Radioactive Decay and Half-Life
5. Dating Using Radioactivity
Learning outcomes
- I can explain how radioactive dating works.
- I can describe carbon-14 dating.
- I can explain when carbon dating is appropriate.
- I can identify other radioactive dating methods.
- I can solve simple radioactive dating problems.
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.
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
Parent Isotope After Each Half-Life
Suppose a sample begins with 100% parent isotope.
| Half-Lives | Parent Remaining | Parent Decayed |
|---|---|---|
| 0 | 100% | 0% |
| 1 | 50% | 50% |
| 2 | 25% | 75% |
| 3 | 12.5% | 87.5% |
| 4 | 6.25% | 93.75% |
| 5 | 3.125% | 96.875% |
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.
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
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
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.
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.
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
| Dating System | Approximate Half-Life | Often Useful For |
|---|---|---|
| Carbon-14 | 5730 years | Recent once-living materials |
| Potassium-40 | 1.25 billion years | Suitable volcanic rocks and minerals |
| Uranium-238 | 4.47 billion years | Very old rocks and minerals |
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.
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.
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
- What is radiometric dating?
- Explain why radioactive isotopes can act as natural clocks.
- What is meant by a parent isotope?
- What is meant by a daughter isotope or daughter product?
- What radioactive isotope is used in radiocarbon dating?
- What is the approximate half-life of carbon-14?
- Explain how carbon-14 enters living organisms.
- Why does the carbon-14 clock become useful after an organism dies?
- A sample contains 50% of its expected original carbon-14. Estimate its age.
- A sample contains 25% of its expected original carbon-14. Estimate its age.
- A sample contains 12.5% of its expected original carbon-14. Estimate its age.
- A sample contains 6.25% of its expected original carbon-14. Estimate its age.
- Why is carbon-14 unsuitable for dating most rocks millions of years old?
- Give three examples of materials that may be suitable for carbon-14 dating.
- Explain why fossils are not always dated directly using carbon-14.
- Identify one radioactive dating system suitable for much older geological materials.
- Explain why uranium-lead dating is useful for very ancient rocks and minerals.
- A hypothetical isotope has a half-life of 500 million years. A mineral contains 12.5% of its original parent isotope. Estimate its age.
- Explain how contamination could affect a radiometric age.
- 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.