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Problem 41 Hard Difficulty

One of the many isotopes used in cancer treatment is $_{79}^{198} \mathrm{Au}$, with a half-life of 2.70 $\mathrm{d} .$ Determine the mass of this isotope that is
required to give an activity of 225 $\mathrm{Ci} .$

Answer

0.92$m g$

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Video Transcript

problem. 41 1 of the many isotopes using cancer treatment is gold 1 98 that has 1/2 life of 2.70 days. Question asks to determine the mass of this isotope as required, given activity of 225 curie. There's a very large activity which you often need in cancer treatment. You need a high dose of radiation, um, in order to kill off these rapidly growing cancer cells. So given this ring, given this rate of activity, what is the mass? That's what we're asking for here. So if we think what kind of relationship can you get from activity to mass or something to do with its very specific about the atom? We know that we have the relationship, that the activity is just the Raitt concert or decay constant multiplied by the number of atoms. President in the sample. We don't have did take a concert this scenario, but it is very easy calculable because we have 1/2 life and we know that the relationship between this decay constant and the half life is simply a natural longer them of two over the half life. So I was put in here to be 2.70 days, which would give us a activity or decay constant 0.257 per day. His little sidebar here. So, based on our known activity, we can solve for the number of atoms to be the well, your activity divided by the rate constant. Both these terms are provided to 25 over $250.257. Write that out. It's a cury. What should be hitting that? It's not very useful when we have a 0.257 per day. So if we perform, this calculation would have one of Adam's to be Curie Times Day, which is not room anything, really, it doesn't make too much sense. So we should convert Arcuri to a rate of the case per second or Becquerel. So we're doing that. We have 3.7 times 10 to 10 decays or Adam's per second under the King's off putting Adam's per second. That is all over one curie of the rate procuring. So now, looking an equation, we still have days that don't really cross out with seconds so we can convert that, knowing that, um, I see one d has 24 hours, but council off the hour, The days and then, innit? Count off the hours. 24 hour, one hour has 3600 seconds. That's the unit we need. So, looking at these now, just to double check with units we have, we have Curie's cancel out with rate per Curie D's cancel it with these our cancers, that with ours, one of her seconds can't South one over seconds. And we're left with Adams, which is the desired amount for this equation. So we find out that he never Adam's present in a gold Adam cooled sample with this many curious to point A Doe times 10 to the 18 Adams. Great. So we have the number of atoms. The Army won't find the mass of the sample with this many atoms. So if we think But that formula would look like if you're looking for mass, we say we have number. Adams won't buy that by, uh, multiply that by the amount of nuclear ons we have. Do you have in our gold sample you got and the most Mind and Matthew have her nuclear on for nuclear on for Adam Pretty single Adam Yeah, Captain, This is active for Adam. We're finally having units of just Massport reacting brick, which would just be kilograms. So this is the equation or the logic behind the equation that will use for this part? I was leaving on the same page You thinkit's useful to have altogether. So the massacre sample again. 2.80 times tend the 18 Adams. Well, gold Adam has 108 98 nuclear ones. And you know the mass for nuclear on per each individual, Adam is 1.67 Begin. Look us up. It's in our boat textbook 1.67 times, 10 to the negative, 27 kilograms. So our number of Adam's cancels with our per Adam in our final term nuclear ons. Who cancels that with her nuclear on here? So we're left with just the use of kilograms. As I left in our equation here and not playing this through. We're left with 9.26 times 10 of the native seven kilograms. So you need it. Is that or if you want to turn the grams 9.6, I'm sending you to four grams or you can rewrite it is Gerald 0.9 to 6. Military. Are these air acceptable? That's the same value. Just represented with different units. So the trick is part of it of this question is to recognize this relationship here where we take your number. Adams Most fire the mass per nuclear on for each Adam and most bye bye. Gold's amount of new bones here. Gold 1 98 has 100 90 nuclear ones to get a final massive our sample.