Modelling Radioactive Decay (VCE SSCE Physics): Revision Notes

Modelling Radioactive Decay

Introduction to radiocarbon dating

Radiocarbon dating is a powerful scientific technique that allows us to determine the age of objects containing organic material. This method relies on measuring the decay of radioactive carbon-14 atoms present in the sample.

A remarkable example of radiocarbon dating in action comes from the Warratyi rock shelter in the Flinders Ranges, South Australia. Scientists from the Australian Nuclear Science and Technology Organisation (ANSTO) used radiocarbon dating to discover that Aboriginal peoples settled in this arid region approximately 49,000 years ago. This technique is reliable for dating objects up to 55,000-60,000 years old.

Understanding random decay

The random nature of radioactive decay

Radioactive decay occurs as a completely random process. This means it is impossible to predict exactly when any particular radioactive nucleus will decay, regardless of how long the atom has existed. Think of it like this: you cannot tell which specific atom will decay next, just as you cannot predict which specific person in a crowd will sneeze next.

However, when dealing with large numbers of nuclei, we can make statistical predictions about the behaviour of the entire group. For example, one gram of uranium-238 contains approximately $2.6 \times 10^{21}$ radioactive atoms. With such enormous numbers, we can accurately predict the overall decay rate using the concept of half-life.

Simulating decay with dice

A practical way to understand random decay is through a dice simulation. Here's how it works:

1. Start with 120 dice
2. Roll all the dice
3. Remove all dice showing a particular number (say, 6)
4. Count the remaining dice
5. Repeat the process

This models radioactive decay because:

• You cannot predict which individual dice will show a 6 (randomness)
• Approximately one-sixth of all dice will show a 6 each time (statistical prediction)
• The number of "active" dice decreases with each roll (decay over time)

Half-life concept

Definition of half-life

Half-life is the time taken for half of a group of unstable nuclei to decay. It serves as a measure of the decay rate for any radioactive isotope.

Range of half-lives

Different radioisotopes decay at vastly different rates:

• Very short half-lives: Francium-218 has $t_{\frac{1}{2}} = 1.0$ millisecond
• Very long half-lives: Uranium-238 has $t_{\frac{1}{2}} = 4,500$ million years

Natural and artificial radioisotopes

Radioisotopes can be classified as either natural (occurring in nature) or artificial (human-made). Here's a comparison:

Isotope	Emission	Half-life	Application
Natural
Lead-210	β⁻	22.2 years	Lead dating of sand and soil
Radium-88	α	1600 years	Historical medical uses
Carbon-14	β⁻	5730 years	Carbon dating
Uranium-238	α	4500 million years	Nuclear fuel, geological dating
Artificial
Fluorine-18	β⁺	110 minutes	PET scans
Technetium-99m	β⁻	6.0 hours	Medical tracer
Iodine-131	β⁻	8 days	Medical tracer, radiation therapy
Iodine-125	γ	60 days	Internal radiation therapy
Cobalt-60	γ	5.3 years	External radiation therapy
Americium-241	α	460 years	Smoke detectors
Plutonium-239	α	24,100 years	Nuclear reactor fuel

Properties of half-life

Half-life and applications

The half-life determines the suitability of a radioisotope for different applications:

Medical tracers require short half-lives (e.g., technetium-99m, 6.0 hours) so that radioactivity doesn't remain in the body longer than necessary.

Radiotherapy may need slightly longer half-lives (e.g., iodine-131, 8 days) to provide sustained treatment.

Smoke detectors use isotopes with very long half-lives (e.g., americium-241, 460 years) so the radioactive source never needs replacement, only the battery.

Decay curves

Understanding exponential decay

The decay of radioactive material follows a characteristic pattern called exponential decay. When plotted on a graph, this creates a decay curve that shows the amount of radioactive material remaining versus time.

All radioactive elements produce the same shaped decay curve - only the scale values on the axes differ. This universal pattern is described by the general radioactive decay formula:

$N = N_0 \left(\frac{1}{2}\right)^n$

Where:

• $n$ = the number of half-lives that have passed
• $N_0$ = original amount of radioactive substance
• $N$ = final amount of the substance after $n$ half-lives

Carbon-14 decay curve

Carbon-14 is widely used for dating organic materials. The graph below shows how carbon-14 decays over time:

This decay curve demonstrates that:

• If 50% of the original carbon-14 remains, the object is 5,730 years old (one half-life)
• If 25% remains, the object is 11,460 years old (two half-lives)
• If 12.5% remains, the object is 17,190 years old (three half-lives)

Mathematical modelling of decay

The decay formula

The general formula for calculating radioactive decay is:

$N = N_0 \left(\frac{1}{2}\right)^n$

This equation can be used in two ways:

1. Finding the final amount when you know the number of half-lives
2. Finding the number of half-lives when you know the final amount

Carbon-14 dating techniques

Carbon-14 dating works because living organisms constantly exchange carbon with their environment, maintaining a constant ratio of carbon-14 to carbon-12. When an organism dies, this exchange stops, and the carbon-14 begins to decay while carbon-12 remains stable. By measuring the ratio of carbon-14 to carbon-12, scientists can determine when the organism died.

Materials that can be dated

Radiocarbon dating can be applied to many materials containing organic carbon:

• Charcoal, wood, twigs, seeds
• Bones and shells
• Leather, peat, and soil
• Hair and pottery
• Pollen and wall paintings
• Corals and blood residues
• Fabrics, paper, parchment
• Resins and water samples

Medical applications

Technetium-99m: the medical workhorse

Technetium-99m is the most widely used radioactive isotope for medical diagnostic studies. Hospitals prefer short-lived isotopes because they minimize radiation exposure to patients.

Rather than using expensive particle accelerators, hospitals produce technetium-99m using a "radioactive cow" - a generator containing molybdenum-99 (the "mother" nuclide with a 67-hour half-life). As molybdenum-99 decays, it produces technetium-99m (the "daughter" nuclide with a 6.0-hour half-life):

$^{99}_{42}\text{Mo} \rightarrow ^{99m}_{43}\text{Tc} + ^{0}_{-1}\beta$

$^{99m}_{43}\text{Tc} \rightarrow ^{99m}_{43}\text{Tc} + ^{0}_{0}\gamma$

The technetium-99m is extracted ("milking the cow") and used for medical imaging before it decays.

Activity and strength of radioactive sources

Definition of activity

Activity measures the strength of a radioactive source. It is defined as the number of nuclei that decay each second - essentially the rate of disintegrations.

The unit of activity is the becquerel (Bq), where:

$1 \text{ Bq} = 1 \text{ disintegration per second}$

Cobalt-60 decay and activity

Cobalt-60 is a gamma-ray emitter with a half-life of 5.27 years. It is used for sterilising medical equipment and in external radiotherapy. Consider a medical source containing 10 g of cobalt-60:

• Initial activity: 440 TBq ($4.40 \times 10^{14}$ Bq)
• After one half-life (5.27 years): 5 g remains, activity = 220 TBq
• After two half-lives (10.54 years): 2.5 g remains, activity = 110 TBq
• After three half-lives (15.81 years): 1.25 g remains, activity = 55 TBq

Natural radioactivity on Earth

Sources of radiation

Since Earth formed 4,500 million years ago, two major mechanisms have produced natural radiation:

1. Terrestrial radiation - from radioactive elements in Earth's crust
2. Cosmic radiation - from space

Approximately 82% of the average annual radiation that people receive comes from natural sources:

• Radon gas from the ground: 40%
• Rocks and soil: 15%
• Human body and food: 15%
• Cosmic rays: 12%
• Artificial sources (mainly medical): 18%

Radioactivity in everyday life

We are constantly exposed to low levels of radiation from everyday items:

Item	Radioisotope	Activity
1 kg bananas	Potassium-40	100 Bq
1 kg brazil nuts	Radium-226	250 Bq
Australian home (100 m²)	Radon-222	3 kBq
70 kg adult human	Potassium-40	7 kBq
Smoke detector	Americium-241	30 kBq
Medical diagnosis	Fluorine-18	5 MBq
Medical therapy	Caesium-137	100 TBq

Human radioactivity

Humans are naturally radioactive! A 70 kg person contains potassium-40, a gamma-ray emitter with a half-life of 1.25 billion years, producing approximately 7,000 disintegrations per second (7 kBq). This potassium is essential for proper body function and is obtained from food like bananas. This natural radioactivity has been present throughout human evolution.

Half-lives of common isotopes

Isotope	Half-life
Uranium-238	4.5 billion years
Potassium-40	1.25 billion years
Carbon-14	5,730 years
Radium-226	1,600 years
Americium-241	460 years
Strontium-90	28 years
Iodine-131	8.1 days
Technetium-99m	6.0 hours
Bismuth-214	20 minutes
Polonium-214	0.15 milliseconds

Key exam skills

Using the half-life equation effectively

The decay formula $N = N_0 \left(\frac{1}{2}\right)^n$ can be applied in two directions:

Direction 1: Given the number of half-lives, find the final amount

• Substitute the known values into the equation
• Calculate the result

Direction 2: Given the final amount, find the number of half-lives

• Express the fraction as a power of $\frac{1}{2}$
• The exponent gives you the number of half-lives