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What is the activity of a radioactive sample after six half-lives have passed as a fraction of the initial activity of a sample?

In this question, we’re asked about the activity of a radioactive sample. We’re asked about this activity after six half-lives have passed as a fraction of the initial activity. So before we go about answering this question, let’s look at the most exotic keyword that we’ve underlined in this question. Half-Life is defined as the time taken for the radioactivity of a sample to drop to half its initial value. In other words, how long does it take for the activity of a radioactive sample to drop to half its initial value.

We need to work out the activity of a sample after six half-lives. And a good way of going about this is to set up a nice table. This table will contain the number of half-lives that have passed and the activity of the sample. Now, let’s start with zero half-lives having passed. As we said earlier, a half-life is the amount of time taken for the radioactivity of a sample to drop to half its initial value. So if zero half-lives have passed, then no time has passed. In other words, the activity of a sample must still be at its initial value.

Now, we don’t know what this initial value actually is. But as we’ll see later, it doesn’t matter what the initial activity is. We can choose to call it what we want. So let’s say we call it 𝑥. This is a common thing to call unknown variables. So it makes sense to try 𝑥. We could also label the initial activity as one. One what? Well, it doesn’t matter. And it turns out that this is probably the most convenient way of doing it. But we’ll only be able to explain that properly with hindsight. So for now, let’s run with it. And we’ll find out why one works later.

We could also say that the initial activity is some random number such as, for example, 12800. And it could be in becquerels maybe, why not. This initial activity will show us what to do when we’re given an actual number in the question as the initial activity. So say for example, a question tells us that the initial activity of a sample is 12800 becquerel. What can we do? We can calculate the activity after six half-lives being given an initial activity with the same method that we will use here. So this will serve to show you how to tackle a similar question.

Anyway, so let’s now label the rest of the half-lives. We’ve got one half-life, two half-lifes, three half-lives, four half-lives, five half-lives, and six half-lives. In the bottom part of the table, we’ll be labelling the activity of each one of these samples as one half-life has passed, as two half-lives have passed, and so on and so forth until we get to six half-lives.

So let’s start with 𝑥. Let’s say our initial activity was 𝑥. Well, what is the activity going to be after one half-life. Well, after one half-life, the activity is half the original value. So the activity is going to be half multiplied by 𝑥. Another way of writing this is 𝑥 by two.

Let’s move on to two half-lives. Well, after two half-lives, the activity is half the previous value. In other words, it falls by half again after the second half-life. So we know it’s going to be half the previous value or half of 𝑥 by two. We can write this as 𝑥 by four.

Now, when three half-lives have passed, that’s one extra half-life after two half-lives have passed. That kinda makes sense. So the value of the activity after three half-lives must once again be half of what it was when two half-lives have passed. In other words, it’s half of 𝑥 by four. And that ends up being 𝑥 by eight.

We can carry on this pattern for four half-lives, five half-lives, and six half-lives. And this is the point at which we stop because the question is asking us, what was the activity after six half-lives? So the activity is 𝑥 by 64 after six half-lives. This however is not our final answer. We’ll come back to that in a second.

Instead, let’s look at the activity if our initial activity was one. Well, if our initial activity was one, then after one half-life, the initial activity has dropped a half that value, which is a half. After two half-lives, it drops by half again. So we’ve got a quarter. After three half-lives, it has dropped by half yet again. So it’s an eighth. After four half-lives, it’s a 16th. After five half-lives, it’s one thirty-second. And after six half-lives, it’s one sixty-fourth.

Finally, if we’re looking at the initial activity as being 12800, after one half-life, it drops by half. So it becomes 12800 divided by two and that simplifies to 6400. We divide by two yet again for the activity after the second half-life has passed, which becomes 3200. Carrying on this pattern, we have 1600 as the activity after three half-lives, 800 after four half-lives, 400 after five half-lives, and 200 after six half-lives.

At this point, we found out the final activities of the samples after six half-lives assuming that the initial activity was 𝑥 or one or 12800. So everything in the middle of the table is now useless to us. All we need is the activity at the beginning — the activity after zero half-lives that is — and the activity at the end after six half-lives. So now, here’s a summary of our table.

We’ve got the initial activities that we labelled as anything we wanted: 𝑥, one, or 12800. And we’ve got the final activities after six half-lives. The question is asking us what the activity is after six half-lives as a fraction of the initial activity. In other words, it’s asking us what each one of these values is as a fraction of this — the initial activity. So let’s work out that fraction.

In this case, we’ve got 𝑥 over 64 as a fraction of 𝑥 because we want it as a fraction of the initial activity. So it’s the final activity 𝑥 over 64 divided by the initial activity which is 𝑥. And now, the 𝑥s cancel out leaving us with one over 64. In other words, the final activity is one sixty-fourth of the initial activity. We can do the same for the second row when we said the initial activity was one. So we’ve got the final activity one sixty-fourth as a fraction of the initial activity, which is one. And this simplifies to yet again one over 64.

Applying the same logic to the final row, we’ve got the final activity of the sample after six half-lives, which is 200 in this case becquerel, I think we said earlier, but again, the units are irrelevant, divided by the initial activity which is 12800. Evaluating this gives us surprise, surprise one sixty-fourth. Therefore, the final activity after six half-lives is one sixty-fourth the initial activity of the sample.

And this just goes to show that in this kind of question, it doesn’t matter what we call the initial activity. And in fact, there was a clue for us in the question that what we called the initial activity was not going to be relevant. We were asked to find the final activity after six half-lives as a fraction of the initial activity.

This meant that whatever the final activity was, we were going to have to divide it by the initial activity. And there was a high chance that something would cancel out. And that’s exactly what we saw: everything cancelled out leaving us with one over 64 in every case. And so our final answer is that the activity of a radioactive sample after six half-lives is one sixty-fourth of the initial activity.