Hello, my name's Dr.

George.

This lesson is called Activity and Half-Life Calculations, including complex ones, and it's part of the unit nuclear physics.

The outcome for this lesson is "I can interpret radioactive half-life graphs and make calculations using values of half-life." Here are the key words for this lesson.

I'll introduce activity and half-life later, but first I'd like to remind you what radioactive isotopes are.

Isotopes are different versions of atoms of the same element.

They have the same number of protons in their nucleus, but different numbers of neutrons.

And isotopes that are radioactive have unstable nuclei that will decay over time and emit ionising radiation.

This lesson has two parts.

They're called activity and half-life and radioactive half-life calculations.

The nuclei for radioactive isotope are unstable, and they'll decay over time to form new isotopes.

For example, in alpha decay on the left, a parent nucleus decays to produce a new daughter nucleus and an alpha particle that's emitted.

Similarly, beta decay can happen in which a beta particle is emitted.

The daughter nuclei are more stable than the parent nuclei, and they may be completely stable.

Stable isotopes don't decay.

And so over a period of time there are fewer and fewer decays per second as the number of unstable nuclei goes down.

So as time passes, a radioactive source actually becomes less radioactive because there are fewer unstable nuclei left still to emit radiation.

The number of decays per second is called the activity of the sample.

A sample with a high activity has many decays per second, and a sample with low activity has fewer decays per second.

Activity's measured using the unit becquerel, named after the physicist Henri Becquerel, One becquerel simply means one decay per second.

So 100 becquerels would mean 100 decays per second, and a kilobecquerel would mean 1,000 decays per second, because the prefix kilo always means 1,000, as in kilometre or kilogramme.

In a pure sample of a radioactive isotope, all of the nuclei are identical, so the chance of any one nucleus decaying is the same as for any other.

The activity of the sample is proportional to the number of nuclei that it has, and that's also proportional to the mass.

The more nuclei there are, the more atoms there are, and each atom has the same mass.

So if you double the number of atoms say, then you double mass.

This one milligramme sample has an activity of 200 kilobecquerels, so two milligrammes of it would have an activity twice that.

And if we double the mass again, we double the activity again to 800.

So here's a question for you.

Which of these three samples of the radioactive isotope plutonium-240 has the lowest activity? Each time I ask a question, I'll wait for five seconds, but if you need longer, press pause, and press play when you've chosen your answer.

The correct answer is the sample with the lowest mass.

That's B.

It has the lowest mass, so it has the lowest number of atoms, the lowest number of nuclei, and so the lowest activity, the lowest number of decays per second.

The two masses on the left use the units micrograms, millionths of a gramme, and the one on the right uses milligrammes, thousandths of a gramme.

As a sample decays over time, the number of remaining radioactive nuclei, unstable nuclei, decreases, and that means that the activity of the sample will decrease over time.

Here's an example graph of activity against time for a sample of a radioactive isotope.

And for a pure sample of a radioactive isotope that decays into a stable daughter, this is always the shape of the activity graph against time.

And there is a predictable pattern here.

For this isotope, if we read the activity after two hours, we find that it's half of the original activity.

It drops from 160 to 80.

But after another two hours, it's halved again from 80 to 40.

And after another two hours, so six hours after the start, the activity is halved again from 40 to 20.

And it turns out that the time for the activity to halve has a constant value for any particular isotope.

You could start at any point on this graph, read the activity, and then you'd find that two hours after that, the activity would be half of what it was.

And the time for the activity of a sample of radioactive isotope to halve is its radioactive half-life, also just called half-life.

The activity halves every half-life.

So looking at the graph again, we saw that after one half-life the activity halved.

In the second half-life it halves again.

So it's now a quarter of its original value.

And in the third half-life it halves again.

So the activity is then 1/8 of its original value.

Now take a look at this graph, which is for a sample of a different radioactive isotope.

Can you find the half-life of this isotope? And the answer is C.

And here's a way that you could see that.

The starting activity is 120.

So we could look at how long it takes to fall to half of that, 60, and it's six minutes.

Well done if you got that.

There are two ways that we can define the half-life of a radioactive isotope.

We could say the half-life of a radioactive isotope is a time taken for half of its nuclei to decay.

But we can also say the half-life of a radioactive isotope is a time taken for its activity to fall to half of its initial value.

Both of these take exactly the same amount of time for any particular radioactive isotope.

And so both of these are correct definitions for half-life.

Here's another graph question for you, but notice the different label on the axis.

This time it's not showing activity, it's showing percentage of nuclei remaining.

Now what's the half-life of the radioactive isotope? The answer is B, two and a half years.

Remember, the definition of half-life is also the time taken for half of the nuclei to decay.

So if we look at the time when 50% of the nuclei remain, it's two and a half years.

Now, which of the following are definitions of radioactive half-life? And it's C and D.

As we saw before, the time taken for half of the nuclei to decay or the time taken for the activity to fall to half of its original value.

Half-life isn't half the time it takes for all of the radioactive atoms to decay.

That wouldn't be the same thing.

And by the way, you might like to think about why we don't use the idea of something like whole life, the time taken for all of the nuclei to decay.

Why would that not be a predictable amount of time for a sample? Now here's a task for you.

A scientist monitors the activity of a sample of radioactive isotope over a period of time, and the results are shown in this table.

I'd like you to plot a graph with activity on the y-axis and time on the x-axis for this isotope and choose sensible scales while trying to make sure that your graph fills most of the space on the page.

And then use your graph to determine the half-life of this isotope.

When you draw the graph, don't join the points with straight lines.

Try to draw a best fit curved line.

Press pause while you do the activity and press play when you're ready to check your answers.

Let's take a look at the answers now.

Your graph should be similar to the one shown here, although yours will also show crosses for the plotted points.

The half-life we could read from here.

The time taken for the activity to drop from 80 to 40 is three and a half hours.

If you want to be more sure of one of these measurements from a graph, you could measure the half-life, say three times with different start times, and then find the average.

Now let's move on to the second part of the lesson, radioactive half-life calculations.

The predictable pattern in radioactive decay can be used to determine how much of a sample will remain after a certain length of time and what its activity will be.

This is useful when we are using radioactive samples, for example, in medicine, or also when we're thinking about how long radioactive waste will remain dangerous.

So we've already seen that after one half-life the activity is half of its original value, and after two the activity is half of that, and after three half-lives, the activity is half of that.

And this pattern continues, and we can use this to make predictions.

Try this question.

What happens to the activity of a radioactive isotope with a half-life of two hours if it's left for eight hours? The correct answer is C, it falls to 1/16 of the original value.

You can work that out from seeing that eight hours is four half-lives, four times two hours, and so we need to halve the original activity four times.

If we do that, we'll be finding 1/16 of the original activity.

Well done if you got that.

So we can use this to predict the actual activity in becquerels of a sample of radioactive isotope after a certain time, as long as we know the initial activity and the half-life.

I'll show you how to answer this example question.

The isotope francium-225 has a half-life of four minutes.

A sample of the isotope has an activity of 60 kilobecquerels.

Find the activity of the sample after 12 minutes.

So first we need to think about how many half-lives 12 minutes would be.

And it's three half-lives, 12 divided by 4.

So the activity will have halved three times during three half-lives.

So we need to take the initial activity, halve three times.

That's the same as multiplying by a half three times.

And if we do that we get 7.

5 kilobecquerels.

I'll show you another example, and then I'll give you a question to try.

The half-life of curium-273 is 20 minutes.

What fraction of the initial sample would remain after one hour? One hour is 60 minutes, so that's three half-lives for this isotope.

So the fraction remaining after three half-lives will be half of half of half of the original amount, which is an eighth.

And now a question for you.

The half-life of rutherfordium-263 is 15 seconds.

What fraction of the initial sample would remain after one minute? First of all, we note that one minute is 60 seconds, which is four half-lives, 4 times 15 seconds.

So the fraction remaining after four half-lives is half of half of half of half of the original, and that's 1/16 of the initial sample.

Well done if you got that.

We can also work backwards.

We can work out how long it takes for the activity to fall by a certain fraction of the original activity.

For example, the isotope lead-199 has a half-life of 1.

5 hours.

How long will it take for the activity of a sample of this isotope to fall to 1/8 of the initial value? This time we need to realise that 1/8 is half of a half of a half.

And so three half-lives need to pass.

We know the half-life is 1.

5 hours, and so the time needed is 3 times 1.

5, which is 4.

5 hours.

Now again, I'm going to go through another question for you, and then ask you a question.

The half-life of samarium-145 is 340 days.

How long would it take for the activity of a sample to fall to 1/16 of its original value? It takes one half-life to fall to half of its original value, two to fall to a quarter, three to fall to 1/8, and four half-lives to fall to 1/16 of its original value.

We've been given the half-life, so we need 4 times 340 days, 1,360 days.

And now a question for you.

The half-life of curium-239 is 2.

5 hours.

How long will it take for the number of nuclei in a sample to fall to 1/8 of the original number? Again, one half-life is how long it takes to fall to half of the original value, two to fall to a quarter, three to 1/8.

So we need three half-lives, which is 7.

5 hours.

Well done if you're getting these questions right.

And here are two final questions for you.

In the first one, you're given a table of four isotopes.

I'd like you to try to fill in the missing activities and the missing half-lives.

In question two, you don't have to do calculations in your answer.

You could write a descriptive answer for this one.

Press pause while you do these, and press play when you're ready to check your answers.

And here are the answers.

For isotope A, the half-life is 20 seconds, so we simply need to halve the activity every 20 seconds as shown here.

For isotope B, the half-life is 40 seconds.

We wouldn't be able to calculate the activity after 20 seconds or 60 seconds, but we don't have to because those are shown.

We can work out the activity after 40 seconds.

We halve the original activity of 640, and then we halve again to get the activity after another half-life.

In C, we have all the activities, but we don't know the half-life.

The activity goes from 200 to 141 in the first 20 seconds.

That's not halving, and we can't work out the half-life from that.

But we can also see that the activity falls from 200 to 100 in the first 40 seconds, and it halves again from 100 to 50 in the next 40 seconds.

So 40 seconds must be the half-life.

And in D, we don't know the half-life, but we can see that in 20 seconds, the activity falls from 1,200 to 300.

300 is a quarter of 12, so the activity is halved twice.

That means that 20 seconds must be twice the half-life.

And so the half-life is 10 seconds.

In question two, you were asked to consider lead-186 with a half-life of 4.

8 seconds, and lead-200 with a much longer half-life of 21.

5 hours.

The question asked you to compare the activities of samples of these over a period of a day.

The activity of the lead-186 would decrease very quickly as it has a short half-life.

It would be a tiny fraction of the original after one day, 'cause one day as many half-lives for this isotope.

The activity of the lead-200 would also decrease but much more slowly.

It would've fallen by just over half in one day because one day, 24 hours, is a little more than one half-life for this isotope.

Well done if you've got many or all of these correct.

And now we've reached the end of the lesson.

So here's a summary.

The activity of a radioactive sample is a number of decaying nuclei per second, and it's measured in becquerel, symbol Bq.

The half-life of a radioactive isotope is a time taken for the activity to fall to half of its original value.

After each half-life activity has fallen by half.

After two half-lives, activity is a quarter of the original activity, half times half.

After three half-lives, the activity is 1/8 of the original activity, half times half times half.

An isotope with a high activity has a short half-life.

Well done for working through this lesson.

I hope you enjoyed it, and I hope to see you again in a future lesson.

Bye for now.