a few millimetres of aluminium.

Exam technique

Answering half-life hazard questions

Decide first whether the question is about how dangerous the source is now or about how long it stays dangerous, because half-life answers those two questions in opposite directions.
Write the chain in full: short half-life, so a large fraction of nuclei decay each second, so the activity is high, so the exposure in a given time is large.
In a calculation, work out $n=tT1/2n=\dfrac{t}{T_{1/2}}$ first and show it, because the number of half-lives is usually a separate mark from the final activity.
When two sources are compared, say something about both of them; an answer that only describes the short-lived one cannot reach full marks.
Finish a choice question with a judgement that names the property, for example that the isotope with the shorter half-life is chosen because its activity falls to a safe level within days.

Common Mistake

Do not say a source is safe after one half-life; half of the original nuclei are still there and the activity is still half of what it was.
Do not say the activity reaches zero, because each half-life removes only half of what remains.
Do not claim a long half-life is always more dangerous; it lasts longer, but a short half-life source of the same size gives the larger exposure right now.
Do not treat half-life as the time for the whole sample to decay, or as a time that changes when the sample gets older or warmer.
Do not choose shielding from the half-life; the type of radiation emitted decides whether aluminium or lead is needed.

Self review

Explain why two samples with the same number of undecayed nuclei have different activities if their half-lives differ.
A source of half-life $hours6\ \text{hours}$ has an activity of $Bq4800\ \text{Bq}$ ; calculate its activity one day later.
Describe the main hazard created by a source with a very long half-life.
State two precautions that matter most for a short half-life source and explain why.
State two precautions that matter most for a long half-life source and explain why.
Explain why half-life does not decide which shielding material is used.

6.5.3 Half-life and radiation dangers

Linking half-life to activity

Sources with a short half-life

Sources with a long half-life

Comparing two sources

Matching precautions to half-life

Answering half-life hazard questions

Review 6.5.3 Half-life and radiation dangers by teaching Genie

6.5.3 Half-life and radiation dangers

Linking half-life to activity

Sources with a short half-life

Sources with a long half-life

Comparing two sources

Matching precautions to half-life

Answering half-life hazard questions

Review 6.5.3 Half-life and radiation dangers by teaching Genie

6.5.3 Half-life and radiation dangers

Linking half-life to activity

Sources with a short half-life

Sources with a long half-life

Comparing two sources

Matching precautions to half-life

Answering half-life hazard questions

Review 6.5.3 Half-life and radiation dangers by teaching Genie

6.5.3 Half-life and radiation dangers Revision Guide

6.5.3 Half-life and radiation dangers

Linking half-life to activity

Sources with a short half-life

Sources with a long half-life

Comparing two sources

Matching precautions to half-life

Answering half-life hazard questions

Review 6.5.3 Half-life and radiation dangers by teaching Genie

6.5.3 Half-life and radiation dangers Revision Guide