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What is the half-life of an unknown compound if 17.0% of the compound decomposes in 150 minutes via a first-order process? 1116 minutes 59 minutes 40 minutes 279 mi...

Question

What is the half-life of an unknown compound if 17.0% of the compound decomposes in 150 minutes via a first-order process? 1116 minutes 59 minutes 40 minutes 279 minutes 558 minutes

What is the half-life of an unknown compound if 17.0% of the compound decomposes in 150 minutes via a first-order process? 1116 minutes 59 minutes 40 minutes 279 minutes 558 minutes



Answers

What is the half-life of a compound if 75 percent of a given sample of the compound decomposes in 60 min? Assume first-order kinetics.

So we know that 75% of our total decomposes in 60 minutes. We need to figure out how Maney half lives have occurred in those 60 minutes. So I always like to pretend that if we start with 100 Adams after, ah, 1/2 life, it's gonna be 50. After another half life, it's gonna be 25. And so in this case is telling us that 75% of our total has decomposed meaning we have 25% left, so 25% left of 100. Adams is gonna be 25. So right after that, we know that you've gone through 2/2 lives in order to decompose 75%. So to half lives equals 75% decomposition. So if 2/2 lives occur in 60 minutes equal 60 that means 1/2 life is just gonna be half a 60. And that's 30

Okay, in this problem, we're told the half life of a substance is 2.45 minutes, and we're told that the substance decreases from 132.8 grams to 8.3 grams and were asked to figure out how many half lives that is and how long that takes. And I'm going to show you two different ways to do this problem. The first way works because this problem was conveniently chosen with numbers that work out nicely. But the second way works for any problem, even if the numbers don't work out quite as smoothly. Okay, so the easiest way would be to figure out, ah that it takes 2.45 minutes for the substance to go from 132.8 grams down to half of that, which is 66.4. So that's been two minutes at 2.45 minutes. Now it's going to go from 66.4 grams down to half of that, which is 33.2. So that took another 2.45 minutes. And now it's going to go from 33.2 grams to half of that, which is 16.6, and that takes another 2.45 minutes. And then it also finally goes from 16.6 grams down to 8.3 grams, which took another 2.45 minutes. So that worked out to be 4/2 lives for half lives, we could say, And the total amount of time if you add all those together is 8.18 minutes. Now that worked out great, because it was exactly 4/2 lives. But if it wasn't a number that worked out so perfectly, we might want to do it a different way. So here's what I would suggest. Okay? We're going to use our formula for exponential decay, which is why equals a knot. Okay, times e to the Katie. And if we're talking about half the original amount of substance, then for why we're going to use half of a knot a knot can stay a knot and were given the time the half life as 2.45 so we can substitute that in for tea and saw for Kay. All right, so we're going to divide both sides of the equation by a knot. And that leaves us with 1/2 and I'm just gonna write. That is 0.5 equals E to the 2.45 K. And then we're going to take the natural log of both sides. So now we have natural log of 0.5 equals 2.45 k and divide both sides by 2.45 So now we have a value for K. K equals the natural log of 0.5, divided by 2.45 And that is our constant for this particular substance. Now that we have that we go back again to the equation, why equals? Why not or y equals a not each of the Katie. We substituted our final amount for why, which is 8.3 grams and our initial amount for a knot which was 132.8 grams. And we substitute in our value of K, which is natural log of 0.5 over 2.45 and we saw 14. So we divide both sides by 1 32.8 and that gives us 0.6 to 5 equals E to the natural log of 0.5 over 2.45 T. Now, at this point, you might recognize that 0.0.625 is equivalent to 1/8 or one. Let's see, What is it? 1/16 I think it is. And so that might be a clue that you are going to have an even number of half lives. Okay. From here, we're still trying to sell for tea. So we're going to take the natural love of both sides. And we have natural log of 0.6 to 5 equals the natural log of 0.5 over 2.45 times t. And finally, to get tea by itself, we want to multiply by the reciprocal of natural light 0.5 over 2.4 pipe. So, in essence, that means we're going to multiply by 2.45 and divide by natural audible 0.5. And when you plug that hole thing in the calculator, you get 8.18 minutes, and that tells you the total time it takes. And then if you take that and you divide it by 2.45 The amount of time it takes for each half life you'll get, you will get four. So that tells you you have 4/2 lives and it takes 8.18 minutes total. So same answer two different ways to get it.

In this question, we have that a material goes from ah Hun 1280 decays a minute and then 4.6 hours later 4.6 I was It is at 320 decays a minute, and we want to find the half life. So to do this, we're going to use the equation off an is equal toe and nort e to the minus lambda T. And we're also going to use the equation that half Life T is equal to London to over Lambda. So we're going to use this equation because we need to calculate Lambda to be able to use our half life equation on we've been given and not and n So that's why we need this equation to then be able to use this equation. So we'll begin with the and is equal to and not e to the minus lambda T. And that tells us that, uh, end is 320 R n North is 1280 cause and not is the number of decays at the start. And then we have e to the minus lambda. A lot of time is 4.6 hours. So we rearrange this equation. So we have 320 divided by 1280 is equal to e to mind slander times 4.6 hours. And now I haven't converted these from decays per minute because I knew that they were going to end up in a fraction like this. So if I did any conversion to the 320 to get in decays per hour, for example, because that is the rest of the former of the question I would have to do the same conversion to this 1280 on those conversions would simply cancel out. So there's no need to. So that's why I haven't done a conversion here then, too. So then carrying on to continue to rearrange this equation, we haven't e on this side, and we need to get rid of that. So we take the natural log of both sides of the equation that gives us that we have learned 3 20/1 to 80 is equal to minus Lambda Times 4.6. So gloom 3 20 over 1 to 80 is equal to minus 1.386 So we can simply divide this by minus 4.6. And that tells us our Lunda So Lamda is equal to not 0.3 01 So that is the first part off the question. The next thing we're going to do is utilize our half life equation which says that t is equal to learn to Oh, Verlander. So let's do everything out. T is equal to long to Oh, Verlander, we've just calculated London. So we're going to be putting in 12 over no 0.301 And if you put that into your calculator, you should get that. We get 2.3 hours, 2.3 hours. And I know that this is ours because I calculated our Lambda in terms off hours because of this 4.6 hours here. So the lambda we calculated, was no 0.301 per hour. So that is why our half life isn't ours. So that is how we do that question. If you were slightly confused by where I got this half life equation, I'm now just going quickly run through the derivation of that if you are confused by it and great, you finish the question so I'll just quickly wiped the board and we'll go through that derivation. So on our half lifetime, we know that our number of the case is half that that we that we started with so we couldn't right here we have, ah, half and not equals and not e to the minus lambda t. So if we divide both sides by and not we get that half is equal to e to the minus Lunda t, we then need to get rid of this e So we take the natural log of both sides. So we get the loan off. A half is equal to e is equal to minus lambda t. We then know that we can write a natural log fraction as Lund one minus loan to We also know that the natural lock of one is equal to zero so I can get rid of this. You can also get rid of the minus side from both sides by dividing by minus one. So we're left with 12 is equal to Lambda t. So then simply divided by Lunda. We get that loan to over Lambda is equal to our time when we have at the half life. So that is just are t half. So that is where the deprivation for that equation came from. He was slightly confused, but that is the end of that question.

Everybody's so we need to find the half life. Okay? And what we can dio is use this equation right here. Uh, I didn t Okay, and we need to get this alone. So we're gonna do is notice Over here. So we have one. You have that would you speaks out with Butters and for this you simply this vested takeoff e here. And when we have that you're given this. So this was over, right? So we're getting 0.69 p t half life here. People's no in our there. We need to switch everything around and we do that. What is gonna switch? So we have It's on top, you know? Yeah, I irritable with the wrong symbol there. Okay, so now up here and now get 1/2 the half life year. It's gonna be 0.69 t times. Ellen. Uh, all right, here. So in this problem we are given, so we're gonna plug and chug into this. So we are given this information of 1 20 Okay, minutes? Yeah, you get I'm of 3.6 hours and were given a 1 40 Your initial this one for Okay, so Now we're just gonna plug everything in his half like lesion. 0.693 times time. And that is 3.6 hours. And you do L A In the top one is 1 20 in our units right here are gonna cancel out. So we don't need to worry about those. Okay. And now, if you put all this into your calculator, you're gonna get 1.2 hours, okay? And that is your answer. Thank you.


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