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The following question is from grade $10$ science textbook:

A student wants to measure the half-life of a radioactive isotope. He is told the isotope has a half-life of between $10$ and $20$ minutes. Illustrating your answers as appropriate, describe: a) the measurements that he should take; b) how he should use the measurements to arrive at an estimate of the half-life for the isotope.

I understand that if the quantity of the sample was $x$ units, it would drop to $x/2$ units somewhere between $10$ to $20$ minutes. So, to estimate the half-life, should we measure the sample every minute from $11^{th}$ to $19^{th}$ minute? Also, how are the a) and b) parts of the question different?

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4 Answers 4

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Here is a hint:

The student would also need to decide the 'background radiation'.

That's the approximately constant radiation (that would increase the students measurements), from other sources.

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  • $\begingroup$ using GM counter? Or, is the value for background radiation fixed? and it can be taken up from other experiments? $\endgroup$
    – aarbee
    Apr 12, 2021 at 14:52
  • $\begingroup$ It varies from place to place, so would need to be measured separately, or estimated from the students measurements $\endgroup$ Apr 12, 2021 at 14:54
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If this is a 10th grade question, the purpose is probably to establish whether or not a student understands the concept of 'half life' and the calculations involved to find it. To even consider background radiation is besides the point. Another reason to suspect this is, the question is asking for a description, rather than actual performance of the measurements and calculations involved. It does not ask for formulas! It also concerns an estimate, which is often possible to obtain in more than one way, it being an achievement rather than an acquirement.

As far as the answer goes. A). As many measurements as practically possible, but at least four measurements should be taken. For example: t = 0 , t + 9, t + 15 and t + 21. The first one giving max radiation, the second being conclusively less that minimal half-life, the third being average expected half-life and the fourth being conclusively past max half-life. This is however just an example.

B). From those measurements the rate of decay can be calculated and from that it can be calculated at what point in time the isotope has lost half of its radiation capacity. The result is an estimated half-life.

For some isotopes half-life kicks a fairly violent 'knee' into the decay rate graph. The sharper that knee, the more the accuracy of the estimation becomes dependent on the number of measurements. That sudden change in decay rate is actually the whole reason 'half-life' is being used. Its the rate at which the rate of decay decrease decreases that makes it interesting. (I hope I got the previous sentence correct)

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In question A you should ask yourself what quantity you are really measuring. Do you measure the weight of the remaining atoms by using a scale, a force of the radioactive decay by using a spring (which obviously does not make sense), $\ldots$?

For question B it helps if you start by generating a "feeling/intiution" for the subject. Hence, in my opinion it is clever to write down a table, where we assume the initial atom number to be $N(t=0) = 10\,000$, and then calculate the remaining atoms after different time intervals. Here I choose time steps of $10min$, but you could of course use any number. Now, we could choose three different half lifetime $t_{1/2}=\{10min, 15min, 20min\}$, and calculate the number of radioactive atoms after these time steps for each half lifetime:

enter image description here

I only entered the first column, please do the calculations for the remaining two columns.

Once you have the table you should ask yourself: After which time interval am I able to clearly distinguish the results? Is is maybe wise to calculate the ratio between the remaining atom number and look when this ratio is the largest? E.g., suppose the number of atoms after $T=30min$ for $t_{1/2}=15min$ would be 2500. The ratio between $t_{1/2}=15min$ and $t_{1/2}=10min$ would be $2500/1250 = 2$. Is this ratio a "better" quantifier than the absolute number and why? Does this ratio increase or decrease if we take the measurement at $T=10min$? What is the ratio between these numbers after $T=60min$? Do I understand the why the ratio shows this behaviour? Can I think of an other example where I would use the same reasoning?

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  • $\begingroup$ Reading your answer gave me an idea that maybe the value can be checked at 15th min. If the quantity is already halved, it means half life is in between 10 to 15 mins. If not, then the half life is in between 15 to 20mins. Accordingly, next measurement can be taken at either 12.5 mins or 17.5 mins. This would further narrow us down on the half life. We can do a few more iterations to get pretty close to the half life. $\endgroup$
    – aarbee
    Apr 12, 2021 at 19:37
  • $\begingroup$ No, that's not what I wrote. I believe you are jumping to conclusions without working through the steps I outlined. I know that it involves some effort, however, it will hopefully provide a deeper understanding of this subject, too. $\endgroup$
    – Semoi
    Apr 12, 2021 at 19:48
  • $\begingroup$ No, no, I am saying I got another idea while reading your answer $\endgroup$
    – aarbee
    Apr 13, 2021 at 10:46
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The halflife of a radioactive substance is a constant independent of the amount of the substance as long as there are many atoms present. Let's assume $x=512\times 10^{23}$, roughly $8$ moles. After one halflife, you still have $256\times 10^{23}$ atoms, plenty to exhibit the statistical halflife behavior. So, you don't have to stop at a single halflife.

Also, you can start at any time within your data, at some arbitrary $x,t$ datum and find the time at $x/2$. The change in time will be the halflife.

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  • $\begingroup$ If we started with 8 moles, after one half life, won't we be left with 4 moles? $\endgroup$
    – aarbee
    Apr 12, 2021 at 19:43
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    $\begingroup$ @aarbee Yes, but that's still a huge statistical population. Even 0.0001 moles is a big population and the halflife will be the same. And after 10 halflives you will still have about 0.008 moles, a huge number of atoms. In fact, most laboratory radioactive samples aren't much more than 0.001 mole. $\endgroup$
    – Bill N
    Apr 12, 2021 at 19:50

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