Applications of Radioactivity

Radioactivity has a wide variety of uses in industry, agriculture and medicine
Some of the main uses are
- Nuclear power
- In medicine e.g. radiotherapy, tracers and sterilising equipment
- Radiocarbon dating of archaeological artefacts
- Uranium-lead dating of rock samples
- Radioisotope power systems

Radiocarbon Dating

The isotope carbon-14 is commonly used in radioactive dating
It forms as a result of cosmic rays knocking out neutrons from nuclei, which then collide with nitrogen nuclei in the air:

${}^{1}n + {}^{14}N \to {}^{14}C + {}^{1}p$

All living organisms absorb carbon-14, but after they die they do not absorb any more
The proportion of carbon-14 is constant in living organisms as carbon is constantly being replaced during the period they are alive
When they die, the activity of carbon-14 in the organic matter starts to fall, with a half-life of around 5730 years
Samples of living material can be tested by comparing the current amount of carbon-14 in them and compared to the initial amount (which is based on the current ratio of carbon-14 to carbon-12), and hence they can be dated

Reliability of Carbon Dating

Carbon dating is a highly reliable method for estimating the ages of samples between 500 and 60 000 years old
This range can be explained by looking at the decay curve of carbon-14:

Radiocarbon Decay, downloadable AS & A Level Physics revision notes

Carbon-14 decay curve used for radiocarbon dating

If the sample is less than 500 years old:
- The activity of the sample will be too high to measure small changes accurately
- Therefore, the ratio of carbon-14 to carbon-12 will be too high to determine an accurate age

If the sample is more than 60 000 years old:
- The activity will be too low to distinguish between changes in the sample and background radiation
- Therefore, the ratio of carbon-14 to carbon-12 will be too small to determine an accurate age

Uranium-Lead Dating

For many years, scientists could not agree on the age of the Earth
Until recently, the Earth was believed to be only millions of years old
Over the last century, radiometric dating methods have enabled scientists to discover the age of the Earth is many billions of years old
The most critical of these methods is uranium-lead dating

Uranium atoms decay whilst the number of lead atoms increases

Initially, there is only uranium in the rock, but over time, the uranium decays via a decay chain which ends with lead-206, which is a stable isotope
Uranium-238 has a half-life of 4.5 billion years
Over time, the ratio of lead-206 atoms to uranium-238 atoms increases
The ratio of uranium to lead in a sample of rock can then be used to determine its age

Uranium Decay Chain, downloadable AS & A Level Physics revision notes

Uranium-238 decay chain

Radioisotope Power Systems

The decay of an isotope may release energy as heat
Radioisotopic power systems are designed to transform this heat into electrical power
Such devices have been used to power space probes and satellites
Typically, plutonium-238 is used as fuel, with 1 g generating a power output of about 500 mW

Worked example

A space probe uses a source containing 4.0 kg of plutonium-238.

Plutonium-238 is an alpha-emitter with a half-life of 87.7 years. Each alpha decay releases 5.5 MeV per emission. The space probe converts this into electrical energy with an efficiency of 32%.

The space probe can continue to operate as long as the power output is maintained at 0.4 kW or above.

Estimate the time, in years, the source is expected to supply power to the space probe.

Answer:

Step 1: List the known quantities

Mass of Pu-238 = 4.0 kg = 4000 g
Molar mass of Pu-238 = 238 g mol⁻¹
Avogadro's constant, $N_{A}$ = 6.02 × 10²³ mol⁻¹
Half-life of Pu-238 = 87.7 years
Energy released per alpha decay = 5.5 MeV
1 electronvolt (eV) = 1.6 × 10⁻¹⁹ J
Efficiency = 32% = 0.32
Final power output, P = 0.4 kW = 400 W

Step 2: Calculate the initial number of nuclei present in the source

238 g of plutonium-238 contains 6.02 × 10²³ atoms (Avogadro's number), so in 4 kg:

Number of nuclei: $N = \frac{m a s s \times N_{A}}{m o l a r m a s s}$

Initial number of nuclei: $N_{0} = \frac{4000 \times (6.02 \times 10^{23})}{238} = 1.012 \times 10^{25}$ nuclei

Step 3: Calculate the initial activity of the source

Decay constant: $λ = \frac{\ln 2}{t_{1 / 2}}$

Activity: $A = λ N$

Combining these gives:

Initial activity: $A_{0} = \frac{N_{0} \ln 2}{t_{1 / 2}}$

$A_{0} = \frac{(1.012 \times 10^{25}) \times \ln 2}{87.7 \times (24 \times 60 \times 60 \times 365)} = 2.5357 \times 10^{15}$ Bq

Step 4: Calculate the initial power output of the source

Power output: $P = \frac{∆ E}{∆ t}$

Energy released per decay: $E = (5.5 \times 10^{6}) \times (1.6 \times 10^{- 19})$

Activity represents the decays per second, so:

Initial power output: $P_{0} = A_{0} E$

$P_{0} = (2.5357 \times 10^{15}) \times (5.5 \times 10^{6}) \times (1.6 \times 10^{- 19}) = 2231$ W

The electrical power transferred to the probe is:

$P_{0} = 2231 \times 0.32 = 714$ W

Step 5: Use the exponential decay equation to calculate the time of operation

The power available is proportional to the activity of the isotope, so:

Exponential decay of power: $P = P_{0} e^{- λ t}$

$\frac{P}{P_{0}} = e^{- λ t}$

$\ln (\frac{P}{P_{0}}) = - λ t$

$t = - \frac{1}{λ} \ln (\frac{P}{P_{0}}) = - \frac{t_{1 / 2}}{\ln 2} \ln (\frac{P}{P_{0}})$

$t = - \frac{87.7}{\ln 2} \ln (\frac{400}{714}) = 73.3$ years

Therefore, the source is expected to supply power to the space probe for 73.3 years

Applications of Radioactivity (AQA A Level Physics)

Revision Note