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Math Help - Radioactive decay

  1. #1
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    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
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  2. #2
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    Quote 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,

    the bad news first: I can't give you a complete answer to your problem.

    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
    Attached Thumbnails Attached Thumbnails Radioactive decay-rad_decaypb.gif  
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