left. Similarly after 2 half-lives the number of nuclei will have halved again so

Transcription

1 6 Radioactivity Half-life, decay constant, and carbon dating Specification reference MS 0.1, 0., 0.3, 0.5,.,.3,., 3.1, 3., 3.11, 3.1 Introduction These teacher notes cover worksheets 6.5 on half-life and 6.6 on decay constant and carbon dating. In these worksheets students will re-visit the GCSE work to ensure they have a firm foundation and then look at ways of finding the decay constant and its use in carbon dating. Learning outcomes After completing these worksheets students should be able to: explain the meaning of the term half-life calculate half-life numerically and graphically use the value of half-life to calculate the decay constant and activities or number of nuclei present at different times use half-life within carbon dating questions. Define half-life as the time taken for the number of nuclei to decay to half the initial value or the time taken for the mass of the isotope to decrease to half the initial mass. Explain this means that after one half-life, 1 of the nuclei in the original sample will have decayed and there will be 1 left. Similarly after half-lives the number of nuclei will have halved again so that there will be 1 left and 3 will have decayed. Go on to 3 and half-lives as students are often fine with 1 or but can get confused later. You may wish to show them that, for example, which means half-lives have passed, Non-mathematicians often struggle with this and are better working it out a stage at a time. Oxford University Press This resource sheet may have been changed from the original. 1

2 6 Radioactivity Reinforce with the decay graph in Figure 1 showing half-life. Where N number of nuclei in sample N0 initial number of nuclei in sample N 0 N 0 number remaining after one half-life number remaining after two half-lives Figure 1 Explain how to find the half-life from the graph as shown by the red and blue lines. Remind students to check their answers are similar and to find a mean half-life. Remind students that different isotopes have different half-lives some are in seconds, some in minutes, some in years, or even thousands of years, so the units on the time axis will vary. At A Level new definitions and equations need to be understood. Decay constant (λ) is the probability of the decay of any nucleus per unit time and needs to be linked with the half-life. In Units ( units) 1 (e.g. s 1, min 1, hr 1, etc.) λ must be in s 1 if it is going to be used subsequently in A λn. Activity (A) is the number of nuclei of the isotope which disintegrate each second. Units: Becquerels (Bq) A N t N and A A 0 e t Students need to realise that if a question gives them the half-life they can immediately find the decay constant (even if it is not required as an answer it may be needed for the next stage) and then go on to use it in the activity equation A A 0 e t. These two equations form the basis of many A Level questions. Students need reminding about taking logs of both sides of an equation. ln of e t t ln (fraction or decimal) minus number minus signs cancel Students need reminding about the principles of carbon dating from GCSE and showing how at A Level they can calculate the age of an ancient piece of wood. Students need to be provided with graph paper. Oxford University Press This resource sheet may have been changed from the original.

3 6 Radioactivity Answers Half-life 1 min 0 s ( marks) μg ( marks) 3 a See Figure b i 5 mins ii 5 mins mean 5 mins c 50 mins (8 marks) 00 litres Decay constant and carbon dating 1 a days 1 b 3.5 days (5 marks) Bq ( marks) 3 a.8 10 s 1 b c.0 hours (8 marks) 9060 years (5 marks) Worked solutions Half-life 1 The half-life of thoron is 5 s. Calculate the time taken for the activity of the thoron sample to 1 be reduced to of its initial value half-lives 60 s min 0 s ( marks) A radioactive tracer containing 5 μg of technetium-99 is injected into a patient. Technetium-99 has a half-life of 6 hours. Calculate the mass of technetium-99 which is still radioactive hours later. 6 hours hours 1 1 Mass of technetium = 5 = g 16 ( marks) Oxford University Press This resource sheet may have been changed from the original. 3

4 6 Radioactivity 3 Table 1 shows the variation of the activity of a sample of iodine-18 with time. Time/mins Activity/Bq Table 1 a Plot a graph of activity against time for the sample of iodine-18. b c ( marks) Determine the half-life of iodine-18 by: i calculating the time taken for the activity to fall from 60 Bq to 30 Bq 5 mins ii calculating the time taken for the activity to fall from 0 Bq to 0 Bq. 5 mins Mean half-life 5 mins Calculate the time taken for a sample of iodine-18 to fall from 300 to 75 Bq. Two half-lives 5 50 mins A small quantity of the radioactive isotope sodium- of half-life 15 hours and initial activity Bq is inserted into a central heating system and left for 30 hours until thoroughly Oxford University Press This resource sheet may have been changed from the original.

5 6 Radioactivity mixed with the water. After this time a 100 ml sample is removed from the system and its activity is measured as Bq. Calculate the total volume of water in the system. 30 hours After this time the activity of original sample 000 Bq 100 ml gives.0 Bq Volume of water in system ml 00 litres Decay constant 1 A sample of phosphorus-3 has an activity of Bq. Phosphorus-3 has a half-life of 1.3 days. a Calculate the decay constant. In days 1 ( marks) b Calculate time taken for the activity to fall to 3000 Bq. A A0 e λt e 0.085t e 0.085t ln t t t days A sample of iodine-131 which has a half-life of 8.0 days has an initial activity of Bq. Calculate the activity after 8 days. In days 1 A A0 e λt A e ( Bq Bq mark) ( marks) Oxford University Press This resource sheet may have been changed from the original. 5

6 6 Radioactivity 3 A radioactive source has an initial count rate of 111 counts per second. After 10 minutes the count rate has dropped to 8 counts per second. Background radiation 3 counts per second. a Calculate the decay constant. Subtract background from each activity. A A0 e λt e λ 600 ln s 1 b Calculate the number of nuclei in the original sample if it can be assumed that the detector counts all of the radiation emitted from the source. A λ N N 108 N.8 10 c Calculate the half-life of the source. Oxford University Press ( marks) In s hours An oak roof beam in a barn has an activity of 0.0 Bq. An equal mass of living oak has an activity of 1.30 Bq. Calculate the age of the roof beam. Give your answer in years to three significant figures. The half-life of carbon-1 is 5570 years. The half-life, T, in seconds s s So, the decay constant of carbon T s 1 Using activity A A0 e λt where A 0.0 Bq and A Bq gives This resource sheet may have been changed from the original. 6

7 6 Radioactivity e λt so e λt λt 1.19 t s 9060 years (5 marks) Oxford University Press This resource sheet may have been changed from the original. 7