A scientist has a pure sample of a radioactive isotope of oxygen. He wants to find out which isotope it is and does so by taking readings of the radioactivity of the sample overtime and finding the half-life. He plots the radioactivity of the sample overtime, as shown in the graph.
So, this is the graph that we’re talking about. And we can see that the scientist has plotted the radioactivity of this pure isotope of oxygen that he has over a certain period of time. In other words then, on the vertical axis, we’ve got the activity of the sample in becquerel. And on the horizontal axis, we’ve got the time in seconds. As well as this, we’ve been given a table that will become useful to us later. But the first part of this question asks us what is the half-life of the sample.
So, we’ve been asked to use this graph to find the half-life of this pure isotope of oxygen that this scientist has. But to find the half-life, we first need to know what half-life actually means. In the context of radioactivity, half-life is defined as the time taken for the activity of a sample to fall to half its initial value. In other words, how long does it take for our sample that we’ve got to be half as radioactive as it was before.
And to answer this question, we can use this graph and the conveniently labelled values of the activity and time. So, let’s start by saying that our initial activity is 1200 becquerel. In other words, this is our starting point. Now our, quote, unquote, “starting point” actually occurs 5.5 seconds after the scientist began the experiment. But that doesn’t really matter, as long as we account for this fact, as we’ll see in a second.
So, what we’re saying is that the initial activity of our sample is, let’s say, 1200 becquerel. And this occurs at a time on a scientist’s stopwatch of 5.5 seconds, assuming they started the stopwatch right at the beginning of the experiment. So, to find the half-life of this sample, we need to work out how long it takes for the activity of the sample to fall to half its initial value. And because our initial values is 1200 becquerel, we need to find out how long it takes for the activity to fall to 600 becquerel, which is half of 1200.
Now we can see, once again, from the convenient dotted lines on our graph that the 600 becquerel activity mark occurs at a time of 19 seconds into the experiment. However, this does not mean that the half-life of our sample is 19 seconds. Because, remember, we said that our initial activity of 1200 becquerel occurs at 5.5 seconds into the experiment. And the half-life of our sample is actually the time taken to go from 1200 becquerel of activity to 600 becquerel of activity. In other words then, we need to find this time interval here. The time between 5.5 seconds and 19 seconds.
And so, we can say that the half-life of the sample, which we’ll call 𝑡 subscript half, is equal to 19 seconds minus 5.5 seconds. And this time interval ends up being 13.5 seconds. Therefore, we can say that the half-life of the sample is 13.5 seconds.
And at this point, we can also see that if we decided that our initial activity was 600 becquerel, then the half-life would be the time taken to go from 600 becquerel of activity to 300 becquerel. And that time interval is between 19 seconds and 32.5 seconds, which also happens to be 13.5 seconds. In other words then, it doesn’t matter what we choose our initial activity to be. The half-life will always be the same. And it’s the time taken for the activity to fall to half of the initial value.
So, it takes 13.5 seconds to go from 1200 becquerel of activity to 600 becquerel. And it takes 13.5 seconds, yet again, to go from 600 to 300. Equivalently, it will also take 13.5 seconds to go from, say, 2000 becquerel of activity to 1000 becquerel of activity. But anyway, at this point, we found the answer to the first part of our question. The half-life of the sample is 13.5 seconds. So, let’s now look at the next part of the question.
The table shows the half-lives of different isotopes of oxygen. Which isotope of oxygen does the scientist have?
Okay, so, if we look at the table over here, we can see that the top row shows us the name of the isotope. So, we’ve got oxygen 14, oxygen 15, oxygen 19, oxygen 20, oxygen 21, and oxygen 22. And in the bottom row, we’ve got the half-life in seconds for each one of these isotopes. So, for example, oxygen 14 has a half-life of 70.6 seconds. And we need to use this table to find out which isotope of oxygen the scientist has.
But then we’ve seen already that the isotope that the scientist has has a half-life of 13.5 seconds. So, coming back to this table, we can see that the isotope which has a half-life of 13.5 seconds is oxygen 20. And so, we can state that as the answer to this second part of our question. So, to recap, the half-life of the sample that the scientist has is 13.5 seconds. And this means that the isotope of oxygen that the scientist has is oxygen 20.