So, I already asked a question on the topic, that helped me understand some more.
Now, here i find myself blocked again.
I don't know how to solve the following problem:

In a basement, you notice a radioactive Activity of $A \; Bq\cdot m^{-3}$ of 222Rn nuclids. $A$ is constant.
The question is the following: How many moles of Rn should be added at the end of each day, to keep $A$ constant, assuming no Rn is leaving the Room.

Now, to me it really doesn't make any sense. Since the decay is exponential. Any quantity i add would decay exponentially.

Now, It's clear to me that is i add a small enough quantity with the halflife value of Rn and that activity value, $N(t)$ the quantity of Rn nuclids decreases by such a small amount that i am tempted to just add the quantity that was lost in a day at the start of every new day. Basically saying that on this scale, we can say that $A$ is constant and by replenishing the lost Rn amount, we keep it in that "constant" zone. Is that correct ?
Because mathematically, it really can't be that an exponential process would be kept on a constant level, by adding a constant value at regular intervals.

  • $\begingroup$ If you measure a constant rate of particle decays, it means that the lifetime of particle A is so large, more than a year ( see en.wikipedia.org/wiki/Island_of_stability ) , so for your basement experiment the rate is constant . . If you measure in day intervals 222Rn has a lifetime of 3.8 days, you should see the decay curve. $\endgroup$ – anna v Oct 15 at 9:54

You are right- it isn't possible to maintain a constant level in a leaking water tank except by constantly introducing water that exactly compensates for the amount lost through the leak. However, if the leakage is sufficiently small, you can top up the tank every day so the water level appears constant when viewed over a long time period ignoring short term fluctuations. I think the question is asking you to say how much of a daily top-up you would need to maintain the value of A from day to day.

  • $\begingroup$ coming from a mathematical background, these type of approximations and questions always get me. $\endgroup$ – Pastudent Oct 15 at 9:45
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    $\begingroup$ I think that part of the problem is that the people who write the questions don't take sufficient care over their wording. Taken literally the question has no answer, since it is not possible to keep A constant in an unqualified sense. Perhaps physicists are more used to dealing with sloppy language than mathematicians are! $\endgroup$ – Marco Ocram Oct 15 at 9:53

Yes, I think the objective of the question is to see if you can calculate the amount of radon that will decay in one day. It would be better phrased if it asked how many moles of radon should be added to restore the activity to the level it was at the beginning of the day.

  • $\begingroup$ And the next step in the exercise is to add a constant amount during the day. We basically are comparing approximation by a constant vs linear polynomial right ? $\endgroup$ – Pastudent Oct 15 at 9:41
  • $\begingroup$ I have answered the question "on this scale, we can say that A is constant and by replenishing the lost Rn amount, we keep it in that "constant" zone. Is that correct ?". $\endgroup$ – gandalf61 Oct 15 at 14:30

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