8.3 nuclear physics - radioactive decay 1 - qs

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8.3 Nuclear Physics - Radioactive decay 1 – Questions

Q1. An ancient sealed flask contains a liquid, assumed to be water. An archaeologist asks a scientist to determine the volume of liquid in the flask without opening the flask. The scientist decides to use a radioactive isotope of sodium ( ) that decays with a half-life of 14.8 h.

(a) She first mixes a compound that contains 3.0 × 10–10 g of sodium-24 with 1500 cm3 of water. She then injects 15 cm3 of the solution into the flask through the seal. Show that initially about 7.5 × 1010 atoms of sodium-24 are injected into the flask.

(1)

(b) Show that the initial activity of the solution that is injected into the flask is about 1 × 106 Bq.

activity = _____________Bq (3)

(c) She waits for 3.5 h to allow the injected solution to mix thoroughly with the liquid in the flask. She then extracts 15 cm3 of the liquid from the flask and measures its activity which is found to be 3600 Bq.

Calculate the total activity of the sodium-24 in the flask after 3.5 h and hence determine the volume of liquid in the flask.

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(3)

(d) The archaeologist obtained an estimate of the volume knowing that similar empty flasks have an average mass of 1.5 kg and that mass of the flask and liquid was 5.2 kg. Compare the estimate that the archaeologist could obtain from these masses with the volume calculated in part 4.3 and account for any difference.

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(Total 9 marks)

Q2. (a) A solar panel of area 2.5 m2 is fitted to a satellite in orbit above the Earth. The panel

produces a current of 2.4 A at a potential difference of 20 V when solar radiation is incident normally on it.

(i) Calculate the electrical power output of the panel.

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(ii) Solar radiation on the satellite has an intensity of 1.4 kW m–2. Calculate the efficiency of the panel.

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(b) The back-up power system in the satellite is provided by a radioactive isotope enclosed in a sealed container which absorbs the radiation from the isotope. Energy from the radiation is converted to electrical energy by means of a thermoelectric module.

(i) The isotope has an activity of 1.1 × 1014 Bq and produces α particles of energy 5.1 MeV. Show that the container absorbs energy from the α particles at a rate

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of 90 J s–1.

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(ii) The isotope has a half-life of 90 years. Calculate the decay constant λ of this isotope.

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(iii) The mass number of the isotope is 239. Calculate the mass of isotope needed for an activity of 1.1 × 1014 Bq.

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(Total 11 marks)

Q3. A radioactive nuclide decays by emitting α particles. The graph shows how the rate of decay At of the source changes with time t.

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(a) Determine

(i) the half-life of the nuclide,

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(ii) the decay constant,

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(iii) the initial number of undecayed nuclei present at time t = 0.

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(b) Each decay releases 1.0 × 10–12 J. For the time interval between t = 30 s and t = 80 s, calculate

(i) the number of nuclei which decay,

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(ii) the energy released.

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(Total 9 marks)

Q4.

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(a) The nuclide can decay by electron capture to become an isotope of lead as shown in the following equation,

(i) Explain what is meant by electron capture.

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(ii) Give one reason why electromagnetic radiation is emitted following this process.

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(iii) Give the equation for another process in which is converted into an isotope of lead.

(5)

(b) The nuclide is also an α particle emitter. An initial measurement of the α particle activity of a sample of this isotope gives a corrected count rate of 1200 counts s–1. After an interval of 24 hours the corrected rate falls to 290 counts s–1. Assume that corrections have been made for the radiation both from daughter products and background radiation.

(i) Show that the decay constant of is about 1.6 × 10–5 s–1.

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(ii) Calculate the half-life of this sample.

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(iii) Calculate the number of nuclei in the sample when the corrected count rate was 1200 counts s–1.

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(Total 10 marks)

Q5. (a) When an α particle is emitted from a nucleus of the isotope , a nucleus of

thallium, Tl, is formed. Complete the equation below.

→ α + Tl (2)

(b) The α particle in part (a) is emitted with 6.1 MeV of kinetic energy.

(i) The mass of the α particle is 4.0 u. Show that the speed of the α particle immediately after it has been emitted is 1.7 × 10–7 m s–1. Ignore relativistic effects.

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(ii) Calculate the speed of recoil of the daughter nucleus immediately after the α particle has been emitted. Assume the parent nucleus is initially at rest.

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(Total 8 marks)

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Q6. (a) Sketch a graph to show how the number of neutrons, N, varies with the number of

protons, Z, for stable nuclei over the range Z = 0 to Z = 80. Draw a scale on the N axis.

(2)

(b) On the same graph, enclosing each region by a line, indicate the region in which nuclides are likely to decay, by

(i) α emission, labelling the region A,

(ii) β– emission, labelling the region B,

(iii) β+ emission, labelling the region C. (3)

(c) Complete the table.

mode of decay change in proton number Z change in neutron number N

α emission −2

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β– emission

β+ emission

e capture

p emission 0

n emission 0

(3) (Total 8 marks)

Q7. (a) An α particle source of half-life 3420 years has a rate of decay of 450 kBq.

Calculate

(i) the decay constant, in s–1,

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(ii) the number of radioactive atoms in the source.

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(b) A narrow beam of α particles is directed at a thin gold foil target in an evacuated vessel. Only a very small proportion of the α particles scatter backwards at an angle greater than 90° to the direction from which they came

(i) Describe what happens to the majority of the α particles incident on the gold foil.

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(ii) Several deductions may be made about the structure of gold atoms from the results of α– particle scattering. Write down two of these deductions.

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______________________________________________________________ (3)

(Total 7 marks)

Q8. (a) Suggest two reasons why an α particle causes more ionisation than a β particle of

the same initial kinetic energy.

You may be awarded marks for the quality of written communication in your answer.

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(b) A radioactive source has an activity of 3.2 × 109 Bq and emits α particles, each with kinetic energy of 5.2 Me V. The source is enclosed in a small aluminium container of mass 2.0 × 10–4 kg which absorbs the radiation completely.

(i) Calculate the energy, in J, absorbed from the source each second by the aluminium container.

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(ii) Estimate the temperature rise of the aluminium container in 1 minute, assuming no energy is lost from the aluminium.

specific heat capacity of aluminium = 900 J kg–1 K–1

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(Total 7 marks)

Q9. (a) Sketch, using the axes provided, a graph of neutron number, N, against proton

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number, Z, for stable nuclei over the range Z = 0 to Z = 80. Show suitable numerical values on the N axis.

(2)

(b) On the graph indicate, for each of the following, a possible position of a nuclide that may decay by

(i) α emission, labelling the position with W,

(ii) β– emission, labelling the position with X,

(iii) β+ emission, labelling the position with Y. (3)

(c) The isotope decays sequentially by emitting α particles and β– particles,

eventually forming the isotope . Four α particles are emitted in the sequence.

Calculate the number of β– particles in the sequence.

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(d) A particular nuclide is described as proton-rich. Discuss two ways in which the nuclide may decay. You may be awarded marks for the quality of written communication in your answer.

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(Total 10 marks)

Q10. (a) Calculate the radius of the U nucleus.

r0 = 1.3 × 10–15 m

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(b) At a distance of 30 mm from a point source of rays the corrected count rate is C. Calculate the distance from the source at which the corrected count rate is 0.10 C, assuming that there is no absorption.

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(c) The activity of a source of β particles falls to 85% of its initial value in 52 s. Calculate the decay constant of the source.

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(d) Explain why the isotope of technetium, 99Tc m, is often chosen as a suitable source of radiation for use in medical diagnosis.

You may be awarded additional marks to those shown in brackets for the quality of written communication in your answer.

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(Total 10 marks)

Q11. The age of an ancient boat may be determined by comparing the radioactive decay of

from living wood with that of wood taken from the ancient boat. A sample of 3.00 × l023 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 , which has a decay constant of 3.84 × 10–12 s–1.

(a) What is meant by the decay constant?

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(b) Calculate the half-life of in years, giving your answer to an appropriate number of significant figures.

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1 year = 3.15 × 107 s

answer = ______________________ years (3)

(c) Show that the rate of decay of the atoms in the living wood sample is 1.15 Bq.

(2)

(d) A sample of 3.00 × 1023 atoms of carbon is removed from a piece of wood taken

from the ancient boat. The rate of decay due to the atoms in this sample is 0.65 Bq. Calculate the age of the ancient boat in years.

answer = ______________________ years (3)

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

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___________________________________________________________________ (2)

(Total 11 marks)

Q12. (a) Bi can decay into Pb by a β– followed by an α decay, or by an α followed by a

β– decay. One or more of the following elements is involved in these decays:

Write out decay equations showing each stage in both of these decays.

First decay path Second decay path

(6)

(b) (i) Describe how you would perform an experiment that demonstrates that gamma radiation obeys an inverse square law.

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(ii) Explain why gamma radiation obeys an inverse square law but alpha and beta

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radiation do not.

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(Total 15 marks)

Q13. A nuclide of manganese undergoes beta+ decay to form a nuclide of chromium (Cr).

(a) Complete the equation for this decay process.

Mn Cr + β+ + (2)

(b) State the name of the exchange particle involved in this beta+ decay.

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(Total 3 marks)

Q14. Complete the following equation showing the β+ decay of carbon-11.

(Total 3 marks)

Q15.

Radioisotope thermoelectric generators (RTGs) are electrical generators powered by radioactive decay. As a radioisotope decays some of the energy released is converted into electricity by means of devices called thermocouples. In this way RTGs have been used as power sources in satellites, space probes and heart pacemakers.

The Cassini space probe was launched in 1997. It carried three RTGs each containing 11 kg of a nuclear fuel, plutonium oxide (a compound having two oxygen atoms combined with every plutonium-238 atom). In 1997, when the probe was launched, the power released from one gram of plutonium oxide was 500 mW.

Plutonium-238 is an alpha emitter, decaying into uranium(U).

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The half-life of the decay is 87.7 years.

mass of one mol of plutonium-238 = 238 g mass of one mol of oxygen atoms = 16 g

(a) State and explain why environmentalists might have been concerned by the use of such a large quantity of plutonium-238.

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(b) State and explain whether the activity of a given number of atoms of plutonium is affected when they are in the form of plutonium oxide.

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(c) (i) Calculate the decay constant, in s–1, for plutonium-238.

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decay constant ______________________ s–1

(2)

(ii) Calculate the number of plutonium-238 atoms in the total mass of the plutonium oxide in the Cassini probe at the beginning of its mission.

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number of plutonium-238 atoms ______________________ (5)

(iii) Calculate the initial activity of the plutonium-238 in the Cassini probe. Give a suitable unit for your answer.

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initial activity of plutonium-238 ______________________ unit ___________ (3)

(d) (i) Write a nuclear equation for the decay.

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(ii) Assume the power released by the RTGs’ fuel originated as the kinetic energy

of the alpha particles emitted in the decay of .

Calculate the maximum kinetic energy of each alpha particle.

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kinetic energy of alpha particle ______________________ J (4)

(Total 20 marks)

Q16. The carbon content of living trees includes a small proportion of carbon-14, which is a

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radioactive isotope. After a tree dies, the proportion of carbon-14 in it decreases due to radioactive decay.

(a) (i) The half-life of carbon-14 is 5740 years. Calculate the radioactive decay constant in yr–1 of carbon-14.

decay constant ____________________ yr–1

(1)

(ii) A piece of wood taken from an axe handle found on an archaeological site has 0.375 times as many carbon-14 atoms as an equal mass of living wood. Calculate the age of the axe handle in years.

age ____________________ yr (3)

(b) Suggest why the method of carbon dating is likely to be unreliable if a sample is:

(i) less than 200 years old,

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(ii) more than 60 000 years old.

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(Total 6 marks)

Q17. The isotope of uranium, , decays into a stable isotope of lead, , by means of a series of α and β– decays.

(a) In this series of decays, α decay occurs 8 times and β– decay occurs n times. Calculate n.

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answer = ____________________ (1)

(b) (i) Explain what is meant by the binding energy of a nucleus.

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(ii) Figure 1 shows the binding energy per nucleon for some stable nuclides.

Figure 1

Use Figure 1 to estimate the binding energy, in MeV, of the nucleus.

answer = ____________________ MeV (1)

(c) The half-life of is 4.5 × 109 years, which is much larger than all the other half-lives of the decays in the series.

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A rock sample when formed originally contained 3.0 × 1022 atoms of and no

atoms.

At any given time most of the atoms are either or with a negligible number of atoms in other forms in the decay series.

(i) Sketch on Figure 2 graphs to show how the number of atoms and the

number of atoms in the rock sample vary over a period of 1.0 × 1010 years from its formation. Label your graphs U and Pb.

Figure 2

(2)

(ii) A certain time, t, after its formation the sample contained twice as many

atoms as atoms.

Show that the number of atoms in the rock sample at time t was 2.0 × 1022.

(1)

(ii) Calculate t in years.

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answer = ____________________ years (3)

(Total 10 marks)

Q18. The decay of a radioactive substance can be represented by the equation

A = A0e–λt

where A = the activity of the sample at time t A0 = the initial activity at time t = 0 λ = the decay constant

The half life, T½ of the radioactive substance is given by

T½ =

An experiment was performed to determine the half-life of a radioactive substance which was a beta emitter. The radioactive source was placed close to a detector. The total count for exactly 5 minutes was recorded. This was repeated at 20 minute intervals. The results are shown in the table below.

time, t / minutes

total count, C, recorded in 5 minutes

count rate, R / counts

minute–1

corrected count

rate, RC / counts

minute–1

ln (RC / minute–1)

0 1016 203 183 5.21

20 892 178 158 5.06

40 774 155 135 4.90

60 665 133 113 4.73

80 608 122 102 4.62

100 546 109 89 4.49

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(a) A correction has been made to the count rate, R, to give the corrected count rate, RC. Explain why this correction has been made and deduce its value from the table.

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(b) Draw an appropriate straight line through the plotted points. (1)

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(c) Determine the gradient G of your graph.

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(d) Use your graph to determine the half-life in minutes of the radioactive substance used in this experiment.

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half-life, T½ ______________________ minutes (2)

(e) Due to the nature of a radioactive decay there will be an uncertainty in the total count recorded. What type of error is this called?

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(f) (i) It can be shown that the error in the total count C, is given by

uncertainty in total count C = ± √C

Using data from the table, calculate the uncertainty in the smallest total count, C.

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(ii) Hence calculate the percentage uncertainty in the smallest total count, C.

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(iii) Another student performed the same experiment with identical equipment but took total counts over a 1 minute period rather than a 5-minute period. The total count, C, at 140 minutes was equal to 84 counts. Estimate the percentage uncertainty in this total count, and hence explain the advantage of using a larger time.

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(Total 13 marks)

Q19. A pure sample of nuclide X containing N nuclei has an activity A. The half-life of X is 6000 years.

A pure sample of nuclide Y containing 3N nuclei has an activity 6A.

What is the half-life of nuclide Y?

A 1000 years

B 3000 years

C 12 000 years

D 18 000 years (Total 1 mark)

Q20. A radioactive source contains a nuclide which has a half-life of 12 hours. A detector placed near the source records an average count rate of 180 counts per minute. The average background count rate is 20 counts per minute.

What will be the average count rate after 24 hours?

A 40 counts per minute

B 45 counts per minute

C 50 counts per minute

D 60 counts per minute (Total 1 mark)

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Q21. The actinium series of radioactive decays starts with an isotope of uranium, nucleon (mass) number 235, proton (atomic) number 92.

Which line in the table shows the nucleon number and proton number of the isotope after the emission of 5 α particles and 2 β– particles?

Nucleon number proton number

A 213 82

B 215 80

C 215 84

D 227 87

(Total 1 mark)

Q22. After 64 days the activity of a radioactive nuclide has fallen to one sixteenth of its original value. The half-life of the radioactive nuclide is

A 2 days.

B 4 days.

C 8 days.

D 16 days.

(Total 1 mark)

Q23. The sodium isotope Na is a radioactive isotope that can be produced by bombarding the aluminium isotope Al with neutrons. Which line, A to D, in the table correctly represents the production of Na from the aluminium isotope Al and its subsequent decay?

production decay

A

B

C

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D

(Total 1 mark)

Q24. Which of the following does not give a value in seconds?

A capacitance × resistance

B

C half-life

D

(Total 1 mark)

Q25. Which of the following best describes the decay constant for a radioisotope?

A The reciprocal of the half-life of the radioisotope.

B The rate of decay of the radioisotope.

C The constant of proportionality which links half-life to the rate of decay of nuclei.

D The constant of proportionality which links rate of decay to the number of undecayed nuclei.

(Total 1 mark)

Q26. Nobelium-259 has a half-life of 3500 s.

What is the decay constant of nobelium-259?

A 8.7 × 10–5 s–1

B 2.0 × 10–4 s–1

C 1.7 × 10–2 s–1

D 1.2 × 10–2 s–1

(Total 1 mark)

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Q27. Cobalt-60 has a half-life of 5.27 years.

What is the total activity of 1.0 g of cobalt-60?

A 4.2 × 1013 Bq

B 2.2 × 1014 Bq

C 2.5 × 1015 Bq

D 1.3 × 1021 Bq (Total 1 mark)

8.3 nuclear physics - radioactive decay 1 - qs

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