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A radioactive sample contains 3.50$\mu \mathrm{g}$ of pure $^{11} \mathrm{C},$ which has a half - life of 20.4 min. (a) How many moles of $^{11} \mathrm{C}$ are present initially? (b) Determine the number of nuclei present initially. What is the activity of the sample (c) initially and (d) after 8.00 h?

a. 3.18 \times 10^{-7} \mathrm{mol}

b. 1.91 \times 10^{17} \text { atoms }

c. 1.08 \times 10^{14} B q

d. 8.92 \times 10^{6} B q

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number 51 we have 3.5 micrograms of carbon 11 which is radioactive and has 1/2 life of 20.4 minutes and apart. Eight. Where is to find how many moles that is? I'm just gonna do one big look conversion thing. Some starting with 3.5 micrograms. Um, one microgram the same as one times 10 to the negative sixth grams microgram counsel I can go to. That's pretty sloppy there. Um, I go from grams to use. I know that one U is the same as 1.67 times 10 to the negative 24th grams. So Graham's canceled. And now for this carbon 11 I know that, um, there's 11 point 011 use in each. Adam. That was the mess in the back on independents. Pete. So you canceled alone. And how many atoms are in a bowl? That's other governors number so 6.0 to to times 10 of the 23rd. That's money, Adams. Or in one more So Adam's Kenson. So now I just needed to this math. This times this divided by this divide by this divided by this equals 3.16 times. 10 to the negative self. Where the river Negative seven. Well, for the pure put down here, that's sloppy. But that so many moles I have We heard the Marge my page. Sorry, can't do. But, um, And the Newport bi they want to know how many Adams that is. Well, if I would just stop right here, just go to the same problem up to here. That would be though the Yemeni Adams. I have. And that was 1.90 times 10 to the seventies. Actually, the S committee nuclear I have. And each Adam has one nucleus. So this would be the number of nuclei, so that was partly he was part B apart. See, we wanna know. What was the activity of that sample initially? And what is that? The activity of that sample. After eight hours, you put that Ever heard my last eight hours? Well, the activity I know the number off nuclear I have, and that is related. Bye. This equation where if I took the number of nuclei times the decay constant, that would equal the activity. So I'm trying to get so if I have the original number of nuclear that we've given the original activity. So my initial this remember, with decay constant how you get that The half life of something is the natural log of two, which was 0.693 over that decay constant. So if I have these switch places, so I know that my decay constant will be 0.693 over the half. Life and the half life has to be in seconds. So remember this was my half life over here, So I'm gonna do 20.4 time 60 to convert that into seconds instead of minutes. So that's gonna be my decay constant. So I'm gonna come over him, plug that in. So my original number of new class, which I found over here 1.90 understand the 17th. It was seven. At times my decay constant, Which is this 0.693 divided by 20.4 times 60 to convert that the seconds and that will be my original activity. For that, I get 1108 times 10 to the 14th. And that being becquerels big you. So that's my original activity. Enough. I want to find what it is sometime later I'm gonna use this question that I always used for half life. It's my favorite equation. It's technically gets the number of nuclear I knew. Number of nuclei is the original number of nuclei comes 1/2 raised to the number of half lifes. Now, remember my equation here, though If I would just take these number of nuclear and multiply it by the bikes decay constant, then it's gonna be activity instead. So I'm just gonna plug that in here. You know, I don't there now, with now this is the activity and that's activity instead of number of nuclear. So this is what I want the new activity. So I'm going to call that or to equal the original activity which has just found over here times 1/2 raised to the number of half lives. Now here it doesn't matter, you know. I want I want the number of half life's so the half life is 24 minutes and I want to go for eight hours. So eight hours divided by you know that 24 minutes of they'll tell how many half lifes that is. It doesn't matter what Unity's Aaron as long as that. The same. So for to get this number of half life. So I'm gonna take that eight hours and multiply it by 60. So it's also in minutes over my half life, which is 20.4. So this is 1/2 raised to this powers. We're gonna figure that out. I do all that and I get My new activity is 8.92 times 10 to the sixth, and that's in the back row.

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