Don't understand this question regarding radioactive decay

[Just some student]

1. Create and solve differential equations for the number of different amounts of isotopes, which change through time in the decay chain X -> Y -> Z, where X and Y are radioactive atoms, and Z is a stable atom.

2. Make a mathematical analysis of how the different amounts of isotopes from 1 mg if the isotope Tellurium-131 changes through time.

Again, I don't need help solving these, just understanding what I'm asked to do.

 

[nomadreid]

The most basic equation for radioactive decay is that the change (with respect to time) in the mass m₀ of the isotope present = a constant times the initial mass of the isotope; this is obviously a diff.eq., whose solution will give you the mass m₁ . But now you have another change, so you end up with the rate of change of your mass given an initial mass of m₁ = another constant times m₁, the solution of which will give you the final mass m₂. So, after writing these equations, can you now also write an equation whose solution will yield m₂ given m₀? Then you are asked, if m₀ is 1 mg, and you find the decay sequence for that isotope and the corresponding half-lives which will determine the respective constants, to apply your equations to those values. That is how I see it; if anyone corrects me, that is also good.

 

[haruspex]

the change (with respect to time) in the mass m0 of the isotope present = a constant times the initial mass of the isotope

That could be misinterpreted as referring always to the mass at time 0. Allow me to reword it.

If the mass of the isotope at time t is m(t) then the rate of loss of that mass is a constant (for that isotope species) multiplied by the current mass, m(t).
Can you turn that into a differential equation?

 

[Alloymouse]

I'm so sorry that I'm kinda late, but here goes for the sake of learning:

http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/halfli2.html

Essentially there are a few equations governing the exponential decay behaviour of radioactive isotopes, which can be related to their half life. From these equations, you should be able to get differential equations in t.

 

[SummerStudy]

Get acquainted with the basic equations concerning the behavior of radioactive isotopes. This will help you to understand the whole picture. However, if you still want to find out more concerning your question you can ask a question at https://studydaddy.com/ people there are good at finding solutions to complicated questions.