Now, method three is the easiest one to spot because in this method they're talking about fractions or percentages. In method three we also utilize both the radioactive half-life equation and the radioactive integrated rate law. In this case, the questions will ask for the fraction or percentage remaining while involving half-life. Here we're going to say that the fraction of radioisotope is equal to your final concentration divided by your initial concentration. And remember if we're multiplying a fraction by 100, that will give us a percentage, in this case the percentage remaining of our radioisotope.

Here if we take a look at our examples or one example, it says here the half-life of iodine 131, an isotope used in thyrotherapy, is 8.021 days. What fraction of iodine 131 remains in a sample that is estimated to be 6.25 months old? All right, so we can use this flow chart to help us organize our thoughts. Here we have our starting position. So the first question we ask ourselves is, do you have the decay constant K? Well, if we look here, it doesn't tell us what the K constant K is. So we can't move on to yes. We have to move on to no.

Well, if it's a no, then use the radioactive half-life equation to help us isolate the decay constant K. So here half-life equals ln2 / K. We know what the half-life is. With that information, I could find my K. If I rearrange this equation, K equals ln2 / half-life, so it would be ln2 / 8.021 days. So here that would give me 0.08642 days inverse for K, right? So now I know K, so then I move on. I use the radioactive integrated rate law which is this, and I use it to solve for the initial and final concentration to find our fraction which again is equal to my final divided by my initial.

Now remember we can rearrange this equation for our integrated rate law. Since we're looking for a fraction, it can become lnfinalinitial, and that'll equal -KT. Now here we plug in the information that we know. Our K is -0.08642 days inverse. But remember K and T have to have the same units. Here, K is in days inverse, so time needs to be in days. So I'll have to change these months into days. Now this is going to be an estimate where we're going to say that one month is equal to about 30 days. Of course, some months are 31 days, February is 28 or 29 days. This is just an average, so months here cancel out. I'll get 187.5 days which I plug in here.

When we multiply these two together, that's going to give me -16.20375 which is equal to lnfinalinitial which is my fraction. I need to isolate just that fraction itself. So here, to isolate just that fraction, I'm going to take the inverse of the natural log on both sides. That's going to have the ln drop from my left side and then if I take the inverse of the natural log on the right side it becomes e to this number. And if I were to plug that into my calculator I'll get as my fraction 9 point. And here if we want to do in terms of significant figures, 3 significant figures, so 9.18 times 10 to the -8. So this would be my fraction remaining for this particular example question.