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Problem 23 Hard Difficulty

A freshly prepared sample of a certain radioactive isotope has an activity of 10.0 mCi. After 4.00 h, the activity is 8.00 mCi. (a) Find the decay constant and half - life of the isotope. (b) How many atoms of the isotope were contained in the freshly prepared sample? (c) What is the sample’s activity in mCi 30.0 h after it is prepared?

Answer

a. 12.4 h
b. 2.39 \times 10^{13} \text { nuclei }

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Top Physics 103 Educators
Christina K.

Rutgers, The State University of New Jersey

Andy C.

University of Michigan - Ann Arbor

LB
Liev B.

Numerade Educator

Zachary M.

Hope College

Video Transcript

for number 23. We have a radioactive sample that has an activity of 10 militaries. And then after four hours, it has an activity of eight. Milic yours and where has to find his dick? A constant. And they have life. Should be. Yeah. Okay. I like to use this equation. I like to use, um, where n is. It's actually the number of nuclei, and this is 1/2 raised to the number of half life's. Now, I know we don't have the number of nuclear. We have the activity, but if I just multiply both sides by the decay rate, this is activity now. So this is, um the new activity is the original activity times 1/2 raised to the number of half life. So I'm gonna plug in my values. I know my, uh, my new activities heat. My original activity is 10 and this will be 1/2 raised to the number of half Life's. Why? No, it's four hours so that some, you know, it's four hours divided by the number of half life's. So if 1/2 life was for this would be one of the one they have life. If 1/2 life was eight years. This will be 1/2 of 1/2 life. Oh, so now I just need to solve. I'm going to divide both sides by 10. So I get 0.8 equals 1/2 1.5 0.5 raised to the four over to Forever T. And I'm just gonna take the log of both sides. Move this for now. Um, so this will be logged of 0.8 a. The Sylvie, the log of 0.5. I remember going to export of logs. I can really bring that out in front and multiply instead. So this is for over tea. So now if I just multiplies both sides by t and then divide both sides by a log of eight, I have but t my half life is four times the log, a 0.5 over the log appointed, and I do that and I get 12.4. Um, this is just a fraction ratio. So that was an hour's. This isn't ours. So there's my health life. The other port of port, eh is finding, um, the decay constant. Well, no, I mean no half life. That's pretty simple, because the half life of something. The half life a cz 0.639 0.693 over the decay constant. That's the natural log of two. Um, I just found this was too upon for hours. So until I'm going to take that over there, I'm gonna bring this over here. Divide so much decay rate. No, my decay constant. His 0.5 five nine per hour important be I'm to find the original, you know, find the original number of nuclei some doing that look in. No, you really want. Well, the equation it relates those, um, pretty straightforward. The original activity is equal to the decay constant some of the original number of nuclear. So my original activity. I'm gonna rearrange my equation here. Um, so I'm looking for the number of nuclear, So the number of nuclei, the original never nuclear will equal the original activity divided by the decay constant. So my original activity was a peer of the 10. But that's Millie Curie. My dick a constant is this, but that's per hour. I do need to get those in the right units to her to make a new for my end to come out a nuclear. So I'm just gonna throw my conversions in here. Oh, so to make Pierre Curie, I'm gonna change my Millie Curie to regular career. Uh, when Milic Yuri is 10 to the native three Curie's. But then also, I need to change curious debate girls. So one curie is the same as 3.7 times Turn to the 10th. That girl's so you can tell me Unit's gonna work out there on top because military canceled and Curie canceled down here in the bottom. I just need hours, two seconds. This is on the bottom. So I needed on the top to cancel. So one hour is the same as 3600 seconds. So no, mike hours cancels. So when they do that with my conversions, I get that my number of nuclei is 2.38 times 10 of 13 nuclear enough reports. See, we want to know, um, what is the reaction? What is the activity after 30 hours? I'm just gonna go back to this this equation of here that I came up with going to use this. So I want to know what my new or is my original arms to get back to. This was 10. So then it's 1/2 raised to the number of half lives. So now we're thinking about 30 hours. So 30 hours invited by my half life, which I found here is 12.4. That's this is an expert. It's raised of this sinologist new calculator 0.5 raise to the parentheses, 30 divided by 12.4 and Prentiss e equals times 10 and I get 1.86 and that would be in Millie Curie's.

University of Virginia
Top Physics 103 Educators
Christina K.

Rutgers, The State University of New Jersey

Andy C.

University of Michigan - Ann Arbor

LB
Liev B.

Numerade Educator

Zachary M.

Hope College