AQA A Level Physics

Topic Questions

8.2 Radioactive Decay

1a1 mark

Radioactive decay is a phenomenon that revolutionised our understanding of atomic physics. 

A list of words is provided below. 

Underline the word that best describes the nature of radioactive decay.  

Predictable

Infinite

Random

Dangerous

1b3 marks
(i)
Define the activity of a radioactive sample 
(ii)
State the standard international unit of activity.
1c3 marks

The isotope Lanthanum–138 has a decay constant of 2.0 × 10–19 s–1.  

Calculate the number of nuclei of Lanthanum–138 that would have a measured activity of 40 Bq.

 

1d4 marks

Using the information given in part (c), calculate the half–life of Lanthanum–138 in years.

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2a4 marks

The decay constant of a radioactive isotope is used to calculate its half–life. 

Define: 

   (i)   The decay constant 

   (ii)   The half–life

2b4 marks

The activity of a radioactive isotope of iodine is measured over 24 days. 

Figure 1 shows a plot of the data. 

Figure 1

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Use Figure 1 to:

(i)

State the initial activity, A subscript 0 of the radioactive isotope of iodine, giving an appropriate unit with your answer

(ii)

Determine the half–life in days

2c6 marks

The equation of the graph has the following form: 

            A =A subscript 0 e to the power of negative lambda t end exponent 

Complete the data in Table 1 below for the equation of the graph. 

The first line has been completed for you.

 

Symbol

Meaning

Unit

A

Activity

Bq

A subscript 0

 

 

lambda

 

 

t

 

 

2d4 marks

Use your answer to part (b) to calculate the decay constant, in s–1, of the radioactive isotope of iodine.

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3a1 mark

Carbon dating is a method used by scientists to estimate the age of organic materials. 

The reliability of carbon dating depends on the age of the sample being dated. 

Place a tick  in one of the boxes below to select the range of ages for which an age determined by carbon dating would be reliable. 

8-2-s-q--q3a-easy-aqa-a-level-physics

3b3 marks

The half–life of carbon–14 is 5740 years.  

Calculate the decay constant, in year–1, of carbon–14.

3c3 marks

A wooden chest discovered at an archaeological site is recovered. 

A piece of wood from the chest has 0.25 times as many carbon–14 atoms as an equal mass of living wood.

(i)

State the number of half–lives of carbon–14 that have passed since the wooden chest was created

(ii)

Calculate the age of the wooden chest, in years

3d3 marks

In living wood and all other living materials, the proportion of carbon–14 remains approximately constant.

(i)

Identify one source of carbon–14 for living organisms.

(ii)

State and explain what happens to the proportion of carbon–14 inside an organism following its death

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4a3 marks

A sample X contains 1.25 × 1026 radioactive nuclei. 

The half–life of sample X is 1500 years. 

Calculate the number of nuclei remaining after 4500 years.

4b5 marks
(i)

Calculate the decay constant, in year–1, of sample X

(ii)

Hence, or otherwise, calculate the number of nuclei remaining in sample X after 10 000 years

4c3 marks

When sample X has 1.25 × 1026 nuclei, it has a molar mass of 94.0 g mol–1. 

Show that the number of moles in sample X is about 208.

4d3 marks

Sample Y has a half–life twice as long as sample X, at 3000 years. 

Without calculation, state and explain how the decay constant of sample Y compares to the decay constant of sample X.

 

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5a2 marks

Protactinium has a half–life of 70 s.   

A sample of containing 0.5 moles of protactinium is prepared and monitored over a period of time. 

Determine the percentage of protactinium remaining after 210 s.

5b2 marks

The molar mass of protactinium is 231 g mol–1. 

Calculate the initial number of nuclei in the prepared sample of protactinium.

5c4 marks

Using the information given in part (a), calculate the decay constant of the prepared protactinium. 

State an appropriate unit for your answer.

5d3 marks

Using your answers to part (b) and part (c), calculate the initial activity of the prepared sample of protactinium. 

State an appropriate unit for your answer.

 

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1a4 marks

The radioisotope uranium-238 open parentheses U presubscript 92 presuperscript 238 close parentheses decays through a decay chain to the radioisotope lead-206 open parentheses P presubscript 82 presuperscript 206 b close parentheses, which is stable. 

During the decay chain, 8 alpha particles are emitted, and X beta particles are emitted. 

Calculate the value of X.

1b4 marks

The half-life of uranium-238 is so long in comparison to any of the isotopes in its decay chain that we can assume the number of lead-206 nuclei, N subscript P b end subscript at any time is equal to the number of uranium-238 that have decayed. 

            The number of uranium-238 nuclei begin mathsize 16px style N subscript U end style at time t is given by the equation: 

                  N subscript U equals N subscript 0 e to the power of negative lambda t end exponent 

            where begin mathsize 16px style N subscript 0 end style is the number of uranium-238 nuclei at t = 0. 

Show that the ratio N subscript P b end subscript over N subscript U is given by: 

                  N subscript P b end subscript over N subscript U equals e to the power of lambda t end exponent minus 1

1c3 marks

Enriched uranium fuel is a mixture of the fissionable uranium-235 with the more naturally abundant uranium-238. Mixtures of radioactive nuclides such as this are very common in the nuclear power industry.  

Two samples of radioactive nuclides X and Y each have an activity of A0 at t = 0. They are subsequently mixed together. 

Show that the total activity of the mixture at time t = 48 years is equal to begin mathsize 16px style 9 over 64 end styleA0.  

The half-life of X and Y are 16 and 8 years respectively.

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2a4 marks

Table 1 shows measurements of activity for two imaginary radioactive isotopes. 

Table 1

Imaginary radioactive isotope

Initial activity / MBq

Final activity / MBq

Time taken / hours

X

70

35

2.0

Y

70

38

3.5

 

The decay constant lambda gives the probability (per second) that a radioactive isotope will decay. 

Show that the ratio of decay constants lambda subscript X over lambda subscript Y, for the two radioactive isotopes in Table 1 is approximately equal to 2.

2b4 marks

A physics postgraduate wants to compare the activity of the two isotopes in Table 1 graphically. She decides to sketch a graph as shown in Figure 1. 

Figure 1

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Add labels to the spaces provided in Figure 1 to show which line corresponds to isotope X and isotope Y and justify your answer.

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3a2 marks

Rubidium (Rb) was found in samples of moon rock which were retrieved from the Apollo missions. The isotope R presubscript 37 presuperscript 87 b is known to have a half-life of 4.90 × 1010years.

 

One of the moon rock samples contains 1.2 mg of R presubscript 37 presuperscript 87 b. 

Show that the mass of R presubscript 37 presuperscript 87 b that the rock sample contained when the moon was formed 4.47 ×109 years ago was about 1.3 mg.

3b3 marks

Calculate the activity of 1.2 mg of R presubscript 37 presuperscript 87 b.

3c3 marks

Radioactive carbon dating is a method for determining the age of samples containing organic material. This is a commonly used method for terrestrial objects. 

Pieces of ancient wood found in a fireplace at an archaeological site can be dated by measuring the activity of carbon-14. A sample contains one carbon-14 atom per 8.0 × 1010 carbon-12 atoms. In living wood, the concentration of carbon-14 atoms is greater at one carbon-14 atom per 3.0 × 1010 carbon-12 atoms.

The half-life of carbon-14 is 1.8 × 1011 s. 

Explain why the concentration of carbon-14 is greater in living wood and calculate the age, in years, of the sample taken from the archaeological site.

3d2 marks

This method of radioactive carbon dating, which uses a ratio of carbon-14 atoms to carbon-12 atoms, can be improved by using all the carbon-14 atoms, rather than only those which happen to decay when dating is carried out.

Suggest why using all the carbon-14 atoms in the sample of ancient wood would give a more reliable value for its age.

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4a2 marks

The radioactive isotope uranium-238 decays in a decay series to the stable lead-206. 

The half-life of U presubscript 92 presuperscript 238 is 4.5 × 109 years, which is much larger than all the other half-lives of the decays in the series. 

A rock sample, when formed originally, contained 6.0 × 1022 atoms of U presubscript 92 presuperscript 238 and no P presubscript 82 presuperscript 206 b atoms. 

At any given time, most of the atoms are either U presubscript 92 presuperscript 238 or P presubscript 82 presuperscript 206 b with a negligible number of atoms in other forms in the decay series. 

Sketch on Figure 1the variation of number of U presubscript 92 presuperscript 238 atoms and the number of P presubscript 82 presuperscript 206 b atoms in the rock sample vary over a period of 1.0 × 1010 years from its formation.

Label your graphs U and Pb. 

Figure 1

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4b2 marks

A certain time, t, after its formation the sample contained twice as many U presubscript 92 presuperscript 238 atoms as P presubscript 82 presuperscript 206 b atoms. 

Show that the number of U presubscript 92 presuperscript 238 atoms in the rock sample at time t was 4.0 × 1022.

4c3 marks

The ratio of the number of lead nuclei NPb to the number of uranium nuclei NU at some time t is given by: 

            N subscript P b end subscript over N subscript U equals e to the power of lambda t end exponent minus 1 

Calculate the time taken (in years) for there to be twice as many U presubscript 92 presuperscript 238 atoms as P presubscript 82 presuperscript 206 b atoms.

4d4 marks

Lead-214 is an unstable isotope of lead-206. It decays by emitting a β particle to form bismuth-214 (Bi)

Bismuth is also unstable and has two decay modes:

  • Emitting an α particle to form thallium-210 (Tl) + energy
  • Emitting a β particle to form polonium-214 (Po) + energy 

Write decay equations for the decay chain of lead -214 to thallium -210 and to polonium -214 and comment on the nature of the energy released.

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1a2 marks

The radioisotope iodine-131open parentheses I presubscript 53 presuperscript 131 close parentheses  is used in medicine to treat overactive thyroid glands. It has a half-life of 8.02 days. 

Calculate the decay constant of  I presubscript 53 presuperscript 131.

1b2 marks

Calculate the number of atoms of I presubscript 53 presuperscript 131  necessary to produce a sample with an activity of 8.0 × 105 disintegrations s–1.

1c3 marks

Calculate the time taken, in hours, for the activity of the same sample of I presubscript 53 presuperscript 131  to fall from 8.5 × 105 disintegrations s–1 to 8.0 × 105 disintegrations s–1.

1d3 marks

Calculate the time (in days) for a sample of iodine to decay to 1% of its initial activity.

1e2 marks

Give two reasons why radioisotopes such as I presubscript 53 presuperscript 131  are particularly suitable for use as a treatment for overactive thyroid glands.

 

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2a3 marks

The age of an ancient boat may be determined by comparing the radioactive decay of C presubscript 6 presuperscript 14 from living wood with that of wood taken from the ancient boat.

A sample of 4.30 × 1023 atoms of carbon is removed for investigation from a block of living wood. In living wood one in 1012 of the carbon atoms is of the radioactive isotope C presubscript 6 presuperscript 14, which has a decay constant of 3.84 × 10–12 s–1. 

Calculate the half-life of C presubscript 6 presuperscript 14 in years, giving your answer to an appropriate number of significant figures.

2b2 marks

Show that the rate of decay of the C presubscript 6 presuperscript 14 atoms in the living wood sample is 1.65 Bq.

2c3 marks

A sample of 4.30 × 1023 atoms of carbon is removed from a piece of wood taken from the ancient boat. The rate of decay due to the C presubscript 6 presuperscript 14  atoms in this sample is 1.15 Bq. 

Calculate the age of the ancient boat in years.

2d4 marks
(i)

Give two reasons why it is difficult to obtain a reliable age of the ancient boat from the carbon dating described.

(ii)

Suggest why the method of carbon dating is likely to be unreliable if the sample is:

  • Less than 200 years old.
  • More than 60 000 years old.

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3a5 marks

The radioisotope bismuth (B presubscript 83 presuperscript 203 i) can decay to become an isotope of lead by two different decay methods. These are partially completed in Table 1. 

Complete the decay equations and decay methods in Table 1. 

Table 1

Decay equation

Decay method

 

B presubscript 83 presuperscript 203 i space plus space e presubscript times times times end presubscript presuperscript 0 minus space rightwards arrow space P presubscript times times times end presubscript presuperscript 203 b space plus space... space plus space Q

 

 

Electron capture

 

B presubscript 83 presuperscript 203 i space rightwards arrow space P presubscript 82 presuperscript 203 b space plus space... space plus space...

 

 

3b2 marks

B presubscript 83 presuperscript 203 i is also an alpha particle emitter. 

A student carries out an experiment to measure alpha particle activity in a sample of this isotope. They take two readings of the corrected count rate over 24 hours. Their data is as follows: 

  • Initial corrected count rate = 1500 counts s–1­­­­
  • Final corrected count rate = 270 counts s–1­­­­ 

Show that the decay constant of B presubscript 83 presuperscript 203 i is about 2.0 × 10–5 s­–1.

3c2 marks

Calculate the half-life of this sample.

3d2 marks

Give two major sources of radioactivity that the student would need to account for when measuring the activity of the sample. 

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4a3 marks

An isotope of technetium is a gamma emitter used by doctors as a tracer in the human body. It is injected into the patient’s blood stream. Scanners outside the body measure the gamma activity, enabling the blood flow to be monitored. 

Figure 1 shows the variation of activity with time, t, for a sample of the isotope. 

Figure 1

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Use data from the graph to determine the half-life of the technetium isotope.

4b3 marks

Use data from the graph to calculate the number of nuclei of the radioactive technetium isotope present at time t = 0.

4c3 marks

Figure 2 shows the variation of the number of nuclei remaining with time, t, for the same sample of the technetium isotope. 

Figure 2

8-2-q4c-sq-aqa-al-physics

Use Figure 2 to show that, after a time of 50000 s, about 1 × 1021 nuclei are decaying every second.

4d4 marks

All nuclei of technetium-99 have the same decay probability. 

Explain how this statement accounts for the observation that the number, N, of radioactive technetium atoms in the sample varies with time, t, as shown in Figure 2.

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5a2 marks

The half-life of lead–214 is 26.8 minutes. 

Show that the decay constant of lead–214 is approximately 4 × 10–4 s–1.

5b4 marks

Calculate the percentage of the original number of nuclei of lead–214 left in a sample after a period of 120 minutes.

5c3 marks

At t = 0, a sample contains 5.63 × 107 nuclei of lead–214. 

Calculate the number of nuclei which decay in the time interval between t = 60 min and t = 120 min.

5d1 mark

Each decay releases 2.0 × 10–10 J of energy. 

Calculate the energy released in the time interval between t = 60 min and t = 120 min.

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