# Math Help - Radioactive decay

Need some help on a homework problem:

At time t = 0 a vessel is initially loaded with 1 kg. of pure Lead-214, which subsequently decays to Bismuth-214 and Polonium-214. Let L, B, and P denote the decay constants for these 3 elements.

Write a system of equations that gives the amounts of these 3 elements over time, and solve the system exactly.

I know the equation for the decay of a radioactive element is so I have the equation for Lead-214:

x(t) = 1 kg. * e ^ (-L * t)

I can't figure out the equations for the other two elements though. I do know Lead-214 decays to Bismuth-214 which decays to Polonium-214.

Any help would be awesome.

Thanks,
charps

2. Originally Posted by charps43
Need some help on a homework problem:

At time t = 0 a vessel is initially loaded with 1 kg. of pure Lead-214, which subsequently decays to Bismuth-214 and Polonium-214. Let L, B, and P denote the decay constants for these 3 elements.

Write a system of equations that gives the amounts of these 3 elements over time, and solve the system exactly.
I know the equation for the decay of a radioactive element is so I have the equation for Lead-214:
x(t) = 1 kg. * e ^ (-L * t)

I can't figure out the equations for the other two elements though. I do know Lead-214 decays to Bismuth-214 which decays to Polonium-214.

Any help would be awesome.

Thanks,
charps

Hello,

I've attached a drawing of the problem as I understand it.

If $x_{Pb}(t)=1kg \cdot e^{-L \cdot t}$is the amount of lead, which is left after a time t, then $x_{non-lead}(t)=1kg-x_{Pb}(t)$.
That means $x_{Bi}(t)=\left(1 kg-1kg \cdot e^{-L \cdot t} \right)\cdot e^{-B \cdot t}$

And if $x_{Bi}(t)$ is the amount of bismuth, which is left after a time t, then $x_{Po}(t)=\left( x_{non-lead}(t)-x_{Bi}(t) \right) \cdot e^{-P \cdot t}$.

(Note: I've to split the following equation - it's too big)

$x_{Po}(t)=\left( \left(1 kg-1kg \cdot e^{-L \cdot t} \right)-\left(1 kg-1kg \cdot e^{-L \cdot t} \right)$
$\cdot e^{-B \cdot t}\right) \cdot e^{-P \cdot t}$ (I suppose that this equation isn't correct, it looks somehow ill to me. But I cann't find my mistake)

For all three amounts of material the equation $x_{Pb}(t)+x_{Bi}(t)+x_{Po}(t)=1 kg$ is true. (Probably the mass isn't constant, because you can observe a loss of material during the process of decay)

And now I have reached the point where I believe that I have made somewhere a severe mistake, because I can't imagine what you should calculate exactly.
But nevertheless I send you this text because maybe you can use it as a hint how or where to start.

Good luck!

EB