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Problem 25 Hard Difficulty

Chromium’s radioactive isotope $^{51} \mathrm{Cr}$ has a half - life of 27.7 days and is often used in nuclear medicine as a diagnostic tracer in blood studies. Suppose a $^{51} \mathrm{Cr}$ sample
has an activity of 2.00 $\mu \mathrm{Ci}$ when it is placed on a storage shelf. (a) How many $^{51} \mathrm{Cr}$ nuclei does the sample contain? (b) Calculate the sample’s activity in Bq when it is removed from storage one year later.

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Top Physics 103 Educators
Elyse G.

Cornell University

Andy C.

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Aspen F.

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Video Transcript

from over 25. We have the radioactive curry, um, and has 1/2 life of 27.7 days. So it's the half life, and initially, it hasn't activity of two micro curies. And we're as to find how many nuclear other artist start. And what is the activity after being on the shelf for one year? First of all, finding in Well, I'm gonna have my equation. Liver. You're ready to go. These are pretty much the equations I always use. I know there's different derivations of these. I like these. Um, I'm going to use this equation, but just rearrange it for em. So n will equal home or over Linda, and I'm gonna rearrange this equation. Bye. Rearrange this one. You can get that. Um well, I am Joe. Decay rate equal 0.693 which is a natural look of two over the half life. So I'm going to substitute that in for this now, okay? And plug in my values. I have to get these old in the right units. I'm gonna put everything in normal s i units. So I'm just gonna come up here and do my conversions. I don't need? Well, I do. I need this in seconds. So I know one day is 24 hours, one hour ISS, 60 minutes. Well, let's go all the way to 3600 seconds. We'll be done because days canceled on ours cancels, and I'll be left in seconds. So for that, I get 2.393 times, 10 to the sixth. And then here I'm gonna change my micro curies to back Carell's, um, first of all, just going to get rid of that prefix. I know that one, Milic Yuri is the same as one times 10 Negative. Sixth. Curious. So now I'm curious. And one curie is the same as 3.7 times. 10 to the 10th. That girl's so curious. Cancel and I'm have 74,000. That girl's okay. So now I'm gonna put these in my right. My s I units here, So or my activity is 74,000 natural longer, too. Job. I'm half life in seconds. 2.3 and I get 2.55 times 10 to the 11th sits and that's winning. Nuclear. I have online for Port B. Use this equation. This is for number of nuclear. Um, I'm gonna write my equation. But if you look at this equation, um, if you take the number of declare and multiply it by the decay rate, then you have ours. So I'm just gonna imagine I'm multiply both of these. Those suds, the equation by that. And now my equation is what I want were the new activity will be the original activity times 1/2 raised to the number of half lifes. So this is what I'm looking for. The new activity after a year. My original activity in becquerel is 74,000 1/2 raised to the number of half lives. I'm going to go back to this regular days. I don't need to use seconds for this because as long as these air in the same unit, so it works out. So ah, year will be 365 days divided by 27.7 because that's how many days are 1/2 life. So this will tell me how many half lifes I have. So this is 1/2 raised to this. A lot of time to 74,000 and I get 7.99 back. Girls

University of Virginia
Top Physics 103 Educators
Elyse G.

Cornell University

Andy C.

University of Michigan - Ann Arbor

Zachary M.

Hope College

Aspen F.

University of Sheffield