# 4.3 Half-life

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## The 'half-life' of an element

Half-life

The half-life of an element is the time it takes for half the atoms of a radioisotope to decay into other atoms.

[link] gives some examples of the half-lives of different elements.

 Radioisotope Chemical symbol Half-life Polonium-212 Po-212 0.16 seconds Sodium-24 Na-24 15 hours Strontium-90 Sr-90 28 days Cobalt-60 Co-60 5.3 years Caesium-137 Cs-137 30 years Carbon-14 C-14 5 760 years Calcium-41 Ca-41 100 000 years Beryllium-10 Be-10 2 700 000 years Uranium-235 U-235 7.1 billion years

So, in the case of Sr-90, it will take 28 days for half of the atoms to decay into other atoms. It will take another 28 days for half of the remaining atoms to decay. Let's assume that we have a sample of strontium that weighs 8g. After the first 28 days there will be:

1/2 x 8 = 4 g Sr-90 left

After 56 days, there will be:

1/2 x 4 g = 2 g Sr-90 left

After 84 days, there will be:

1/2 x 2 g = 1 g Sr-90 left

If we convert these amounts to a fraction of the original sample, then after 28 days 1/2 of the sample remains undecayed. After 56 days 1/4 is undecayed and after 84 days, 1/8 and so on.

## Group work : understanding half-life

Work in groups of 4-5

You will need:

16 sheets of A4 paper per group, scissors, 2 boxes per group, a marking pen and timer/stopwatch.

What to do:

• Your group should have two boxes. Label one 'decayed' and the other 'radioactive'.
• Take the A4 pages and cut each into 4 pieces of the same size. You should now have 64 pieces of paper. Stack these neatly and place them in the 'radioactive' box. The paper is going to represent some radioactive material.
• Set the timer for one minute. After one minute, remove half the sheets of paper from the radioactive box and put them in the 'decayed' box.
• Set the timer for another minute and repeat the previous step, again removing half the pieces of paper that are left in the radioactive box and putting them in the decayed box.
• Repeat this process until 8 minutes have passed. You may need to start cutting your pieces of paper into even smaller pieces as you progress.

Questions:

1. How many pages were left in the radioactive box after...
1. 1 minute
2. 3 minutes
3. 5 minutes
2. What percentage (%) of the pages were left in the radioactive box after...
1. 2 minutes
2. 4 minutes
3. After how many minutes is there 1/128 of radioactive material remaining?
4. What is the half-life of the 'radioactive' material in this exercise?

A 100 g sample of Cs-137 is allowed to decay. Calculate the mass of Cs-137 that will be left after 90 years

1. The half-life of Cs-137 is 30 years.

2. If the half-life of Cs-137 is 30 years, and the sample is left to decay for 90 years, then the number of times the quantity of sample will be halved is 90/30 = 3.

3. 1. After 30 years, the mass left is 100 g $×$ 1/2 = 50 g

2. After 60 years, the mass left is 50 g $×$ 1/2 = 25 g

3. After 90 years, the mass left is 25 g $×$ 1/2 = 12.5 g

Note that a quicker way to do this calculation is as follows:

Mass left after 90 years = (1/2) ${}^{3}$ $×$ 100 g = 12.5 g (The exponent is the number of times the quantity is halved)

An 80 g sample of Po-212 decays until only 10 g is left. How long did it take for this decay to take place?

1. Fraction remaining = 10 g/80 g = 1/8

2. ${\left(\frac{1}{2}\right)}^{x}=\frac{1}{8}$

Therefore, x = 3

3. The half-life of Po-212 is 0.16 seconds. Therefore if there were three periods of decay, then the total time is 0.16 $×$ 3. The time that the sample was left to decay is 0.48 seconds.

## Looking at half life

1. Imagine that you have 100 g of Na-24.
1. What is the half life of Na-24?
2. How much of this isotope will be left after 45 hours?
3. What percentage of the original sample will be left after 60 hours?
2. A sample of Sr-90 is allowed to decay. After 84 days, 10 g of the sample remains.
1. What is the half life of Sr-90?
2. How much Sr-90 was in the original sample?
3. How much Sr-90 will be left after 112 days?

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