4. Activity and Decay Rate

Activity and Decay Rate

Radioactive activity describes how rapidly radioactive nuclei are decaying in a sample.

Activity is defined as:

the number of nuclear decays occurring per unit time

The symbol commonly used for activity is:

A

The SI unit of activity is the:

becquerel (Bq)

1 Bq = 1 decay per second

Therefore, a radioactive source with an activity of:

500 Bq

undergoes, on average:

500 nuclear decays every second.

---

Activity Is a Rate

Activity tells us how quickly nuclear transformations are occurring.

For example:

Source A

Activity = 100 Bq

This means:

100 decays/s

Source B

Activity = 5000 Bq

This means:

5000 decays/s

Source B therefore has a much greater decay rate.

---

Calculating Activity from Decays

For a short interval over which the activity is approximately constant, the average activity can be calculated:

A ≈ number of decays / time

For example, suppose:

2400 nuclear decays occur during 8 seconds.

Average activity:

A = 2400/8

A = 300 Bq

---

Example

A radioactive sample undergoes:

18,000 decays

during:

60 seconds.

Average activity:

A = 18,000/60

A = 300 Bq

---

Becquerel

The becquerel is named after French physicist Henri Becquerel, whose work contributed to the discovery of radioactivity.

Because:

1 Bq = 1 decay/s

we can write:

500 Bq = 500 decays/s

10,000 Bq = 10,000 decays/s

2.5 MBq = 2.5 × 10⁶ decays/s

---

Activity and Number of Radioactive Nuclei

Activity depends on:

• the number of radioactive nuclei present
• the probability that each nucleus will decay

The relationship is:

A = λN

where:

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

This equation is fundamental to radioactive decay.

---

Understanding A = λN

Suppose two samples contain the same radioactive isotope.

Because they contain the same isotope, they have the same:

decay constant λ

If Sample A contains twice as many radioactive nuclei as Sample B, then:

A_A = 2A_B

Therefore:

more radioactive nuclei → greater activity

for samples of the same isotope.

---

The Decay Constant

The decay constant, λ, describes the probability of decay per unit time for a radioactive nucleus.

A large decay constant means:

• greater probability of decay per unit time
• faster decay
• greater activity for the same N
• shorter half-life

A small decay constant means:

• smaller probability of decay per unit time
• slower decay
• lower activity for the same N
• longer half-life

---

Activity Changes with Time

As radioactive nuclei decay:

N decreases

Since:

A = λN

activity also decreases.

Therefore:

radioactive activity decreases with time

for a sample in which the radioactive parent isotope is not being replenished.

---

Example

Suppose a source initially has:

A = 8000 Bq

After one half-life:

A = 4000 Bq

After two half-lives:

A = 2000 Bq

After three:

A = 1000 Bq

After four:

A = 500 Bq

Activity follows the same halving pattern as the number of radioactive parent nuclei.

---

Activity and Half-Life

The half-life of an isotope is the time required for its activity to decrease to half its previous value.

For example, suppose:

t₁/₂ = 6 hours

and:

A₀ = 6400 Bq

Then:

Every 6 hours, the activity halves.

---

Activity After Several Half-Lives

After n half-lives:

A = A₀(1/2)ⁿ

where:

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

---

Example: Activity Calculation

A radioactive source initially has an activity of:

12,800 Bq.

Find its activity after four half-lives.

A = 12,800(1/2)⁴

A = 12,800/16

A = 800 Bq

---

Finding the Number of Half-Lives

Use:

n = t/t₁/₂

For example:

half-life = 5 days

elapsed time = 20 days

n = 20/5

n = 4

If initial activity is:

6400 Bq

then:

6400 → 3200 → 1600 → 800 → 400

Therefore:

A = 400 Bq

---

Worked Example

A radioactive isotope has:

initial activity = 9600 Bq

half-life = 4 hours

elapsed time = 12 hours

First:

n = 12/4

n = 3

Then:

A = 9600(1/2)³

A = 9600/8

A = 1200 Bq

---

Activity and Half-Life Are Related

The half-life and decay constant are related:

t₁/₂ = 0.693/λ

Therefore:

λ = 0.693/t₁/₂

This explains the relationship between half-life and activity.

For the same number of radioactive nuclei:

short half-life → large λ → high activity

long half-life → small λ → low activity

---

Comparing Two Isotopes

Suppose two samples each contain the same number of radioactive nuclei.

Isotope X

half-life = 2 hours

Isotope Y

half-life = 20 years

Isotope X has a much larger decay constant.

Therefore, initially:

Isotope X has the greater activity.

Its nuclei are decaying more rapidly.

Isotope Y decays much more slowly.

---

Important Qualification

A short half-life does not automatically mean that any sample of that isotope has greater activity than any sample of a long-lived isotope.

Activity depends on both:

λ and N

because:

A = λN

A very small amount of a short-lived isotope could have less total activity than a very large amount of another isotope.

Always consider both:

• decay probability
• number of radioactive nuclei

---

Activity and Exponential Decay

Activity follows an exponential decay relationship:

A = A₀e^(−λt)

where:

• A = activity at time t
• A₀ = initial activity
• λ = decay constant
• t = elapsed time

This produces an exponential decay curve.

---

Interpreting an Activity Graph

An activity graph normally has:

x-axis → time

y-axis → activity (Bq)

The curve:

• begins at the initial activity
• decreases rapidly at first
• becomes progressively less steep
• approaches zero

The graph is not a straight line because the same fraction, rather than the same amount, decreases during each half-life.

---

Why the Graph Becomes Less Steep

Suppose:

A₀ = 8000 Bq

During the first half-life:

8000 → 4000 Bq

Decrease:

4000 Bq

During the second:

4000 → 2000 Bq

Decrease:

2000 Bq

During the third:

2000 → 1000 Bq

Decrease:

1000 Bq

The activity always halves, but the absolute decrease becomes smaller.

Therefore, the graph gradually becomes less steep.

---

Finding Half-Life from an Activity Graph

Suppose an activity graph begins at:

1200 Bq.

Half of this is:

600 Bq.

Find the time when the curve reaches:

600 Bq.

Suppose this occurs after:

7 hours.

Then:

t₁/₂ = 7 hours

You can confirm this by checking when the activity reaches:

300 Bq.

Ideally, this should occur approximately another:

7 hours later.

---

You Do Not Have to Start at Zero

Suppose an activity graph shows:

800 Bq at 5 hours

and:

400 Bq at 11 hours.

The activity has halved.

Therefore:

t₁/₂ = 11 − 5

t₁/₂ = 6 hours

This method is useful when the initial part of a graph is difficult to read.

---

Comparing Activity Curves

Suppose two isotopes begin with the same activity.

Isotope A

short half-life

Isotope B

long half-life

The activity of isotope A falls more rapidly.

Its graph is:

steeper

The activity of isotope B falls more slowly.

Its graph is:

less steep

---

Finding Original Activity

The equation:

A = A₀(1/2)ⁿ

can be rearranged:

A₀ = A × 2ⁿ

Suppose:

A = 250 Bq

after:

4 half-lives.

Then:

A₀ = 250 × 2⁴

A₀ = 250 × 16

A₀ = 4000 Bq

---

Finding Elapsed Time

Suppose activity decreases from:

6400 Bq

to:

800 Bq.

Halve:

6400 → 3200 → 1600 → 800

Therefore:

3 half-lives

If:

t₁/₂ = 5 hours

then:

t = 3 × 5

t = 15 hours

---

Finding Half-Life

Suppose activity decreases from:

4800 Bq

to:

600 Bq

during:

24 hours.

Count the halvings:

4800 → 2400 → 1200 → 600

This represents:

3 half-lives

Therefore:

t₁/₂ = 24/3

t₁/₂ = 8 hours

---

Fractional Half-Lives

Activity can also be calculated when the elapsed time is not an exact number of half-lives.

Use:

A = A₀(1/2)^(t/t₁/₂)

For example:

A₀ = 2000 Bq

t₁/₂ = 8 h

t = 20 h

Then:

A = 2000(1/2)^(20/8)

A = 2000(1/2)^2.5

A ≈ 2000 × 0.1768

A ≈ 354 Bq

---

Activity vs Count Rate

Activity and count rate are related, but they are not the same quantity.

Activity

Number of nuclear decays occurring in the source each second.

Measured in:

Bq

Count rate

Number of radiation events recorded by a detector per unit time.

Often measured in:

• counts/s
• counts/min

A detector usually records only a fraction of all the decays occurring in the source.

---

Why Count Rate Is Lower Than Activity

Suppose:

source activity = 10,000 Bq

This means:

10,000 decays/s.

However, the detector might record:

500 counts/s.

This can happen because:

• radiation travels in many directions
• some radiation is absorbed
• some radiation misses the detector
• the detector is not 100% efficient

Therefore:

count rate ≠ activity

unless detector efficiency and geometry are appropriately accounted for.

---

Background Radiation

A detector records environmental radiation even when the radioactive source being studied is absent.

This is called:

background radiation

Therefore:

corrected count rate = measured count rate − background count rate

Corrected count rate can then be used as an experimental indicator of the source's activity when the detector geometry and efficiency remain unchanged.

---

Example with Background Radiation

Suppose:

measured count rate = 720 counts/min

background = 40 counts/min

Then:

corrected count rate:

= 720 − 40

= 680 counts/min

---

Activity and Count Rate Have the Same Half-Life

If the detector setup remains unchanged:

corrected count rate ∝ activity

Therefore, as activity halves:

the corrected count rate also halves.

Suppose:

corrected count rate = 800 counts/min

After one half-life:

400 counts/min

After two:

200 counts/min

After three:

100 counts/min

---

Background Radiation and Raw Detector Readings

Suppose:

background = 50 counts/min

Initial corrected source count:

800 counts/min

The detector initially reads:

800 + 50 = 850 counts/min

After one half-life, the source contributes:

400 counts/min.

But the detector reads:

400 + 50 = 450 counts/min

Notice:

850/2 = 425

not 450.

Therefore:

the raw detector reading does not halve

because background remains.

The background-corrected count rate halves.

---

Worked Detector Example

A detector initially measures:

1640 counts/min.

Background:

40 counts/min.

Corrected rate:

1640 − 40 = 1600 counts/min

After some time, the detector reads:

240 counts/min.

Corrected:

240 − 40 = 200 counts/min

Now:

1600 → 800 → 400 → 200

This represents:

3 half-lives

If the half-life is:

4 hours

then:

elapsed time = 3 × 4

= 12 hours

---

Activity and Number of Nuclei Example

Suppose:

λ = 0.002 s⁻¹

N = 500,000 nuclei

Use:

A = λN

A = 0.002 × 500,000

A = 1000 Bq

Therefore, the sample undergoes on average:

1000 decays/s

at that moment.

---

Finding Number of Nuclei from Activity

Rearrange:

A = λN

to:

N = A/λ

Suppose:

A = 2400 Bq

λ = 0.004 s⁻¹

Then:

N = 2400/0.004

N = 600,000 nuclei

---

Finding Decay Constant from Activity

Rearrange:

A = λN

to:

λ = A/N

Suppose:

A = 500 Bq

N = 100,000 nuclei

Then:

λ = 500/100,000

λ = 0.005 s⁻¹

---

Activity and Mass

For the same radioactive isotope:

more radioactive material → more radioactive nuclei → greater activity

For example, if two pure samples contain:

10 mg

and:

20 mg

of the same radioactive isotope, then under the same conditions the 20 mg sample contains approximately twice as many radioactive nuclei.

Therefore, its activity is approximately:

twice as large

---

Specific Activity

Scientists sometimes compare radioactive materials using:

specific activity

Specific activity describes activity per unit mass.

For example:

Bq/kg

or:

Bq/g

Isotopes with short half-lives generally have greater specific activity than long-lived isotopes because a larger fraction of their nuclei decay per unit time.

---

Activity Does Not Measure Radiation Energy

A source with:

10,000 Bq

has 10,000 decays/s.

But this does not tell us the energy released by each decay.

Two sources with the same activity may emit:

• different radiation types
• different radiation energies

Therefore:

activity alone does not completely describe radiation hazard.

---

Activity Does Not Directly Measure Danger

The biological risk associated with radioactive material depends on many factors, including:

• activity
• radiation type
• radiation energy
• exposure time
• distance
• shielding
• whether material is outside or inside the body
• which tissues are exposed

Therefore:

higher activity generally means more decays per second, but activity alone does not determine overall risk.

---

Short Half-Life and High Activity

Consider two samples containing the same number of radioactive nuclei.

Sample A

t₁/₂ = 1 hour

Sample B

t₁/₂ = 100 years

Sample A has a much greater decay probability per unit time.

Therefore:

Sample A has much greater activity initially.

But it also loses activity much more rapidly.

This creates an important relationship:

short half-life → high activity but rapid decrease

long half-life → lower activity per nucleus but slow decrease

---

Medical Applications

In nuclear medicine, activity is important because it determines how many radioactive decays occur each second.

Medical professionals must consider:

• appropriate activity
• isotope half-life
• radiation type
• radiation energy
• biological behaviour

An isotope should remain active long enough to perform its intended function without creating unnecessary prolonged exposure.

---

Activity and Radioactive Waste

Activity is also important when managing radioactive waste.

A short-lived isotope may:

• initially have high activity
• lose activity relatively rapidly

A long-lived isotope may:

• have lower activity per nucleus
• remain radioactive for much longer

Waste management therefore considers both:

activity and half-life

along with radiation type and other properties.

---

Reading an Activity Table

Suppose:

The activity halves every:

4 hours

Therefore:

t₁/₂ = 4 hours

After:

20 hours

the expected activity would be:

200/2

= 100 Bq

---

Interpreting Experimental Activity Data

Real experimental data may look more like:

Experimental values do not necessarily halve at convenient measurement times.

Instead, scientists:

• plot the data
• examine the decay curve
• determine the approximate half-life
• use mathematical modelling when greater precision is required

Random radioactive decay causes experimental points to fluctuate around the expected curve.

---

Exponential vs Linear Activity

A linear decrease might look like:

1000 → 900 → 800 → 700 → 600

A radioactive exponential decrease might look like:

1000 → 500 → 250 → 125 → 62.5

The radioactive sequence loses the same:

fraction

during each half-life.

It does not lose the same:

amount

---

Problem-Solving Strategy

When solving activity problems:

Identify the quantity

Is the question giving:

• activity in Bq?
• detector count rate?
• number of nuclei?
• time?

Check for background

If detector data are used:

corrected count = measured count − background

Determine the number of half-lives

Use:

n = t/t₁/₂

Calculate the activity

For whole or fractional half-lives:

A = A₀(1/2)ⁿ

If N and λ are given

Use:

A = λN

Check your units

Activity should normally be expressed in:

Bq

---

Worked Mixed Problem

A radioactive source has:

initial activity = 24,000 Bq

half-life = 6 hours

Find the activity after:

18 hours.

Number of half-lives:

n = 18/6

n = 3

Activity:

24,000 → 12,000 → 6000 → 3000

Therefore:

A = 3000 Bq

---

Worked Mixed Problem: Original Activity

A radioactive source has:

activity = 125 Bq

after:

5 half-lives.

Original activity:

A₀ = 125 × 2⁵

A₀ = 125 × 32

A₀ = 4000 Bq

---

Worked Mixed Problem: Finding Half-Life

A source decreases from:

6400 Bq

to:

400 Bq

during:

20 hours.

Count the halvings:

6400 → 3200 → 1600 → 800 → 400

This is:

4 half-lives

Therefore:

t₁/₂ = 20/4

t₁/₂ = 5 hours

---

Common Mistakes

Mistake: Saying activity is the amount of radioactive material.
Activity is the rate of nuclear decay.

Mistake: Confusing Bq with counts/min.
Bq describes decays occurring in the source; counts/min describes events recorded by a detector.

Mistake: Assuming every decay is detected.
A detector usually records only some of the radiation emitted.

Mistake: Saying activity decreases by the same number each half-life.
Activity decreases by the same fraction.

Mistake: Forgetting background radiation when using detector data.
Use corrected count rate when investigating source decay.

Mistake: Saying a long half-life means high activity.
For equal numbers of nuclei, a shorter half-life corresponds to greater activity.

Mistake: Assuming activity alone determines radiation danger.
Radiation type, energy, exposure and biological factors also matter.

Mistake: Treating an activity graph as linear.
Radioactive activity follows an exponential decay relationship.

---

Key Terms

Activity (A) — number of nuclear decays occurring per unit time.

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

Decay rate — rate at which radioactive nuclei undergo nuclear decay.

Decay constant (λ) — probability-related constant describing radioactive decay per unit time.

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

Radioactive nucleus — unstable atomic nucleus capable of undergoing radioactive decay.

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

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

Background radiation — environmental radiation detected independently of the source being studied.

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

Specific activity — activity per unit mass of radioactive material.

---

Key Takeaways

• Radioactive activity measures the rate of nuclear decay.
• Activity is represented by A.
• The SI unit of activity is the becquerel (Bq).
• 1 Bq = 1 decay/s.
• Activity depends on the number of radioactive nuclei and their decay probability.
• The fundamental relationship is A = λN.
• As radioactive nuclei decay, N decreases and activity decreases.
• Activity follows an exponential decay pattern.
• After each half-life, activity falls to half its previous value.
• Activity after n half-lives is A = A₀(1/2)ⁿ.
• The decay constant and half-life are related by t₁/₂ = 0.693/λ.
• For equal numbers of radioactive nuclei, a shorter half-life corresponds to greater activity.
• Short-lived isotopes lose their activity more rapidly.
• Long-lived isotopes retain their activity for longer.
• Activity is not the same as detector count rate.
• Detector count rate is usually lower than the source's activity.
• Background radiation should be subtracted when interpreting detector measurements.
• Corrected count rate follows the same half-life as source activity when the detector setup remains unchanged.
• Activity alone does not determine the hazard of a radioactive source.
• Activity graphs can be used to determine and compare radioactive half-lives.

Check Your Understanding

1. Define radioactive activity.
2. What is the SI unit of activity?
3. What does an activity of 1 Bq mean?
4. What does an activity of 2500 Bq mean?
5. A sample undergoes 12,000 decays in 4 seconds. Calculate its average activity.
6. State the relationship between activity, decay constant and number of radioactive nuclei.
7. Explain why activity decreases with time.
8. A source has an initial activity of 6400 Bq. What is its activity after three half-lives?
9. A source has an activity of 12,800 Bq and a half-life of 4 hours. Calculate its activity after 16 hours.
10. A source decreases from 8000 Bq to 1000 Bq. How many half-lives have elapsed?
11. If the half-life in Question 10 is 6 hours, determine the elapsed time.
12. A source has an activity of 250 Bq after four half-lives. Calculate its original activity.
13. Explain the difference between activity and count rate.
14. Why might a 10,000 Bq source produce a detector reading much lower than 10,000 counts/s?
15. A detector measures 650 counts/min while background radiation is 50 counts/min. Calculate the corrected count rate.
16. Explain why the raw detector reading may not halve during one half-life when background radiation is present.
17. Two samples contain the same number of radioactive nuclei. Sample A has a half-life of 2 hours and Sample B has a half-life of 20 hours. Which initially has the greater activity? Explain.
18. Explain why an activity-versus-time graph is curved rather than a straight line.
19. A sample contains 500,000 radioactive nuclei and has λ = 0.004 s⁻¹. Calculate its activity.
20. Explain why activity alone cannot be used to determine how dangerous a radioactive source is.