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.

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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.

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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.

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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.

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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.

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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.

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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:

125.  

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.

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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.

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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.

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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.

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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.

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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:

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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.

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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

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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.

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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

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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.

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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.

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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

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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.

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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

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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.

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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.

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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.

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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.

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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

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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

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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.

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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

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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.

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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%

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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.

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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.

The exact number removed during each round varies.

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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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%

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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.

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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

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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.

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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

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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

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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.

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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.

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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."