The initial activity of a sample of radioactive material is 800 becquerel. What is the activity of the sample after three half-lives have passed?
Okay, so this is a question about radioactivity. And in this question, we’ve got a sample of radioactive material. We’ve been told that this sample has an initial activity of 800 becquerel. What we’ve been asked to do is to find out the activity of the sample after three half-lives have passed.
To answer this question, we first need to know what half-life actually means. Half-life is the time taken for the activity of a sample to drop to half its initial value. So if the initial activity of the sample is 800 becquerel, then after one half-life, its activity is going to drop to half that value. So it’s gonna go from 800 to 800 divided by two becquerels of course. And 800 divided by two is 400. So after one half-life, the activity of the sample is now 400 becquerel.
So what happens when we allow enough time for another half-life to pass? Well, after the second half-life has passed, so one more half-life, the activity once again drops by half. So we got 400 divided by two. And 400 divided by two is 200. And so the activity now is 200 becquerel.
Now we let another half-life pass. Well, the activity drops again to half its previous value, 200 divided by two. Now this happens to be 100. So the activity of the sample is now 100 becquerel. And at this point, we have waited until all three half-lives have passed.
So the sample started with 800 becquerel of activity. After one half-life, it dropped to 400. After a second half-life, it dropped to 200. And after a third half-life, it dropped to 100 becquerel. So we found our final answer. The activity of the sample after three half-lives is 100 becquerel.
Now before we go, there’s another way to write this in mathematical notation. Remember that we’re trying to find the activity of a sample after a certain number of half-lives have passed. So we start with the initial activity, let’s say 800. And then when one half-life has passed, the activity drops by half. In other words, we need to multiply this by one-half or divide it by two. It’s the same thing.
So the activity after one half-life is 800 times a half. The activity after a second half-life, well we multiply this by half once again. And the activity after a third half-life, we multiply by a half again. In other words, every time a half-life passes, we multiply our activity by one-half.
But there’s another way of writing one-half times one-half times one-half. It’s one-half cubed. So we can say that the activity of a sample after 𝑛 half-lives have passed is equal to the initial activity of that sample multiplied by one-half to the power of 𝑛.
And once again, applying this to our question, we’ve got the initial activity multiplied by one-half to the power of 𝑛. And that’s the activity after 𝑛 half-lives, where in this case 𝑛 is equal to three. So once again, to reiterate, our final answer is that the activity of the sample after three half-lives have passed is 100 becquerel.