9 replies to this topic

[Mark]

Element X decays into Element Y, with a half-life of λ₁.
Element Y decays into Element Z, with a half-life of λ₂.

there exists a pure sample of element X at time t=0, and nothing interferes with the system, at what time will there be the most of Element Y?

- try doing it without CAS.
it is doable.

Edited by Mark, 22 March 2010 - 05:42 PM.

[arunma]

Interesting question. I guess we'd have:

X(t) = X(t=0)*exp(-λ₁t) (I'm assuming that λ₁ was actually meant to be the decay constant rather than the half-life).

Y(t) = X(t=0)(1-exp(-λ₁t)) - (loss of Y to Z)

Oh dear...now I need to solve the differential equation for Y(t). Sadly I've got to do physics right now (the kind that I get paid for), but I'll post this as a way of saving my work, and see if I can finish it up later.

Interesting questions, Mark!

Edited by arunma, 23 March 2010 - 10:47 AM.

[Egann]

Urg. Calculus. I was afeard this would happen; calculus is the only math yours truly truly sucks balls at.


Let me see if I can translate arunma's answer (which, to be frank, I don't really get) to get how we get the answer, even if I can't do it on my own;

We start with X's and Y's decay curve equation, which we can combine (how did you do that? algebraic substitution?) to arrive at Y's supply curve for this problem. Then we take the derivative of Y's supply curve and find the time for the point where the slope is Zero; there should be two points, where there is the most element Y, and an asymptote at infinity.


...Yeah....I should probably learn calculus....

[Mark]

I will have to get round to posting a solution.

[arunma]

Don't do it yet though! I think if I just stare at it for fifteen minutes (when I actually have fifteen free minutes), I'll be able to get the answer.

I think the issue here is that you need to write out the decay rate of Y to Z. However, the initial decay rate is zero, since the initial quantity of Y is zero. Strangely, I had this question on my PhD qualifier, and I was able to do it then. Looks like I've forgotten a bit now that I don't take classes anymore.

Edited by arunma, 24 March 2010 - 03:41 PM.

[GuardianNinja]

[Mark]

I will post solution soon.

[arunma]

Shit, I completely forgot about this.

[Mark]

time is up:

Let lambda1, lambda2 be decay constants
(we could work with half-lives if we wanted too)
let x1 be amount of substance X
let x2 be amount of substance Y
let x3 be amount of substance Z

[arunma]

Ah, that makes sense. You just set the decay rate of 2 equal to the sum of the rates for 1 and 3.