Radioactive Decay Problem
I've been working with this problem for almost two weeks trying to find a good equation for the decay of Bismuth to no avail.
Can someone give me insightful comments:
Here's the problem:
1. The problem statement, all variables and given/known data
In the radioactive decay series of Uranium (238, 92), isotopes of lead, bismuth occur as products of two successive Beta decays with half -lives of 19.7 minutes and 26.8 minutes, respectively.
U --> Lead --> Bismuth
Decays are each proportional to the amount of isotope present.
Assume initially at time =0, we have 100 mg of lead and 150 mg of bismuth?
We are asked to find the amount of lead and bismuth at any time?
2. Relevant equations
So first we have to formulate a DE for the decay.
The one equation for Lead is simple.
Let L(t) be the amount of lead at any time, then the DE model is:
dL/dt = -a.L</b> (a: constant of decay)
after integration and get L(t) = 100.e^(-a.t)
Using the Initial Value Problem and half-life value we get an equation:
Now the Differential Equation for Bismuth is: (B(t): the amount of Bismuth at any time)
dB/dt= -b.B + a.L</b> (a.L: quantity of lead decayed added to the Bismuth which equals 100.e^(-.035185.t)) and b: constant of decay for Bismuth
Solving this 1st Order DE we get:
[100.e^(-.035185.t) + C.100.e^(-b.t)]
C is a constant of integration.
First: Is my B(t) equation correct based on the problem we have?
And second, how can I solve for b and C based on the IVP I have?
Can you guys help?