# Video: Calculating the Activity of a Radioactive Source

In this video, we will learn how to calculate the activity of a radioactive sample after a given amount of time using the sample’s half-life.

14:43

### Video Transcript

In this video, we will be calculating just how radioactive a radioactive source is.

So, let’s start by recalling that a radioactive source emits ionizing radiation, either 𝛼 particles like the ones shown here, or 𝛽 particles, or 𝛾 rays. However, each source of radiation is different, and some sources emit more radiation than others do. So, how do we go about comparing different radioactive sources in terms of how radioactive they are? Well, we do this by, defining a quantity known as the activity of a radioactive source.

To understand this, we need to recall that regardless of the type of radiation emitted by a source 𝛼, 𝛽, or 𝛾, the radiation itself is emitted from the source because nuclei in the source end up decaying. For example, in the first of the three equations that we’ve written down here, we can see that a uranium nucleus is decaying to a thorium nucleus and also producing an 𝛼 particle. And when lots of these decays happen, lots of 𝛼 particles are released. And 𝛼 radiation consists of these 𝛼 particles.

Similarly, we can see that a thorium nucleus, for example, can decay to a protactinium nucleus plus a 𝛽⁻ particle. And in the third example, we can see an excited state cobalt nucleus decaying into its ground state, which basically means it just loses energy. And that energy is lost in the form of 𝛾 rays. And so, in all of the three examples that we’ve written here, we can see that some nucleus decays in order for radiation, whether that’s 𝛼, 𝛽, or 𝛾 radiation, to be emitted.

Well, we can use the fact that nuclei have to decay to emit radiation in order to define the activity of a radioactive source. Now, the activity of a radioactive source is simply a measure of how many nuclei in that source are decaying per unit time. And so, we can write down this definition of the activity. We can say that it’s the number of nuclei decaying per unit time. And of course, it goes without saying that we’re referring to the number of nuclei decaying in our radioactive source per unit time. Because remember, a radioactive source might just be a lump of radioactive material, which consists of millions of nuclei that are radioactive.

And so, looking at this definition of activity, we can see that if a larger number of nuclei are decaying per unit time, let’s say per second, then the activity of the sample that we’re considering is higher. And in fact, activity even has its own units. Activity is measured in a unit known as becquerel, or capital B lower case q for short. Where one becquerel is defined as one nucleus decaying per second.

So, using this definition of activity as well as the unit of activity, the becquerel, we can find the activities of different radioactive sources and compare them to see which source emits more radiation per unit time. But here’s the thing. Let’s say we consider a very small sample of radioactive material. Let’s say this is our sample, and it consists of 100 radioactive nuclei. In other words, each orange dot represents a nucleus that could undergo radioactive decay.

And at this point, it doesn’t matter what kind of decay, whether it’s 𝛼, 𝛽, or 𝛾. But the point is that each one of these orange nuclei can, at some point, decay and emit radiation. And we could, for example, represent that by the nucleus turning pink and emitting, let’s say, an 𝛼 particle. But we’ll even ignore the 𝛼 particle for now because, like we said, it doesn’t matter what kind of radiation we’re thinking about.

But anyway, so here’s our sample of radioactive material with 100 nuclei that could decay. And let’s say that in the first second, 20 of those nuclei decay. Now, obviously, in reality, those 20 would be spread out over the entire sample, but we’re gonna say, for argument’s sake, that these 20 have decayed just to make it easier for us to visualize what’s going on. Now, if we were to plot the number of nuclei decaying per second over time, then our axes could look something like this. We could have on the vertical axis the number of decays and on the horizontal axis we could have time in seconds. Now, remember, what we’re plotting is the number of nuclei decaying every second.

And so, because the number of nuclei decaying in the first second is 20, we could plot 20 decays against either zero seconds or one second, depending on how we think about it. Because if we plot it against zero seconds, then we could say that between zero seconds and the next second, 20 decays have occurred. Or if we plot it against one second, then we can think about it as after one second has passed, 20 nuclei have decayed. It doesn’t really matter as long as we interpret the graph correctly. But for simplicity’s sake, let’s plot it against zero seconds. In other words, 20 nuclei decay between zero seconds and the next second.

But then, what happens after this? We’ve seen that in the first second, 20 nuclei decay. Well, in the next second, there are now fewer nuclei left over that are able to decay. Because remember, at the beginning we had 100 nuclei, but after one second has passed, 20 of them have decayed and therefore can no longer decay again. And the active sample now only consists of these 80 nuclei. And each nucleus has the same probability of decaying as before. But because we’ve now got fewer nuclei left over that can decay, this means that in this second, less than 20 will decay.

As it turns out, roughly 16 nuclei will decay in the second second. And so, we can plot that on our graph as 16 nuclei decaying between one second and the next. But then, what happens after this? Well, the same situation as before. Now, altogether, 36 nuclei have decayed, and we’ve got 64 left over that can decay. And once again, each nucleus has the same probability of decaying as earlier. But because there’s fewer nuclei left that could decay, in this second, less than 16 will decay. And it turns out that about 13 nuclei will decay in that second. It could be 12 or 14, but it’s most likely to be 13.

And we can continue this pattern and see that within the next second about 10 nuclei decay. And we can continue this pattern and plot points as time progresses. And we’ll see a graph shaped something like this. In fact, if we were to start with a much larger sample with many more nuclei, let’s say millions of nuclei, and plot the number of nuclei against time, then our graph would look something like this. We can see that the curve approaches the horizontal axis as time progresses, but never quite touches the horizontal axis.

And what this means is that at the beginning, a large number of nuclei decay. And then, as time progresses, every second, let’s say, a fewer number of nuclei decay, and so on and so forth. And we can see that at each second, the number of nuclei decaying within that second is decreasing. And therefore, we can say that the activity of the sample is decreasing. Because, remember, the activity of a sample is defined as the number of nuclei decaying per unit time, in this case per second.

And it’s actually very important to remember this graph shape. Where the number of nuclei decaying per second, or in other words the activity, is high at the beginning, and it gets lower and lower, and the graph approaches the horizontal axis but never quite touches it. In other words, our curve starts out steep and then gets flatter and flatter as time progresses.

And the reason that this is important to remember is because every radioactive sample displays this behavior when we plot the number of nuclei decaying per second against time. In other words, let’s say we started with another sample which had the same number of nuclei at the beginning as our first sample. Now, this sample might have a slightly different-looking curve, something like this, for example. But the idea is that even this curve shows the behavior that lots of nuclei decay at the beginning. And then, as time progresses, the curve gets flatter and flatter, approaching the horizontal axis but never quite touching it.

Another way to think about this is that these two curves are very similar in the sense that they’ve got the same shape but just different values on the axes. And we can choose to think about it whichever way we want. Now, let’s go back for a moment to thinking just about our first sample. Well, looking at this curve, we can define a new quantity. We can define a quantity known as the half-life of a radioactive sample. Now, this might sound a little bit ominous, but in reality, it’s a lot more harmless.

Now, the half-life of a radioactive sample is defined as the amount of time taken for the activity of our radioactive sample to drop to half its initial value. And we can clearly see this on this graph over here. The graph that shows the number of decays per second, or in other words the activity, of our sample, which initially had 100 nuclei that could decay. And the way that we can find the half-life of this sample is the following.

We firstly start with our initial activity. We see that initially, there were 20 nuclei decaying in the very first second. And hence, our activity was 20 becquerel, 20 decays per second. Well, then, the half-life of this particular sample is the amount of time taken, which we’ll try and figure out, for the activity to drop to half its initial value. In other words, it’s the time taken for the activity to fall from 20 becquerel to 10 becquerel, which is half that value. And so, we go back to our graph and see that 10 becquerel of activity occurs at this point in time here. Where this point in time is three seconds later.

And so, what we’re saying is that at the beginning, the activity was 20 becquerel. And then, three seconds later, the activity was 10 becquerel. Therefore, the half-life of this particular radioactive sample is three seconds. So, that’s what we mean by half-life. But here’s where things get interesting. Remember we said earlier that every single radioactive sample will have the same-shaped graph when plotting the activity against time, with the caveat that it could be slightly more squashed or slightly more stretched. Well, the reason for this is the following.

If we were to instead come back to this graph, and if we were to now decide that our initial activity was actually 10 becquerel. So, in other words, we were starting to consider our sample when the activity was 10 becquerel. Then, the half-life would be the time taken for the activity to fall to five becquerel. And we can see on our graph that the activity falls to five becquerel at this point in time here. In other words, at six seconds after we started looking at our sample.

But remember, the half-life of a sample is the time taken for the activity of the sample to fall to half its initial value. And we had arbitrarily chosen that our initial value now was 10 becquerel. But that occurred at this point in time here. And so, the time taken to fall to half its initial value is actually the time between here and here, which also happens to be three seconds. And hence, regardless of what we choose our initial activity to be on this graph, it will show us that the time taken for the activity to fall to half that initial value is always three seconds in this particular case. Which is why we can so confidently define the half-life of our particular sample.

And coming back to the three graphs that we’ve drawn here, we can say that a more squashed graph is one which has a lower half-life because it takes less time to go from the initial value to half the initial value. Whereas the more stretched graph is one which has a higher half-life because it takes longer to go from the initial value to half the initial value. And so, the half-life of the squashed graph sample is this, whereas the half-life of the stretched graph sample is this.

And so, once again, to reiterate, every single radioactive sample will show the same-shaped curve when plotting the number of nuclei decaying every second, or in other words the activity, against time. But each will have its own half-life. And the half-life is simply determined by the kind of substance our sample is made from. Because each isotope of radioactive material has its own half-life. And, of course, as a quick note, the other way to change the way that this graph looks would be to have a sample of a different size.

Because, let’s say we had a larger sample. Then, the number of nuclei at the beginning would be slightly different. But if we were to plot the activity of the sample against time, then the shape of the graph would be the same in that we see large numbers of decays at the beginning, and then the number of decays would decrease as time progresses. But the half-life that we’d find on the graph for that particular sample will be dependent on what the sample is made from. But anyway, so, now that we’ve defined the activity of a sample and the half-life of a sample, let’s take a look at an example question.

The initial activity of a sample of radioactive material is 1200 becquerel. What is the activity of the sample after one half-life has passed?

Well, to answer this, let’s start by recalling that the activity of a sample is defined as the number of nuclei decaying per unit time, in other words how many nuclei decay every second. And we can also recall that one becquerel is the unit of activity and is defined as one nucleus decaying per second. And therefore, if we’re told that the initial activity of our sample is 1200 becquerel, then we know that initially 1200 nuclei are decaying every second.

Now, what we’ve been asked to do is to find the activity of the sample after one half-life has passed. So, at this point, we can recall that the half-life of a radioactive sample is defined as the amount of time taken for the activity of the sample to fall to half its initial value. In other words, half-life is an amount of time. Now, we haven’t been given this amount of time in the question itself, but we’ve been told that one half-life has passed, which means that we don’t need to know that amount of time. All we know is that the amount of time equivalent to one half-life has passed.

So, if the half-life of our sample happened to be five seconds. Then, we’ve been told that five seconds have passed. If it happened to be 10 million years, then we’ve been told that 10 million years have passed. And the point is that that amount of time has passed after we measured the activity to be 1200 becquerel. And since the half-life of a sample is defined as the time taken for the activity to fall to half its initial value, we can say that after one half-life has passed, the activity will have fallen to half the value we measured at the beginning. In other words, we can say that the activity of the sample after one half-life has passed is equal to the initial activity multiplied by half, or divided by two if we want to do it that way.

And this ends up being 600 becquerel. And so, what this question is trying to tell us is that at the beginning, we measured our activity to be 1200 becquerel, or 1200 nuclei decaying every second. And then, we allowed some time to pass. After the amount of time equivalent to one half-life of that particular sample had passed, the activity must have fallen to half its initial value because that’s the definition of half-life. And so, our answer to this question is 600 becquerel.

So, now, that we’ve had a look at an example question, let’s summarize what we’ve talked about in this lesson. We firstly saw that the activity of a radioactive sample is defined as the number of nuclei in that sample decaying per unit time. We also saw that the activity is measured in a base unit of becquerel, represented as uppercase B lowercase q, where one becquerel is equivalent to one nucleus decaying per second. And finally, we saw that the half-life of a sample is defined as the amount of time taken for the activity of the sample to drop to half its initial value. And so, using these newly learned terms, we can compare the level of radioactivity of different radioactive samples.