half life

What is the half life of a radioactive element?

Half-life:

1. Radioactive decay is a random process. This means that all the unstable nuclei have the same probability to decay at a certain instant. It is not possible to predict when a particular nucleus is going to decay.
2. The unstable nuclei will decay at different times, some decay earlier while others will decay at a much later time. Therefore, the number of undecayed nuclei decreases with time.
3. The half-life of a radioactive sample is the time taken for the number of undecayed nuclei in the sample to be reduced to half of its original number.
4. However, this applies only when the number of undecayed nuclei in the sample is very large. If the number is small, then the number of undecayed nuclei left after one half-life is only expected to be halved. This means that the actual number left may be slightly more or slightly less.
5. Xenon-140 has a half-life of 40 minutes.
   It takes 40 minutes for the number of undecayed nuclei to decrease to half its original number. Figure illustrates the decay of xenon-140 over 3 half-lives.
   
6. Some radioactive nuclides have very short half-lives while others have very long half lives. Table gives the half-lives of some radioactive nuclides.
   

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Half-life and Activity

1. The activity of a radioactive substance is the number of decays per second of the unstable nuclei.
2. The activity is also called the decay rate.
3. As the number of undecayed nuclei decreases with time, the activity of the radioactive substance also decreases with time.
4. For a radioactive substance with a relatively short half-life, the change in the activity can be measured by a GM tube connected to a ratemeter as shown in Figure.
   
5. The activity of the radioactive source is given by the count rate recorded by the ratemeter.
6. The count rate is recorded at regular time intervals and a graph of the count rate against time is plotted to obtain the decay curve.
7. Figure shows the decay curve for antimony-133 which has a half-life of 2.5 minutes.
   
8. A radioactive nuclide with a short half-life will decay at a faster rate than another radioactive nuclide that has a longer half-life.
9. Figure shows the decay curve for two radioactive nuclides, P and Q.
   

Half Life Problems and Answers

1. Tin-108 has a half-life of 10.3 minutes. A radioactive sample contains 72 g of tin-108. Determine the mass of tin-108 that has decayed and that has not decayed after 30.9 minutes.
Solution:
The mass of tin-108 that has not decayed will be halved every 10.3 minutes.
30.9 minutes is equivalent to 30.9/10.3 = 3 half-lives.
The decay process is as follows:

After 30.9 minutes, 9 g of tin-108 has not decayed and 63 g of tin-108 has decayed.

2. The radioactive atoms in a substance decay to become stable atoms. It was found that after 136 s, 93.75% of the atoms have decayed. What is the half-life of the substance?
Solution:

3. A sample of protactinium-234 of mass 100 g has a half-life of of 6.7 hours.
(a) What fraction of the sample has not decayed after 20.1 hours?
(b) What is the mass of undecayed protactinium-234 after this period of time?
Solution:

4. Figure shows the decay curve of a radioactive sample.

(a) What is the half-life of the sample?
(b) State the value of T.
Solution: