1. Random Nature of Radioactive Decay

Random Nature of Radioactive Decay

Radioactive decay is a spontaneous and random nuclear process. An unstable nucleus can decay by emitting radiation, but it is impossible to predict exactly which individual nucleus will decay next or precisely when it will decay.

This creates an important idea in nuclear physics:

Individual decays are random, but very large populations of radioactive nuclei behave predictably.

This is why scientists can accurately describe quantities such as half-life even though individual radioactive decays cannot be predicted.

 

 

 

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What Does Random Mean?

In radioactive decay, random means that we cannot predict the exact moment when a particular unstable nucleus will decay.

Imagine three identical radioactive nuclei:

A, B and C.

We may know that each has the same probability of decaying during the next minute.

However, we cannot predict:

• whether A will decay
• whether B will decay
• whether C will decay
• which will decay first
• exactly when any one will decay

The decay of an individual nucleus is fundamentally probabilistic.

Radioactive Decay Is Spontaneous

Radioactive decay is also described as:

spontaneous

This means that an unstable nucleus can decay without an external trigger.

Under ordinary conditions, nuclear radioactive decay is generally not caused by:

• heating the sample
• cooling the sample
• increasing pressure
• changing its chemical state
• shining ordinary light on it

The instability is a property of the nucleus itself.

Decay Probability

Each unstable nucleus has a certain probability of decaying per unit time.

This probability is described using the:

decay constant

symbol:

λ

A larger decay constant means a nucleus has a greater probability of decaying during a given interval of time.

A smaller decay constant means the nucleus is more likely to survive for longer.

The Decay Constant

For a very short time interval Δt, the approximate probability that a particular nucleus decays is:

P ≈ λΔt

where:

• P = probability of decay
• λ = decay constant
• Δt = time interval

For example, if:

λ = 0.02 s⁻¹

then over a sufficiently short interval of 1 s, the approximate decay probability is:

P ≈ 0.02 × 1

P ≈ 0.02

or approximately:

2%

This does not tell us whether a particular nucleus actually will decay during that second.

It tells us the probability.

Probability Does Not Predict an Individual Event

Suppose a nucleus has a 10% probability of decaying during a particular time interval.

That does not mean:

"It will definitely decay after 10 intervals."

Instead, during each interval there is a probability that it will decay.

The nucleus might:

• decay almost immediately
• survive several intervals
• survive for much longer

Probability describes likelihood, not a predetermined schedule.

Radioactive Nuclei Do Not Have Clocks

An important feature of radioactive decay is that a nucleus does not become increasingly "due" to decay simply because it has survived for a long time.

For a given radioactive isotope, a surviving nucleus has the same characteristic decay probability as another surviving nucleus of that isotope.

A nucleus cannot be described as:

"old and almost ready to decay."

This is very different from many everyday processes.

A Useful Analogy: Coin Tossing

Imagine tossing 1000 coins.

For each coin:

• heads → nucleus survives
• tails → nucleus decays

After one round, we expect approximately:

500 coins to remain.

After another round:

approximately 250.

Then:

Then approximately:

62 or 63.

We cannot predict exactly which individual coins will show tails.

But with many coins, we can predict the overall pattern quite well.

This is similar to radioactive decay.

 

 

 

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Large Samples Behave Predictably

A radioactive sample may contain an enormous number of nuclei.

Even though individual nuclei decay randomly, the large population produces a statistically predictable pattern.

Suppose there are:

1,000,000 radioactive nuclei

and each has the same probability of decaying.

We cannot identify which particular nuclei will decay.

However, we can predict approximately how many nuclei will remain after a certain time.

This is an example of how:

random microscopic events can produce predictable macroscopic behaviour.

Small Samples Show More Random Variation

Suppose we have only 10 radioactive nuclei.

After one particular time interval, perhaps:

• 3 decay
• 7 remain

In another identical experiment:

• 6 decay
• 4 remain

The percentage variation can be large.

Now imagine:

1,000,000 nuclei.

Random fluctuations still occur, but they represent a much smaller fraction of the total.

Therefore:

larger samples show smoother and more predictable decay trends.

Random Counts in Radiation Experiments

The random nature of radioactive decay can be observed using a radiation detector.

Suppose a Geiger-Müller counter measures a source for one minute several times.

The readings might be:

248 counts

259 counts

241 counts

253 counts

246 counts

The readings are similar but not identical.

This variation is expected because radioactive decay is random.

Random Variation Does Not Mean Experimental Error

Students sometimes assume different count readings mean something went wrong.

For example:

Measurement 1 = 312 counts/min

Measurement 2 = 298 counts/min

Measurement 3 = 307 counts/min

Measurement 4 = 321 counts/min

These differences can occur even when:

• the source remains fixed
• the detector remains fixed
• the counting time remains fixed
• the equipment works correctly

The variation can arise naturally from the statistical nature of radioactive decay.

Predictable Half-Life

Although individual decay events are random, the half-life of a large radioactive sample is predictable.

The half-life is:

the time required for the number of undecayed radioactive nuclei in a large sample to fall to half its initial value.

It can also be described as the time for the activity of a large sample to fall to approximately half its initial value.

Successive Half-Lives

Suppose a sample initially contains:

8000 radioactive nuclei

After one half-life:

8000 → 4000

After two half-lives:

4000 → 2000

After three half-lives:

2000 → 1000

After four half-lives:

1000 → 500

The expected pattern is:

Half-Life Does Not Mean All Nuclei Decay at the Same Time

Suppose an isotope has a half-life of:

10 years.

This does not mean each nucleus survives exactly 10 years and then decays.

Instead, after 10 years, we expect approximately half of a sufficiently large initial population to remain.

Some nuclei may decay:

• almost immediately
• after one year
• after nine years
• after 20 years
• after much longer

The half-life describes a population, not a scheduled lifetime for an individual nucleus.

Half-Life and Probability

After one half-life, an individual nucleus has a:

50% probability of surviving

and therefore a:

50% probability of having decayed

over that interval.

If it survives the first half-life, it still has the same probability of surviving the next half-life.

Therefore:

Probability of surviving 2 half-lives:

1/2 × 1/2 = 1/4

After 3:

1/2 × 1/2 × 1/2 = 1/8

After 4:

1/16

Fraction Remaining

After n half-lives:

fraction remaining = (1/2)ⁿ

For example, after 5 half-lives:

fraction remaining = (1/2)⁵

= 1/32

As a percentage:

1/32 × 100

= 3.125%

So approximately 3.125% of the original radioactive nuclei remain.

Example: Number Remaining

A sample initially contains:

6400 radioactive nuclei.

After three half-lives:

First:

6400 → 3200

Second:

3200 → 1600

Third:

1600 → 800

Therefore:

800 nuclei are expected to remain.

Alternatively:

N = 6400(1/2)³

N = 6400/8

N = 800

Exponential Radioactive Decay

Radioactive decay follows an approximately exponential decay law for a large population:

N = N₀e^(−λt)

where:

• N = number of undecayed nuclei remaining
• N₀ = initial number of radioactive nuclei
• λ = decay constant
• t = time

This produces a curved decay graph rather than a straight line.

 

 

 

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Understanding the Decay Graph

A typical radioactive-decay graph has:

x-axis: time

y-axis: number of undecayed nuclei, activity or corrected count rate

The graph falls rapidly at first and then becomes less steep.

However, it never suddenly drops to zero at a particular half-life.

Instead, the expected quantity repeatedly halves.

Why the Graph Is Curved

At the beginning, many radioactive nuclei are present.

Therefore, many nuclei can decay during a given time interval.

Later, fewer radioactive nuclei remain.

Therefore, fewer nuclei decay during the same length of time.

For example:

8000 → 4000

means 4000 nuclei decayed.

Then:

4000 → 2000

means 2000 decayed.

Then:

2000 → 1000

means 1000 decayed.

The same fraction decays, but the actual number decaying becomes smaller.

This produces an exponential curve.

Constant Fraction, Not Constant Number

This distinction is extremely important.

Radioactive decay does not normally lose the same number of nuclei during each equal time interval.

Instead, the same expected fraction decays over each half-life.

Example:

Initial = 1600

After one half-life:

1600 − 800 = 800 decay

Next:

800 − 400 = 400 decay

Next:

400 − 200 = 200 decay

The number decaying changes.

The fraction remains:

1/2

Reading Half-Life from a Graph

Suppose a graph begins at:

800 counts/min

Find half:

800/2 = 400 counts/min

Find where the curve reaches 400 counts/min.

Suppose this occurs at:

6 hours.

Then:

half-life = 6 hours

You can check again:

400 → 200 counts/min

should take approximately another:

6 hours

for ideal decay data.

Example: Reading a Decay Graph

Suppose a radioactive sample has:

t = 0 h → 1200 counts/min

t = 4 h → 600 counts/min

t = 8 h → 300 counts/min

t = 12 h → 150 counts/min

The count rate halves every:

4 hours

Therefore:

half-life = 4 hours

Activity and Radioactive Decay

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

Activity is measured in:

becquerels (Bq)

1 Bq = 1 decay/s

Activity is related to the number of undecayed nuclei:

A = λN

Therefore, as N decreases:

activity also decreases

This means activity follows the same exponential pattern.

Activity Decay Equation

Because:

A = λN

activity can be written:

A = A₀e^(−λt)

where:

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

Therefore, activity has the same half-life as the number of radioactive nuclei.

Count Rate and Half-Life

A radiation detector may be used to estimate radioactive decay.

If detector geometry remains unchanged:

corrected count rate ∝ activity

Therefore, corrected count rate can also show an exponential decrease.

However, background radiation must first be considered.

Background Radiation on Decay Graphs

Suppose the detector records:

source radiation + background radiation.

As the radioactive source becomes weaker, the measured count rate approaches the background level.

If background is not subtracted, the graph may appear to level off above zero.

 

 

 

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Correcting Count Rate

Use:

corrected count rate = measured count rate − background count rate

Suppose:

Measured rate = 340 counts/min

Background = 40 counts/min

Corrected:

340 − 40 = 300 counts/min

Half of this is:

150 counts/min

To find the half-life from a graph, we should look for when the corrected rate reaches 150 counts/min.

Example with Background Radiation

Suppose:

background = 20 counts/min

At t = 0:

measured = 820 counts/min

Corrected:

820 − 20 = 800 counts/min

After 5 hours:

measured = 420 counts/min

Corrected:

420 − 20 = 400 counts/min

The corrected count rate has halved.

Therefore:

half-life = 5 hours

Decay Constant and Half-Life

The decay constant and half-life are related by:

t₁/₂ = ln(2)/λ

Since:

ln(2) ≈ 0.693

we can write:

t₁/₂ = 0.693/λ

or:

λ = 0.693/t₁/₂

A large λ means:

• high decay probability
• rapid decay
• short half-life

A small λ means:

• lower decay probability
• slower decay
• long half-life

Example: Finding Decay Constant

Suppose an isotope has a half-life of:

10 hours.

Then:

λ = 0.693/10

λ = 0.0693 h⁻¹

This decay constant describes the statistical decay probability of the nuclei.

Example: Finding Half-Life

Suppose:

λ = 0.035 min⁻¹

Then:

t₁/₂ = 0.693/0.035

t₁/₂ ≈ 19.8 min

Therefore, the half-life is approximately:

20 minutes

Probability of Survival

The probability that an individual nucleus survives until time t follows:

P_survive = e^(−λt)

The probability that it has decayed by time t is:

P_decay = 1 − e^(−λt)

These equations describe probability for individual nuclei and expected fractions for large populations.

Example: Survival Probability

Suppose:

λ = 0.10 h⁻¹

After:

t = 5 h

P_survive = e^(−0.10 × 5)

= e^(−0.5)

≈ 0.607

Therefore, the nucleus has approximately a:

60.7% probability of surviving

for at least five hours.

Probability of having decayed:

1 − 0.607 = 0.393

or:

39.3%

Why Large Numbers Produce Smooth Trends

Imagine a radioactive sample containing only:

10 nuclei.

One unexpected decay can change the population by:

10%.

Now imagine:

1,000,000 nuclei.

One decay changes the population by only:

0.0001%.

Individual randomness is still present, but its relative effect on the total becomes much smaller.

Therefore:

large populations produce much smoother statistical behaviour.

Simulation Using Dice

Radioactive decay can also be modelled using dice.

Suppose each die represents one radioactive nucleus.

Roll all the dice.

Any die showing:

6

is considered to have decayed.

Remove those dice.

Roll the remaining dice again.

Continue repeating the process.

 

 

 

10

The exact number removed during each round varies.

But when many dice are used, the overall pattern resembles exponential decay.

Why the Dice Model Works

For each roll, every die has the same probability:

1/6

of "decaying."

We cannot predict which particular die will show 6.

However, with 600 dice, we expect approximately:

100

to show 6 during the first roll.

The individual outcomes are random.

The population trend is predictable.

This is the same key principle underlying radioactive decay.

Limitations of the Dice Model

The model is useful but not perfect.

Real nuclei:

• do not literally roll dice
• can have vastly different decay probabilities
• decay continuously in time rather than in classroom "rounds"
• undergo specific nuclear transformations

The simulation represents the probabilistic mathematics, not the physical mechanism.

Predicting a Population vs Predicting a Nucleus

Consider an isotope with a half-life of:

20 years.

For one particular nucleus:

we cannot say:

"This nucleus will decay in exactly 20 years."

For a sample containing billions of nuclei:

we can predict:

"After approximately 20 years, about half will remain."

This distinction is central to radioactive decay.

Does a Nucleus Remember Its Age?

Suppose a radioactive nucleus survives for three half-lives.

If it survives to the beginning of the next half-life, its probability of surviving that next half-life is still:

1/2

It does not "remember" that it has already survived a long time.

Radioactive decay is approximately memoryless for a given unstable state.

A Surprising Example

Suppose an isotope has a half-life of:

100 years.

A particular nucleus has already existed for:

500 years.

Can we say it is about to decay?

No.

If it has survived until now, its probability of surviving another 100 years is still:

50%

Its previous survival does not make immediate decay inevitable.

Interpreting Scatter Around a Decay Curve

Experimental measurements may not fall perfectly on a smooth exponential curve.

Instead, data points might appear:

• slightly above the expected curve
• slightly below the expected curve

This is normal.

The smooth exponential curve represents the:

expected statistical trend

while individual measurements contain random variation.

Short vs Long Counting Times

Suppose a source averages:

120 counts/min.

If we measure for only 5 seconds, the number of detected events may vary considerably.

If we measure for:

10 minutes

we collect many more events.

The average rate is generally more statistically reliable.

Therefore:

longer measurement times reduce relative random fluctuations.

Random Error and Radioactive Counting

For sufficiently large counts N, radioactive counting approximately follows Poisson statistics.

A useful estimate of the statistical uncertainty in a count is:

uncertainty ≈ √N

For example:

N = 10,000

uncertainty ≈ √10,000

= 100

Relative uncertainty:

100/10,000 × 100%

= 1%

Example with a Smaller Count

Suppose:

N = 100

Approximate uncertainty:

√100 = 10

Relative uncertainty:

10/100 × 100%

= 10%

Compare this with 10,000 counts:

relative uncertainty ≈ 1%

This demonstrates why collecting more counts improves statistical precision.

Large Samples and the Law of Large Numbers

The predictable behaviour of radioactive samples is an example of a broader statistical principle.

When a very large number of independent random events occur, the average outcome tends to become increasingly predictable.

Therefore:

individual event → unpredictable

but:

large population → predictable statistical pattern

This idea appears throughout science, including:

• radioactive decay
• gas particle motion
• genetics
• statistical mechanics
• quantum physics

Radioactive Decay and Quantum Physics

At the microscopic level, radioactive decay is a quantum process.

Quantum mechanics predicts the:

probability of decay

rather than a predetermined exact decay time for an individual nucleus.

This is why randomness is not simply caused by poor measuring equipment.

It is a fundamental feature of the physical process as described by modern physics.

Interpreting Decay Graphs

When examining a radioactive-decay graph, ask:

What is on the x-axis?

Usually:

time

What is on the y-axis?

It might be:

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

Is background included?

If using detector data, determine whether background has already been subtracted.

Where does the y-value halve?

Measure the time required for:

N → N/2

or:

A → A/2

or:

corrected count rate → half

That time is the:

half-life

Example: Graph Interpretation

Suppose a decay graph shows:

At 0 days:

1600 Bq

At 3 days:

800 Bq

At 6 days:

400 Bq

At 9 days:

200 Bq

The activity halves every:

3 days

Therefore:

half-life = 3 days

After 12 days:

200 → 100 Bq

Common Mistakes

Mistake: Saying radioactive decay is completely unpredictable.
Individual decay events are unpredictable, but large samples follow predictable statistical patterns.

Mistake: Saying half-life tells us when an individual nucleus will decay.
Half-life describes the statistical behaviour of a large population.

Mistake: Saying all nuclei decay after one half-life.
Approximately half of a large initial population remains after one half-life.

Mistake: Saying the same number of nuclei decay during every half-life.
The same fraction decays, not the same number.

Mistake: Thinking an old nucleus is more likely to decay because it has survived for a long time.
For a given radioactive state, its characteristic decay probability does not increase simply because it has survived longer.

Mistake: Expecting experimental count rates to lie perfectly on a smooth decay curve.
Random statistical variation produces scatter around the expected trend.

Mistake: Forgetting background radiation when interpreting count-rate data.
Background should normally be accounted for before determining the source's decay behaviour.

Key Terms

Radioactive decay — spontaneous transformation of an unstable atomic nucleus accompanied by emission of radiation.

Random — unpredictable for an individual event even though probabilities can be known.

Spontaneous — occurring without requiring an external trigger.

Probability — numerical measure of the likelihood that an event will occur.

Decay constant (λ) — quantity describing the probability of radioactive decay per unit time.

Half-life — time required for the expected number of undecayed nuclei or activity of a large sample to fall to half its initial value.

Activity — number of nuclear decays occurring per second.

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

Exponential decay — process in which a quantity decreases at a rate proportional to the amount remaining.

Background radiation — radiation detected from environmental sources rather than the particular radioactive source being investigated.

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

Statistical fluctuation — random variation around an expected value.

Poisson statistics — probability model often used to describe radioactive counting events.

Key Takeaways

• Radioactive decay is spontaneous and random.
• It is impossible to predict exactly when a particular unstable nucleus will decay.
• Each nucleus has a characteristic probability of decaying during a given time interval.
• The decay constant λ describes this probability.
• A nucleus that has survived for a long time is not simply "overdue" to decay.
• Individual decay events are unpredictable.
• Large populations of radioactive nuclei behave statistically predictably.
• Larger samples show smaller relative random fluctuations.
• Half-life describes the behaviour of a population rather than the lifetime of an individual nucleus.
• After one half-life, approximately half of a large initial population remains.
• After two half-lives, approximately one-quarter remains.
• After three half-lives, approximately one-eighth remains.
• The expected remaining fraction after n half-lives is (1/2)ⁿ.
• Radioactive decay follows an exponential pattern.
• The same expected fraction, not the same number, decays during each half-life.
• Activity decreases with the same half-life as the number of undecayed nuclei.
• Corrected detector count rate can also be used to investigate half-life.
• Experimental radioactive counts fluctuate naturally because decay is random.
• Repeated measurements and longer counting times improve statistical reliability.
• The smooth decay curve represents the expected behaviour of a large population.

Check Your Understanding

1. What does it mean to say radioactive decay is random?
2. Why can't scientists predict exactly when a particular radioactive nucleus will decay?
3. What is meant by spontaneous radioactive decay?
4. What does the decay constant describe?
5. Explain why probability does not tell us exactly when an individual nucleus will decay.
6. Why can radioactive decay be random while half-life is predictable?
7. Define half-life.
8. Does a nucleus with a half-life of 10 years necessarily decay after 10 years? Explain.
9. A sample contains 6400 radioactive nuclei. How many are expected to remain after one half-life?
10. How many would be expected to remain after four half-lives?
11. What fraction of a sample remains after five half-lives?
12. Explain why radioactive decay produces an exponential rather than linear graph.
13. Why does the number of nuclei decaying during each successive half-life become smaller?
14. A sample's activity falls from 1200 Bq to 600 Bq in 8 hours. Determine its half-life.
15. If the activity is initially 1600 Bq and the half-life is 5 days, determine the expected activity after 15 days.
16. Explain why repeated Geiger-counter readings from an unchanged source are not exactly identical.
17. Why do larger radioactive samples produce smoother decay curves than very small samples?
18. Explain how a coin-toss or dice experiment can model radioactive decay.
19. Why should background radiation be subtracted before using detector data to determine half-life?
20. Explain the statement: "Radioactive decay is unpredictable for an individual nucleus but predictable for a large population."

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.

Successive Half-Lives

A useful pattern is:

Each half-life multiplies the remaining amount by:

1/2

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.

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.

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

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

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

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

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.

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.

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

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.

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.

Activity Equation

After n half-lives:

A = A₀(1/2)ⁿ

where:

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

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

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.

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.

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

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

 

 

 

8

The curve:

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

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.

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

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

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.

Example: Interpreting a Graph

Suppose a graph gives:

The activity halves every:

5 days

Therefore:

half-life = 5 days

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

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

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

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

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

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%

Percentage Decayed

If 12.5% remains, then:

percentage decayed:

= 100% − 12.5%

= 87.5%

Be careful to distinguish between:

percentage remaining

and:

percentage decayed

Example: Percentage Decayed

After four half-lives:

fraction remaining = 1/16

percentage remaining:

= 6.25%

Therefore:

percentage decayed:

= 100 − 6.25

= 93.75%

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.

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

Example: Decay Constant

An isotope has:

t₁/₂ = 8 hours

Then:

λ = 0.693/8

λ ≈ 0.0866 h⁻¹

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.

 

 

 

8

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.

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.

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.

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

 

 

 

7

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

shorter half-life

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

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.

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.

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.

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.

 

 

 

8

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.

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.

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

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.

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.

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.

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

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

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

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

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.

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.

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.

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.

3. Decay Calculations

Decay Calculations

Radioactive decay calculations allow us to determine how the amount or activity of a radioactive isotope changes with time.

The most important idea is that after every half-life, the amount of radioactive parent material remaining is divided by:

2

For example:

800 → 400 → 200 → 100 → 50

Each step represents one half-life.

Decay calculations can involve:

• number of radioactive nuclei
• mass of a radioactive isotope
• activity
• corrected count rate
• elapsed time
• original quantity
• percentage or fraction remaining

 

 

 

8

---

The Basic Half-Life Pattern

The fraction remaining follows this pattern:

Half-Lives Elapsed	Fraction Remaining	Percentage Remaining
0	1	100%
1	1/2	50%
2	1/4	25%
3	1/8	12.5%
4	1/16	6.25%
5	1/32	3.125%
6	1/64	1.5625%

This table can solve many introductory decay problems without using logarithms.

---

Finding the Number of Half-Lives

If the elapsed time and half-life are known:

n = t/t₁/₂

where:

• n = number of half-lives
• t = elapsed time
• t₁/₂ = half-life

---

Example: Number of Half-Lives

An isotope has a half-life of:

6 hours

How many half-lives occur during:

24 hours?

Use:

n = t/t₁/₂

n = 24/6

n = 4

Therefore:

4 half-lives have elapsed.

---

Example: Days

An isotope has a half-life of:

8 days.

How many half-lives occur in:

40 days?

n = 40/8

n = 5 half-lives

---

Keep Units Consistent

Before dividing, make sure the units are the same.

For example:

half-life = 3 hours

elapsed time = 1 day

Convert:

1 day = 24 hours

Then:

n = 24/3

n = 8 half-lives

Do not calculate:

1/3

because days and hours are different units.

---

Calculating the Remaining Quantity

After n half-lives:

remaining quantity = initial quantity × (1/2)ⁿ

For the number of nuclei:

N = N₀(1/2)ⁿ

where:

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

---

Example: Number of Nuclei

A radioactive sample initially contains:

12,000 radioactive nuclei.

Three half-lives pass.

Use:

N = 12,000(1/2)³

N = 12,000 × 1/8

N = 1500 nuclei

You can also halve repeatedly:

12,000 → 6000 → 3000 → 1500

---

Example: Mass Remaining

A sample initially contains:

320 mg

of a radioactive isotope.

After four half-lives:

320 → 160 → 80 → 40 → 20

Therefore:

20 mg remains

Using the equation:

m = 320(1/2)⁴

m = 320/16

m = 20 mg

---

Example: Activity Remaining

A source initially has an activity of:

9600 Bq.

After three half-lives:

9600 → 4800 → 2400 → 1200

Therefore:

A = 1200 Bq

Remember:

1 Bq = 1 decay/s

---

Example: Corrected Count Rate

A source has an initial corrected count rate of:

4800 counts/min.

After five half-lives:

4800(1/2)⁵

= 4800/32

= 150 counts/min

The same half-life mathematics applies because corrected count rate is proportional to activity when the detector setup remains unchanged.

---

Combining Time and Remaining Quantity

Many questions give:

• initial quantity
• half-life
• elapsed time

The best strategy is:

Step 1: Find the number of half-lives.

Step 2: Halve the quantity that many times.

---

Worked Example

A radioactive isotope has:

initial mass = 640 mg

half-life = 5 years

Find the remaining mass after:

20 years.

Find the number of half-lives

n = 20/5

n = 4

Halve four times

640 → 320 → 160 → 80 → 40

Therefore:

40 mg remains.

---

Worked Example: Activity

A source has:

initial activity = 16,000 Bq

half-life = 3 hours

elapsed time = 12 hours

Number of half-lives:

n = 12/3

n = 4

Activity:

16,000 → 8000 → 4000 → 2000 → 1000

Therefore:

A = 1000 Bq

---

Finding the Original Amount

Sometimes the amount remaining is known, but the original amount is unknown.

In this case, work backwards.

Instead of halving:

double once for each half-life.

---

Example: Finding Original Mass

After three half-lives:

25 mg remains.

Work backwards:

25 → 50 → 100 → 200

Therefore:

original mass = 200 mg

---

Example: Finding Original Activity

A source has an activity of:

375 Bq

after four half-lives.

Work backwards:

375 → 750 → 1500 → 3000 → 6000

Therefore:

original activity = 6000 Bq

---

Original Quantity Equation

Starting with:

N = N₀(1/2)ⁿ

we can rearrange:

N₀ = N × 2ⁿ

For example:

N = 500

n = 4

N₀ = 500 × 2⁴

N₀ = 500 × 16

N₀ = 8000

---

Finding Elapsed Time

Sometimes the question gives:

• initial amount
• final amount
• half-life

but asks for:

elapsed time

First determine how many times the amount has halved.

Then:

t = n × t₁/₂

---

Example: Finding Time

A radioactive sample decreases from:

640 mg

to:

40 mg.

Count the halvings:

640 → 320 → 160 → 80 → 40

That is:

4 half-lives

If:

t₁/₂ = 7 days

then:

t = 4 × 7

t = 28 days

---

Example: Activity and Time

A source decreases from:

6400 Bq

to:

800 Bq.

Halve:

6400 → 3200 → 1600 → 800

That is:

3 half-lives

If:

t₁/₂ = 12 hours

then:

t = 3 × 12

t = 36 hours

---

Finding the Half-Life

Sometimes elapsed time and the change in quantity are known.

First determine how many half-lives occurred.

Then:

t₁/₂ = t/n

---

Example: Finding Half-Life

A radioactive sample decreases from:

4800 Bq

to:

600 Bq

during:

18 hours.

Find the number of halvings:

4800 → 2400 → 1200 → 600

Therefore:

n = 3

Now:

t₁/₂ = 18/3

t₁/₂ = 6 hours

---

Another Example

A sample decreases from:

160 mg

to:

10 mg

during:

32 days.

Halving:

160 → 80 → 40 → 20 → 10

Therefore:

n = 4

Half-life:

t₁/₂ = 32/4

t₁/₂ = 8 days

---

Fraction Remaining

The fraction remaining after n half-lives is:

(1/2)ⁿ

For example:

after 4 half-lives:

(1/2)⁴ = 1/16

So:

1/16 of the original radioactive material remains.

---

Percentage Remaining

Convert the fraction to a percentage:

percentage remaining = (1/2)ⁿ × 100%

After 4 half-lives:

percentage remaining:

= 1/16 × 100%

= 6.25%

---

Percentage Decayed

Do not confuse:

percentage remaining

with:

percentage decayed

Use:

percentage decayed = 100% − percentage remaining

After four half-lives:

percentage remaining = 6.25%

Therefore:

percentage decayed:

= 100 − 6.25

= 93.75%

---

Example: Five Half-Lives

After five half-lives:

fraction remaining:

= (1/2)⁵

= 1/32

Percentage remaining:

= 3.125%

Percentage decayed:

= 100 − 3.125

= 96.875%

---

Finding Half-Lives from a Fraction

Sometimes the fraction remaining is given.

For common fractions, simply recognize the pattern.

If:

fraction remaining = 1/2

then:

1 half-life

If:

fraction remaining = 1/4

then:

2 half-lives

If:

fraction remaining = 1/8

then:

3 half-lives

If:

fraction remaining = 1/16

then:

4 half-lives

If:

fraction remaining = 1/32

then:

5 half-lives

---

Example: Percentage Remaining

A sample has:

12.5%

of its original radioactive isotope remaining.

Recognize:

12.5% = 1/8

and:

1/8 = (1/2)³

Therefore:

3 half-lives have elapsed.

---

Exponential Decay

Radioactive decay follows an exponential relationship.

For a large population:

N = N₀e^(−λt)

where:

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

 

 

 

7

The equivalent activity equation is:

A = A₀e^(−λt)

---

Why Radioactive Decay Is Exponential

The rate of radioactive decay depends on how many radioactive nuclei remain.

At the beginning:

many radioactive nuclei are present.

Therefore:

many decays occur.

Later:

fewer radioactive nuclei remain.

Therefore:

fewer decays occur during the same length of time.

This produces a curved exponential decay graph.

---

Constant Fraction Rather Than Constant Amount

Consider:

1600 → 800 → 400 → 200 → 100

The amount lost during each interval is:

800

400

200

100

The amount lost is not constant.

But the fraction lost is always:

1/2

Therefore radioactive decay is:

exponential

rather than:

linear

---

Linear vs Exponential Decay

A linear decrease might look like:

1000 → 900 → 800 → 700 → 600

The same amount is removed each time:

100

Radioactive exponential decay might look like:

1000 → 500 → 250 → 125 → 62.5

The same fraction is removed:

1/2

This distinction is essential when interpreting decay graphs.

---

Decay Constant

The decay constant, λ, describes the probability of decay per unit time.

Half-life and decay constant are related:

λ = 0.693/t₁/₂

and:

t₁/₂ = 0.693/λ

A larger λ means:

• faster decay
• shorter half-life

A smaller λ means:

• slower decay
• longer half-life

---

Example: Decay Constant

An isotope has:

t₁/₂ = 10 days

Then:

λ = 0.693/10

λ = 0.0693 day⁻¹

---

Example: Finding Half-Life from λ

Suppose:

λ = 0.035 h⁻¹

Then:

t₁/₂ = 0.693/0.035

t₁/₂ ≈ 19.8 hours

So the half-life is approximately:

20 hours

---

Calculations with Partial Half-Lives

Not every problem involves an exact whole number of half-lives.

Suppose:

t₁/₂ = 8 hours

but:

t = 20 hours

Then:

n = 20/8

n = 2.5 half-lives

We can still use:

remaining = initial × (1/2)ⁿ

---

Example: Partial Half-Life

Initial mass:

100 mg

Half-life:

8 hours

Elapsed time:

20 hours

Number of half-lives:

n = 20/8

= 2.5

Then:

m = 100(1/2)^2.5

(1/2)^2.5 ≈ 0.1768

Therefore:

m ≈ 100 × 0.1768

m ≈ 17.7 mg

---

Half-Life Form of the Decay Equation

Instead of using λ, decay can be written directly using half-life:

N = N₀(1/2)^(t/t₁/₂)

This equation works for:

• whole numbers of half-lives
• fractional numbers of half-lives

Similarly:

A = A₀(1/2)^(t/t₁/₂)

and:

m = m₀(1/2)^(t/t₁/₂)

when m represents the amount of radioactive parent isotope.

---

Worked Example: Partial Half-Life

A radioactive isotope has:

initial activity = 2000 Bq

half-life = 12 hours

Find its expected activity after:

30 hours.

Use:

A = A₀(1/2)^(t/t₁/₂)

A = 2000(1/2)^(30/12)

A = 2000(1/2)^2.5

A ≈ 2000 × 0.1768

A ≈ 354 Bq

---

Finding Time Using Logarithms

For more advanced problems:

N = N₀(1/2)^(t/t₁/₂)

Divide by N₀:

N/N₀ = (1/2)^(t/t₁/₂)

Taking logarithms allows t to be calculated:

t = t₁/₂ × ln(N/N₀) / ln(1/2)

Because both logarithms are negative when N < N₀, the calculated time is positive.

---

Example: Finding Time with a Non-Simple Fraction

A sample begins with:

1000 mg

and decreases to:

300 mg.

Half-life:

5 years.

Use:

t = 5 × ln(300/1000) / ln(1/2)

t = 5 × ln(0.3) / ln(0.5)

t ≈ 5 × 1.737

t ≈ 8.68 years

So approximately:

8.7 years have elapsed.

---

Finding the Original Amount Using the Equation

Suppose:

remaining mass = 18 mg

half-life = 4 years

elapsed time = 12 years

First:

n = 12/4

n = 3

Use:

m₀ = m × 2ⁿ

m₀ = 18 × 2³

m₀ = 18 × 8

m₀ = 144 mg

---

Count Rate Calculations

Radiation experiments often provide count rate instead of activity.

Remember:

measured count rate = source count rate + background count rate

Therefore:

corrected count rate = measured count rate − background count rate

Use the corrected rate for decay calculations.

---

Worked Example with Background Radiation

A detector initially records:

820 counts/min

Background:

20 counts/min

Therefore:

initial corrected rate:

820 − 20 = 800 counts/min

After some time, the detector records:

120 counts/min

Corrected:

120 − 20 = 100 counts/min

Now compare:

800 → 400 → 200 → 100

That represents:

3 half-lives

If the isotope has:

t₁/₂ = 4 hours

then:

t = 3 × 4

t = 12 hours

---

A Common Background-Radiation Trap

Suppose:

initial measured rate = 500 counts/min

background = 100 counts/min

The corrected initial rate is:

400 counts/min

After one half-life, the source contribution should be:

200 counts/min

But the detector will measure:

source + background

= 200 + 100

= 300 counts/min

Therefore, the raw detector reading does not simply halve from 500 to 250.

Background must be accounted for.

---

Interpreting Decay Graphs

A decay graph commonly plots:

x-axis → time

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

The graph should show:

• rapid decrease initially
• progressively slower decrease
• a smooth exponential curve
• equal half-life intervals for successive halvings

 

 

 

10

---

Finding Half-Life from a Graph

Suppose a graph shows:

800 Bq at 2 hours

400 Bq at 8 hours

The activity has halved.

Therefore:

t₁/₂ = 8 − 2

t₁/₂ = 6 hours

Notice that you do not have to begin at t = 0.

---

Checking a Half-Life on a Graph

Once you find one halving interval, check another.

For example:

800 Bq at 2 h

400 Bq at 8 h

200 Bq at 14 h

Each halving takes:

6 hours

This confirms:

t₁/₂ = 6 hours

---

Comparing Decay Curves

Suppose two isotopes begin with the same amount.

Isotope A

Half-life = 2 hours

Isotope B

Half-life = 10 hours

Isotope A's graph falls more steeply.

Isotope B's graph falls more gradually.

Therefore:

steeper decay curve → shorter half-life

shallower decay curve → longer half-life

 

 

 

9

---

Example: Comparing Two Isotopes

Both isotopes begin with:

6400 nuclei.

Isotope A:

t₁/₂ = 2 h

Isotope B:

t₁/₂ = 8 h

After 8 hours:

Isotope A

n = 8/2 = 4

N = 6400(1/2)⁴

N = 400

Isotope B

n = 8/8 = 1

N = 6400(1/2)

N = 3200

Therefore, isotope A decays much more rapidly.

---

Decay Calculations in Radiometric Dating

Decay calculations can be used to estimate how much time has passed since a radioactive system began evolving under the assumptions of a particular dating method.

For example, carbon-14 has a half-life of approximately:

5730 years

If an appropriate sample contains:

1/4

of the expected original carbon-14 proportion, then:

1 → 1/2 → 1/4

This represents:

2 half-lives

Estimated elapsed time:

2 × 5730

= 11,460 years

 

 

 

7

---

Example: Carbon-14

A suitable sample contains:

12.5%

of its expected original carbon-14 proportion.

Convert:

12.5% = 1/8

Then:

1 → 1/2 → 1/4 → 1/8

Therefore:

3 half-lives

Elapsed time:

3 × 5730

= 17,190 years

---

Choosing the Best Method

For simple whole half-lives:

repeated halving is fastest

Example:

800 → 400 → 200 → 100

For a known number of half-lives:

N = N₀(1/2)ⁿ

For elapsed time:

n = t/t₁/₂

For fractional half-lives:

N = N₀(1/2)^(t/t₁/₂)

For general exponential calculations:

N = N₀e^(−λt)

Choose the simplest method that fits the information provided.

---

A Full Problem-Solving Strategy

When solving a radioactive decay problem:

Identify what is given

Look for:

• initial amount
• final amount
• half-life
• elapsed time
• activity
• count rate
• background radiation

Check units

Make sure time units match.

Correct for background if necessary

Use:

corrected count = measured count − background

Find the number of half-lives

If possible:

n = t/t₁/₂

Choose the calculation

For whole half-lives:

halve repeatedly

or:

remaining = initial × (1/2)ⁿ

Work backwards if necessary

To find the original amount:

original = remaining × 2ⁿ

Check your answer

For ordinary decay without production of the parent isotope:

• remaining quantity should not exceed the initial quantity
• quantity should decrease with time
• units should be correct
• longer elapsed time should normally mean less parent isotope remains

---

Worked Mixed Problem

A source has:

measured initial count rate = 1640 counts/min

background = 40 counts/min

half-life = 3 hours

What measured count rate would be expected after:

9 hours?

First correct the initial count:

1640 − 40 = 1600 counts/min

Number of half-lives:

n = 9/3

n = 3

Source count after three half-lives:

1600 → 800 → 400 → 200

Corrected source rate:

200 counts/min

Now add the background back to predict the detector reading:

200 + 40 = 240 counts/min

Therefore, the expected measured count rate is:

240 counts/min

---

Worked Mixed Problem: Original Amount

A radioactive isotope has:

half-life = 12 years

remaining mass = 15 mg

elapsed time = 48 years

Number of half-lives:

n = 48/12

n = 4

Original mass:

m₀ = 15 × 2⁴

m₀ = 15 × 16

m₀ = 240 mg

---

Worked Mixed Problem: Finding Half-Life

A source has an initial corrected activity of:

7200 Bq

After:

15 days

its activity is:

900 Bq

Halve:

7200 → 3600 → 1800 → 900

Therefore:

3 half-lives

Then:

t₁/₂ = 15/3

t₁/₂ = 5 days

---

Common Mistakes

Mistake: Subtracting half of the original amount repeatedly.
Each half-life halves the amount currently remaining.

Mistake: Using n = half-life/elapsed time.
Correct:

n = elapsed time/half-life

Mistake: Mixing time units.
Convert days, hours, minutes or years before calculating.

Mistake: Multiplying by 1/2 only once when several half-lives have passed.
Use:

(1/2)ⁿ

Mistake: Halving when working backwards.
To find the original quantity, double for each previous half-life.

Mistake: Assuming radioactive decay is linear.
It is exponential.

Mistake: Forgetting background radiation in count-rate problems.
Correct the measured rate before applying half-life calculations.

Mistake: Confusing percentage remaining with percentage decayed.
They must add to:

100%

Mistake: Assuming the amount becomes exactly zero after several half-lives.
The exponential model approaches zero rather than suddenly reaching it.

---

Key Terms

Half-life (t₁/₂) — time required for the expected amount or activity of radioactive parent material to fall to half its previous value.

Elapsed time (t) — amount of time that has passed.

Number of half-lives — number of half-life intervals that have elapsed.

Initial quantity — amount of radioactive parent material at the beginning of the calculation.

Remaining quantity — amount of radioactive parent material still undecayed.

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 — measured count rate after subtracting background radiation.

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

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

Background radiation — radiation detected from environmental sources rather than only the source being investigated.

---

Key Takeaways

• Radioactive quantities halve during each half-life.
• The number of elapsed half-lives is n = t/t₁/₂.
• After n half-lives, the fraction remaining is (1/2)ⁿ.
• The remaining quantity is N = N₀(1/2)ⁿ.
• The original quantity can be found using N₀ = N × 2ⁿ.
• Elapsed time can be calculated using t = n × t₁/₂.
• Half-life can be calculated using t₁/₂ = t/n.
• After 1 half-life, 50% remains.
• After 2, 25% remains.
• After 3, 12.5% remains.
• After 4, 6.25% remains.
• Each half-life halves the remaining quantity, not the original quantity.
• Radioactive decay is exponential rather than linear.
• The same fraction decays during equal half-life intervals.
• Fractional numbers of half-lives can be handled using N = N₀(1/2)^(t/t₁/₂).
• The general exponential equation is N = N₀e^(−λt).
• Half-life and decay constant are related by t₁/₂ = 0.693/λ.
• Count-rate calculations should normally use background-corrected data.
• Decay graphs can be used to determine half-life by measuring the time between successive halvings.
• A shorter half-life produces a steeper decay curve.

Check Your Understanding

1. An isotope has a half-life of 5 hours. How many half-lives occur in 25 hours?
2. An isotope has a half-life of 8 days. How many half-lives occur in 40 days?
3. A sample begins with 6400 radioactive nuclei. How many remain after three half-lives?
4. A radioactive isotope initially has a mass of 320 mg. How much remains after four half-lives?
5. A source initially has an activity of 12,800 Bq. What is its activity after five half-lives?
6. What fraction of a radioactive sample remains after six half-lives?
7. What percentage remains after three half-lives?
8. What percentage has decayed after three half-lives?
9. A sample has a half-life of 4 days and an initial mass of 800 mg. How much remains after 16 days?
10. A source decreases from 6400 Bq to 400 Bq. How many half-lives have elapsed?
11. If the source in Question 10 has a half-life of 3 hours, how much time has elapsed?
12. A sample decreases from 960 mg to 120 mg in 18 years. Determine its half-life.
13. After four half-lives, 25 mg of a radioactive isotope remains. Determine the original mass.
14. A source has an activity of 125 Bq after five half-lives. Determine its original activity.
15. A detector records 850 counts/min and the background is 50 counts/min. What is the corrected count rate?
16. The corrected count rate in Question 15 falls to 100 counts/min. How many half-lives have elapsed?
17. An isotope has a half-life of 6 hours and an initial activity of 2000 Bq. Calculate its expected activity after 15 hours.
18. Explain why a radioactive-decay graph is exponential rather than linear.
19. A carbon-14 sample contains 1/8 of its expected original carbon-14 proportion. Using a half-life of 5730 years, estimate the elapsed time.
20. A detector initially reads 1640 counts/min with a background of 40 counts/min. The isotope has a half-life of 4 hours. Calculate the expected measured detector reading after 12 hours.

 

 

 

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.

 

 

 

7

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

 

 

 

6

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.

 

 

 

9

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

 

 

 

8

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.

 

 

 

7

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.

 

 

 

8

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.

5. Dating Using Radioactivity

Dating Using Radioactivity

Radioactive dating, also called radiometric dating, uses the predictable decay of radioactive isotopes to estimate the age of materials.

Although the decay of an individual radioactive nucleus is random, very large numbers of nuclei follow predictable patterns. Because each radioactive isotope has a characteristic half-life, the amount of radioactive parent isotope remaining can act like a natural clock.

Scientists use different radioactive isotopes for different timescales and materials.

 

 

 

10

The Basic Principle of Radioactive Dating

Imagine that a material initially contains a radioactive isotope.

As time passes:

radioactive parent isotope → daughter product

The amount of parent isotope gradually decreases.

If scientists know:

• the isotope's half-life
• the amount or proportion remaining
• the assumptions and initial conditions appropriate to the dating method

they can estimate how much time has passed.

For simple half-life problems:

age = number of half-lives × half-life

Parent and Daughter Isotopes

The original radioactive isotope is called the:

parent isotope

The product formed by radioactive decay is called the:

daughter isotope or daughter product

For example, carbon-14 eventually becomes nitrogen-14 through beta-minus decay.

We can represent this generally as:

parent → daughter + radiation

As time passes:

• parent decreases
• daughter produced by the decay increases

 

 

 

8

Parent Isotope After Each Half-Life

Suppose a sample begins with 100% parent isotope.

This predictable change provides the basis for many radiometric dating methods.

Carbon-14 Dating

One of the best-known radiometric dating techniques is:

carbon-14 dating

It is also called:

radiocarbon dating

Carbon-14 is written:

¹⁴C

Carbon-14 is a radioactive isotope of carbon.

It undergoes beta-minus decay and eventually becomes nitrogen-14:

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄ₑ

The mass number remains 14, while the atomic number increases from 6 to 7.

 

 

 

7

Where Carbon-14 Comes From

Carbon-14 is continually produced naturally in Earth's atmosphere.

High-energy cosmic radiation produces particles that interact with atmospheric nuclei. Neutrons produced in these interactions can react with nitrogen-14.

A simplified nuclear reaction is:

¹⁴₇N + ¹₀n → ¹⁴₆C + ¹₁p

The newly formed carbon-14 becomes incorporated into atmospheric carbon compounds, including carbon dioxide.

Carbon-14 Enters Living Organisms

Plants take in atmospheric carbon dioxide during:

photosynthesis

Animals obtain carbon by:

• eating plants
• eating other animals

Therefore, carbon moves through food webs.

While an organism is alive, it continually exchanges carbon with its environment.

Its carbon therefore includes:

• stable carbon isotopes
• a small amount of radioactive carbon-14

What Happens When an Organism Dies?

When an organism dies:

carbon exchange with the environment largely stops.

The organism no longer continually replaces its carbon-14 through normal biological processes.

The carbon-14 already present continues to decay.

Therefore:

amount of ¹⁴C decreases with time

This decrease acts as a radioactive clock.

Carbon-14 Half-Life

Carbon-14 has a half-life of approximately:

5730 years

Therefore:

After 5730 years:

50% remains

After 11,460 years:

25% remains

After 17,190 years:

12.5% remains

After 22,920 years:

6.25% remains

After 28,650 years:

3.125% remains

 

 

 

6

Simple Carbon-14 Dating

Suppose a once-living sample contains:

25%

of the carbon-14 level expected for comparable living material under the dating model.

Recognize:

25% = 1/4

and:

1 → 1/2 → 1/4

Therefore:

2 half-lives have passed

Age:

= 2 × 5730

= 11,460 years

Example: 12.5% Remaining

An archaeological sample contains:

12.5%

of its expected original carbon-14 level.

Recognize:

12.5% = 1/8

Sequence:

100% → 50% → 25% → 12.5%

Therefore:

3 half-lives

Age:

= 3 × 5730

= 17,190 years

Example: 6.25% Remaining

A sample contains:

6.25%

of its expected original carbon-14.

Sequence:

100 → 50 → 25 → 12.5 → 6.25

Therefore:

4 half-lives

Age:

= 4 × 5730

= 22,920 years

Finding Age from the Fraction Remaining

For simple whole-half-life problems:

fraction remaining = (1/2)ⁿ

where:

n = number of half-lives.

Then:

age = n × t₁/₂

Example

A sample contains:

1/16

of its expected original radioactive isotope.

Since:

1/16 = (1/2)⁴

then:

n = 4

If the isotope's half-life is:

2000 years

then:

age = 4 × 2000

age = 8000 years

When Is Carbon-14 Dating Appropriate?

Carbon-14 dating is mainly useful for dating material that was once part of a living organism.

Examples include:

• wood
• charcoal
• seeds
• plant fibres
• paper
• textiles made from natural fibres
• bone, when suitable carbon-bearing material is preserved
• leather
• shells in appropriate contexts
• other suitable organic remains

 

 

 

8

Carbon Dating and Fossils

The statement:

"Carbon-14 dates fossils"

needs qualification.

Carbon-14 can sometimes be used on relatively recent organic remains or preserved carbon-bearing material.

However, many fossils are:

• far too old for useful carbon-14 dating
• mineralized
• no longer composed primarily of the organism's original organic material

Scientists may instead date suitable rock layers or volcanic material associated with the fossil using other radiometric methods.

Carbon-14 Is Not Suitable for Ancient Rocks

Carbon-14 has a relatively short half-life compared with geological timescales.

After many half-lives, extremely little carbon-14 remains.

For example, after 10 half-lives:

fraction remaining = (1/2)¹⁰

= 1/1024

≈ 0.098%

Eventually the remaining carbon-14 becomes too small for reliable dating in many contexts.

Therefore, carbon-14 is not the appropriate radioactive clock for rocks that are millions or billions of years old.

Useful Carbon-14 Timescale

Radiocarbon dating is generally most useful for once-living materials up to roughly:

50,000 years old

The practical limit varies depending on:

• sample quality
• measurement technique
• contamination
• laboratory methods
• environmental history

For much older materials, scientists usually use longer-lived radioactive isotope systems.

Why Different Isotopes Are Needed

Different isotopes have different half-lives.

An isotope with a short half-life is useful for relatively recent events.

An isotope with a very long half-life can act as a clock over geological timescales.

A useful dating isotope should have a half-life appropriate to the expected age of the material.

Potassium-Argon Dating

One important geological dating system involves:

potassium-40 (⁴⁰K)

Potassium-40 has a half-life of approximately:

1.25 billion years

One of its decay pathways produces:

argon-40 (⁴⁰Ar)

Potassium-argon dating can be particularly useful for dating suitable volcanic rocks.

 

 

 

7

Why Potassium-Argon Dating Is Useful

Molten rock can release argon gas.

When volcanic rock cools and solidifies, newly produced radiogenic argon can become trapped within suitable minerals.

Scientists can analyze the potassium-argon system to determine the time since the rock or mineral system effectively closed to argon loss, subject to the assumptions of the method.

This is useful in:

• geology
• volcanology
• studies of human evolution
• dating volcanic layers associated with fossils or archaeological sites

Uranium-Lead Dating

Another major dating method is:

uranium-lead dating

One important decay chain begins with:

uranium-238

and eventually produces:

lead-206

Uranium-238 has a half-life of approximately:

4.47 billion years

This makes it useful for dating extremely old geological materials.

 

 

 

7

Zircon Crystals

The mineral zircon is especially valuable in uranium-lead dating.

When zircon crystals form, uranium can enter their crystal structure while initial lead is generally strongly excluded.

As uranium decays over time, radiogenic lead accumulates.

Scientists can measure uranium and lead isotopes in zircon to determine when the crystal formed, while using additional isotope relationships and checks to test whether the system remained suitable for dating.

Some zircon crystals have preserved evidence from very early in Earth's history.

Comparing Dating Methods

The dating system must match:

the material + expected age + geological or archaeological context

Choosing the Appropriate Radioactive Clock

Suppose scientists want to date:

A wooden archaeological object

Likely method:

carbon-14

because the wood was once living.

A volcanic rock millions of years old

A suitable potassium-based method may be appropriate.

An ancient zircon crystal billions of years old

uranium-lead dating may be appropriate.

The correct method depends on both:

• timescale
• material

Radioactive Dating and Decay Curves

Radiometric dating relies on exponential decay.

The remaining parent isotope follows:

N = N₀(1/2)^(t/t₁/₂)

where:

• N = amount remaining
• N₀ = initial amount
• t = elapsed time
• t₁/₂ = half-life

The fraction remaining is:

N/N₀ = (1/2)^(t/t₁/₂)

Reading a Dating Graph

A radioactive dating graph usually shows:

x-axis → time

y-axis → parent isotope remaining

The graph falls exponentially.

Important points are:

100% → 50% → 25% → 12.5% → 6.25%

Each step corresponds to one additional half-life.

 

 

 

7

Example from a Graph

Suppose a graph represents an isotope with:

t₁/₂ = 10,000 years

A sample has:

25%

of the expected original parent isotope remaining.

25% corresponds to:

2 half-lives

Therefore:

age = 2 × 10,000

= 20,000 years

Dating Using Activity

For carbon-14 dating, the activity of a sample can also be compared with an appropriate reference.

Suppose comparable living material has an activity represented as:

100%.

An archaeological sample has:

50%

of that activity.

This represents approximately:

1 half-life

For carbon-14:

age ≈ 5730 years

If the sample has:

25%

then:

2 half-lives

age ≈ 11,460 years

Example with Activity

A carbon-containing archaeological sample has a carbon-14 activity that is:

12.5%

of an appropriate reference level.

12.5% = 1/8

Therefore:

3 half-lives have passed.

Age:

3 × 5730

= 17,190 years

More Advanced Dating Calculation

Sometimes the fraction remaining is not a convenient value such as:

1/2, 1/4 or 1/8.

Use:

N/N₀ = (1/2)^(t/t₁/₂)

For example, suppose:

N/N₀ = 0.30

For carbon-14:

t₁/₂ = 5730 years

We can solve using logarithms:

t = t₁/₂ × ln(N/N₀) / ln(1/2)

Worked Example

Suppose a suitable sample contains:

30%

of its expected original carbon-14.

Then:

N/N₀ = 0.30

t = 5730 × ln(0.30)/ln(0.5)

ln(0.30) ≈ −1.204

ln(0.5) ≈ −0.693

Therefore:

t ≈ 5730 × 1.737

t ≈ 9950 years

So the estimated age is approximately:

10,000 years

Parent-Daughter Relationships

Some radiometric dating techniques measure both parent and daughter isotopes.

Imagine a simplified closed system beginning with radioactive parent nuclei.

As time passes:

parent decreases

while:

radiogenic daughter increases.

After one half-life:

parent ≈ 50%

After two:

parent ≈ 25%

After three:

parent ≈ 12.5%

The parent-daughter relationship can provide information about the time elapsed, but real dating methods must also account for initial daughter material and whether the system remained closed.

Closed Systems

Many radiometric dating methods depend on the sample behaving as a sufficiently:

closed system

after the event being dated.

This means the relevant parent and daughter isotopes have not been significantly added or removed in ways that invalidate the dating model.

If material enters or leaves the sample, the measured isotope ratios may be altered.

Contamination

Contamination is especially important in carbon-14 dating.

Suppose an ancient sample becomes contaminated with:

modern carbon

The added modern carbon contains relatively more carbon-14.

The sample may appear:

younger than it really is

Contamination with carbon from other sources can produce different biases depending on its age and composition.

Therefore, careful sample preparation is essential.

Calibration of Carbon-14 Dates

Simple classroom problems often assume the atmospheric carbon-14 level has remained constant.

In reality, atmospheric carbon-14 concentrations have varied over time.

Scientists therefore calibrate radiocarbon dates using independent records such as:

• tree rings
• corals
• lake and marine records
• other well-dated materials

Tree rings are especially valuable because annual growth rings can provide independently dated samples.

 

 

 

8

Dating Does Not Usually Date the Fossil Directly

Suppose a fossil lies between two volcanic ash layers.

If suitable volcanic minerals are present, scientists may date:

• the layer below
• the layer above

If the lower layer is:

3.2 million years old

and the upper layer is:

2.9 million years old

then the fossil likely formed between those times, assuming the geological sequence has not been disturbed.

Therefore, radiometric dating can help bracket the age of fossils.

Relative Dating vs Absolute/Numerical Dating

Relative dating

Determines whether something is:

• older
• younger

than something else.

Example:

A rock layer beneath another undisturbed layer is generally older.

Radiometric dating

Can provide a numerical age estimate, such as:

125 million years

Using both types of evidence provides a stronger reconstruction of geological history.

Accuracy and Reliability

Radiometric dating is not simply:

"measure isotope → obtain perfect age."

Scientists consider:

• sample contamination
• measurement uncertainty
• initial isotope conditions
• whether the system remained closed
• decay constants
• geological context
• calibration where necessary
• agreement with other dating methods

Multiple independent measurements can strengthen confidence in an age estimate.

Why Radioactive Dating Works

Radiometric dating is possible because:

• radioactive decay follows predictable statistical behaviour
• isotopes have characteristic half-lives
• decay rates are generally stable under ordinary environmental conditions
• modern instruments can measure isotope ratios very precisely
• independent methods can be compared

Radioactive isotopes therefore function as natural clocks.

Dating Earth's History

Long-lived radioactive isotopes allow scientists to investigate events spanning enormous periods of time.

Radiometric dating has helped scientists determine ages for:

• ancient minerals
• meteorites
• volcanic eruptions
• archaeological materials
• evolutionary events
• geological formations

Evidence from several independent radiometric systems contributes to the conclusion that Earth formed approximately:

4.54 billion years ago.

Worked Problem: Carbon-14

A piece of ancient wood contains:

25%

of its expected original carbon-14.

Carbon-14 half-life:

5730 years

25% = 1/4

Therefore:

2 half-lives

Age:

2 × 5730

= 11,460 years

Worked Problem: Carbon-14

A charcoal sample contains:

6.25%

of its expected original carbon-14.

Sequence:

100 → 50 → 25 → 12.5 → 6.25

Therefore:

4 half-lives

Age:

4 × 5730

= 22,920 years

Worked Problem: Potassium-Based Dating

For a simplified classroom problem, suppose a radioactive parent isotope has a half-life of:

1.25 billion years.

A suitable mineral contains:

25%

of the original parent isotope.

25% means:

2 half-lives.

Age:

2 × 1.25 billion

= 2.50 billion years

Worked Problem: Uranium-238

Suppose a simplified dating problem states that:

25%

of the original uranium-238 remains.

Half-life:

4.47 billion years.

25% means:

2 half-lives.

Age:

2 × 4.47

= 8.94 billion years

Mathematically, that is the correct result for the stated hypothetical. In a real geological context, an inferred age older than the Solar System would signal that the assumptions, interpretation, or sample system need to be reconsidered.

Choosing a Dating Method

When choosing a radiometric dating method, ask:

What material is being dated?

Was it once living?

Is it a mineral?

Is it volcanic?

Approximately how old is it?

Thousands of years?

Millions?

Billions?

Is the isotope's half-life appropriate?

A useful radioactive clock should retain a measurable amount of parent isotope over the timescale being investigated.

Has the system remained suitable for dating?

Consider:

• contamination
• heating
• weathering
• movement of parent or daughter isotopes

Common Mistakes

Mistake: Using carbon-14 to date extremely ancient rocks.
Carbon-14's half-life is too short for geological ages of millions or billions of years.

Mistake: Saying carbon-14 dates every fossil.
Carbon dating is mainly used for suitable once-living materials within its useful age range.

Mistake: Saying all carbon-14 disappears after one half-life.
Approximately half remains.

Mistake: Subtracting the same amount during every half-life.
The remaining parent isotope is repeatedly multiplied by 1/2.

Mistake: Assuming radioactive dating requires knowing exactly when individual nuclei decay.
Individual decay is random; large populations behave predictably.

Mistake: Assuming every sample begins with zero daughter isotope.
Real dating techniques account for initial conditions in ways appropriate to the isotope system.

Mistake: Ignoring contamination.
Adding or removing relevant material can alter isotope ratios.

Mistake: Thinking half-life alone determines the best dating method.
The material being dated and the behaviour of the isotope system are also essential.

Key Terms

Radiometric dating — use of radioactive decay to estimate the age of materials.

Radiocarbon dating — dating method based on carbon-14.

Carbon-14 (¹⁴C) — radioactive isotope of carbon with a half-life of approximately 5730 years.

Parent isotope — radioactive isotope that undergoes decay.

Daughter isotope/product — product formed from radioactive decay.

Half-life — time required for the expected amount or activity of a radioactive parent isotope to fall to half its previous value.

Decay constant — quantity describing the probability of radioactive decay per unit time.

Organic material — material derived from living or once-living organisms.

Potassium-argon dating — radiometric method involving potassium-40 and argon-40, useful for suitable geological materials.

Uranium-lead dating — radiometric dating system using uranium and lead isotopes, especially useful with minerals such as zircon.

Closed system — system in which relevant parent and daughter isotopes have not been significantly added or removed.

Contamination — introduction of outside material that can alter measured isotope ratios.

Calibration — correction or comparison of a dating measurement using independent age information.

Dendrochronology — dating and study using annual tree-growth rings.

Key Takeaways

• Radioactive dating uses predictable radioactive decay to estimate age.
• Different radioactive isotopes provide clocks for different timescales.
• Parent isotopes decrease as radioactive decay occurs.
• Half-life determines how rapidly the parent isotope decreases.
• Carbon-14 has a half-life of approximately 5730 years.
• Carbon-14 dating is mainly appropriate for suitable once-living material.
• After one carbon-14 half-life, approximately 50% remains.
• After two, approximately 25% remains.
• After three, approximately 12.5% remains.
• Carbon-14 is generally unsuitable for dating rocks millions or billions of years old.
• Potassium-based dating methods can be useful for suitable geological and volcanic materials.
• Uranium-lead dating is useful for very old minerals and rocks.
• Zircon is particularly important in uranium-lead dating.
• A dating isotope's half-life should match the expected age of the material.
• Simple dating problems can use age = number of half-lives × half-life.
• More general problems use exponential decay relationships.
• Carbon-14 dates often require calibration.
• Contamination can affect radioactive dating results.
• Many radiometric methods depend on the sample remaining a sufficiently closed system.
• Independent dating methods and geological evidence can be compared to improve confidence in an age estimate.

Check Your Understanding

1. What is radiometric dating?
2. Explain why radioactive isotopes can act as natural clocks.
3. What is meant by a parent isotope?
4. What is meant by a daughter isotope or daughter product?
5. What radioactive isotope is used in radiocarbon dating?
6. What is the approximate half-life of carbon-14?
7. Explain how carbon-14 enters living organisms.
8. Why does the carbon-14 clock become useful after an organism dies?
9. A sample contains 50% of its expected original carbon-14. Estimate its age.
10. A sample contains 25% of its expected original carbon-14. Estimate its age.
11. A sample contains 12.5% of its expected original carbon-14. Estimate its age.
12. A sample contains 6.25% of its expected original carbon-14. Estimate its age.
13. Why is carbon-14 unsuitable for dating most rocks millions of years old?
14. Give three examples of materials that may be suitable for carbon-14 dating.
15. Explain why fossils are not always dated directly using carbon-14.
16. Identify one radioactive dating system suitable for much older geological materials.
17. Explain why uranium-lead dating is useful for very ancient rocks and minerals.
18. A hypothetical isotope has a half-life of 500 million years. A mineral contains 12.5% of its original parent isotope. Estimate its age.
19. Explain how contamination could affect a radiometric age.
20. A suitable sample contains 30% of its expected original carbon-14. Using the exponential decay relationship and a half-life of 5730 years, estimate its age.