So this video is all about half-life problems, particularly the ones that you'll see in a chemistry course. So consider the question that we have. Iodine-131 has a half-life of 8 days.
If there are 200 grams of this sample, how much of I-131 will remain after 32 days? Now there's two ways you can do this. You can use an equation, or if you understand the concepts, you can get the answer. Now you need to know what the meaning of a half-life is. Half-life represents the time it takes for a substance to decay.
So in this case, the half-life is 8 days. So in 8 days, half of the sample will have transformed into something else. The other half will remain iodine-131.
So we have 200 grams. of iodine-131. After 8 days, half of it will remain.
So after 8 days, we're going to have 100 grams of iodine-131 left over. That's one half-life. Now another half-life later, or another 8 days, half is going to decay into some other element, and half will remain. So we're going to have 50 grams left over.
After another 8 days, We're going to have 25 grams. So far, 24 days have elapsed until this point. Now we need to get up to 32 days. We want to find out how much is left over after 32 days.
So in another 8 days, it's going to be half of 25. So we're going to have 12.5 grams left over. So that is the amount of iodine-131 that will remain unchanged. After 32 days.
Now, is there a way that we can get the same answer using an equation? Because sometimes the numbers won't be as nice as what we just had here. Now, the first thing you want to do if you're going to use an equation, you want to find the rate constant K. The rate constant K is equal to the natural log of 2 divided by the half-life. natural log of 2 is about.6931 now this number is not going to change it's always going to be the same for this equation whenever you're dealing with a radioactive decay problem it's always associated with a first-order kinetics reaction so you can always use this equation for the most part now the half-life is eight days so let's divide these two numbers point 6931 divided by 8 is equal to point 0 8 664 now this is a round answer so our final answer won't be exactly 12.5 but it should be very close to 12.5 now once we have the rate constant k We could find the final amount using this equation.
AF represents the final amount. A initial, or AO, is the initial amount. And then it's E raised to the negative KT. The negative means that we're dealing with a decay problem. The amount of iodine that remains unchanged is decreasing over time.
So that's why we have the negative sign. So the initial amount of iodine-131 is 200. The rate constant K is 0.08664. And we want to find out how much is going to remain after 32 days. So T is 32. Now E is the inverse of the natural log function.
So simply type exactly what you see here. But make sure you put this in parentheses. So 200 times E raised to the negative.08664 times 32, and you should get this number. I got 12.501.
Now we know the exact answer is 12.5, but this is close enough. So you can use an equation, or you can use that technique to ballpark how much will remain after 32 days. Let's try this one.
Sodium-24 has a half-life of 15 hours. If there are 800 grams of sodium-24 initially, how long will it take for 750 grams of sodium-24 to decay? So feel free to pause the video as you work on this example.
So if we're starting with 800 grams of sodium-24, how much is left over? How much of sodium-24 changes and how much will remain? Notice that 750 grams of sodium-24 decays.
That means it changes to something else. The other 50 that's left over, that's how much didn't change. That's how much remains. So we need to find out how long it will take for 800 grams to decay. Leaving a final amount of 50 grams.
How can we do that? How can we find the time it takes for 800 to be reduced to 50? So we know that it's going to take 15 hours, which is one half-life, for half the sample to decay.
That means half will be left over. So we're going to have 400 that's unchanged. That's still sodium-24. The other 400 has changed into something else. 15 hours later, we're going to have 200 grams left over.
That's unchanged. And after another 15 hours, we're going to have 100 grams that's going to remain the same. And after another 15 hours, it's going to be down to 50. So then count how many half-lives.
This is 1, 2, 3, 4. It took 4 half-lives. for the sample to decay from 800 to 50. So now we could just simply add 15 four times. 15 plus 15 plus 15 plus 15 is the same as 15 times 4, and that's 60. So it's going to take 60 hours for 50 grams to remain out of the original 800 gram sample.
So now let's use an equation to get that same answer. So the first thing we're going to do is find the rate constant K, which is the natural log of 2 divided by the half-life. So if you type in ln2, which is 0.6913, divided by the half-life of 15 hours, ln2 is 0.6931. I said 13, but 0.6931.
If you divide it by 15, you should get 0.04621. So that's the rate constant K. Now our goal is to find the time. And you want to use a different form of the equation that I gave you before. You want to use this form.
Natural log of the final amount divided by the original amount is equal to negative KT. So the final amount... is 50. It's 800 minus the 750. The 750 is a change.
If 750 of sodium-24 changes, 50 remains. So the final amount that is unchanged is 50. The original amount is 800. That's AO. And K is 0.04621.
So we got to solve for T. LN 50. Divided by 800. If you type it in, you should get negative 2.7725887. And then take that number, divide it by negative.04621, and you should get the time.
What I have is 59.9998, which is approximately 60. So it takes 60 hours for 750 grams to decay. Try this problem. The half-life of oxygen 15 is 2 minutes.
What fraction of a sample of oxygen 15 will remain after 5 half-lives? So how can we do this problem? We have the half-life, we have the number of half-lives, but we don't have the original amount of the sample.
We don't know if it's 300 grams, if it's 500, and we don't know the final amount, nor do we know how much it changes by. So how can we find what fraction of a sample will remain after 5 half-lives? So if you're not given the original amount, start with 100%.
Whatever the sample is, it represents a whole, or 100%. Now, after 1 half-life, or 2 minutes... Half is going to remain, so that's 50%. After another half-life of 2 minutes, we're going to have 25% left over. And then after another half-life, it's going to be half of 25%, which is 12.5%.
And then after another half-life, it's going to be half of that, which is 6.25%. And then after another half-life, it's going to be half of 6.25%. Half of 6 is 3. Half of 25 is 12.5, or 0.125 in the case of 0.25. So this is going to be the percentage left over after 5 half-lives, which equates to 10 minutes.
So how can we use this number? to pick the right answer because the answer is a fraction. Now, one thing that you could do is you can convert the percentage to a decimal. To convert a percentage to a decimal, divide by 100, or you could simply move the decimal point two units to the left. So, 3.125% is the same as.03125 as a decimal.
Now, you can turn this decimal to a fraction, or you can convert each of these fractions into a decimal and see which one matches. 1 over 4. If you type in 1 divided by 4 in your calculator, you're going to get 0.25, so it's not 8. And if you type in 1 divided by 8, you're going to get 0.125, so it's not b. 1 over 16. that's.0625 so it's not C and 1 divided by 32 this is.03125 so the correct answer is D That is the fraction that remains.
Now, there's another way that you can get this answer, and it's a much faster way, if you understand it. So we started with 100%. You need to realize that 100% is 1. And remember, once we had 100%, in one half-life of 2 minutes, we divided it by 2, which gave us 50%.
In the second half-life, we divided it by 2 again, which will give us 25%. So if you want to do that, you can do that. you want to find out what remains after 5 half-lives, you need to divide it by 2 5 times. So 100% represents 1. So we're going to divide 1 by 2 5 times, since we want the fraction that remains after 5 half-lives. Now what's 2 to the 5th power?
If you're not sure, you can break it up. 2 to the 5th is the same as 2 squared times 2 cubed, because... 2 plus 3 adds up to 5. 2 squared, 2 times 2 is 4. 2 to the third power, 2 times 2 times 2, 3 times, that's 8. 4 times 8 is 32. So you get this fraction, 1 over 32. That's another way you can get the same answer, and it's a much faster way if you understand it.
It takes 35 days for a 512 gram sample of element X. to decay to a final amount of 4 grams. That's 4 grams unchanged.
So 508 grams decayed into another element. What is the half-life of element X? So what do you think we should do here?
What's the first thing that we should do to solve this problem? For this one, we can use the same technique as we've always been using. We can start with 512 and see how many half-lives it takes to get to 4. So the first half-life will take us to half of 512. Half of 500 is 250, and half of 12 is 6, so half of 512 is 256. Now, if we divide it by 2 again, this will give us the second half-life.
Half of 200 is 100, half of 56 is 28, so this is 128. Now, what is the third half-life? What is the final amount? After the third half-life is going to be. Half of 100 is 50. Half of 28 is 14. 50 plus 14 is 64. So after the fourth half-life, it's going to be half of 64, half of 60 is 30, half of 4 is 2, so 32, and then half of 32 is 16, half of 16 is 8, and half of 8 is 4. So it takes 7 half-lives to go from 512 to 4. So 7 half-lives, which I'm going to say 7t to the half, the half-life is the time it takes for half the substance to decay. So this variable represents the half-life.
So 7 half-lives should equal to a total time of 35 days. So if we divide 35 by 7, That tells us that the half-life is 5 days. Now let's see if we can use the equations to solve for the half-life.
In order to calculate the half-life, we need to find the rate constant k, since k is equal to ln2 times the half-life. If you rearrange the equation, the half-life is ln2 divided by k. So let's solve for k first, and then we can use that equation.
So let's use this equation. The natural log of the final amount divided by the original amount is equal to negative kT. Now we know the final amount is 4, the initial amount is 512. and the time it takes for the sample to go from 512 to 4 is 35 days so K is equal to the natural log of 4 divided by 512 and then take that whole result divided by negative 35 So natural log of 4 over 512, if you type that in you should get negative 4.852 and then divide that by negative 35. So k is equal to 0.13863.
So now that we have the value of k, we can find the half-life. So the half-life is equal to the natural log of 2 divided by k. So if you type this in, ln2 divided by 0.13863, you should get this answer, 4.999979, which you can say it's about 5 days. So that is it for this video. Thanks for watching, and have a great day.