[willt9494]

Hello,

Sorry if this in the wrong forum. I needed a little bit of help with an assignment.

The question is asking me determine the length of a half-life (t 1/2) for the U-238 to Pb-206 parent-daughter pair. Assuming I have 1,000 atoms of U-238 and 0 atoms of Pb-206 at t0, how many atoms will I have after one, two, three, and four half-lives?


I also feel like I might be vastly over complicating this in my head.


Now, my assumption is that after one half-life (4.468 billion-ish years), I'll have 500 U atoms and 500 Pb atoms. The second half-life is another 4.468 billion years later (8.936 total) leaving us with 250 U and 750 Pb, third half-life (13.4 by) will equal 125 U and 875 Pb, and the fourth half life (17.872 billion) will leave us with 62.5 (63?) Uranium atoms and 937 Lead atoms.

Is that correct or am I mistaken?

Thanks in advance

[Enthalpy]

Welcome, willt9494!

I hadn't heard before "half-life" nor "parent-daughter" for nuclides widely separated on a decay chain, but I don't know everything neither.

What is clear is that the first disintegration, from ²³⁸U, is the slowest by far, so the subsequent ones influence very little the pace at which ²³⁸U disappears. Since ²⁰⁶Pb is the only end product, it's also the pace at which it's created.

The chain is detailed there, especially in a diagram
https://en.wikipedia.org/wiki/Decay_chain#Uranium_series
https://upload.wikimedia.org/wikipedia/commons/4/4e/Uranium_series.gif

Your explanation with 50%, 75% and 25%... is correct.

If computing the products in small amounts after a short time, starting from purified ²³⁸U, the disintegrations with shorter decay times would have an influence too. The half-life from ²³⁴U to ²³⁰Th is a quarter million years, so one year after purifying ²³⁸U, the amount of ²³⁰Th would be limited not only by the decay from ²³⁸U but also from ²³⁴U.

By the way, detectors of ionizing rays catch essentially single particles, so they are damn sensitive, and as they distinguish the energy carried by gamma rays, they can also pick very low proportions of one radioactive nuclide among other ones. So together with mass spectroscopes, they can detect traces of a nuclide that would be impractical by most other means. Then the proportions of very rare nuclides in a decay chain can be significant. Very rare nuclides, and the ability to measure their proportion, makes also dating possible.

If comparing the rays emitted by nuclides in a chain, rather than the proportions of the nuclides, nuclides with a shorter live, which disappear quickly hence are rare, are also more radioactive - this cancels out: at "equilibrium" composition in a chain, as many disintegrations create and destroy an intermediate nuclide.