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Model for series radioactive decay

  1. Dec 29, 2008 #1

    KFC

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    If nucleus A decays to nucleus B in rate a, and B decays to C in rate b, and C is decaying at the rate c. To setup a model for that process, we start from A

    [tex]
    \frac{dA}{dt} = - a A(t)
    [/tex]

    and for B, part for B is dying out from it's own decaying process but some amount of A will decay into B, so

    [tex]
    \frac{dB}{dt} = a A(t) - bB(t)
    [/tex]

    But for C, it is quite confusing, the total number of C is proportional to the number of B directly but also to that of A indirectly, of course it will dying out due to it's own decaying process. My question is should I include a term for A(t) in the different equation for C? That is, should I write

    [tex]
    \frac{dC}{dt} = a A(t) + bB(t) - cC(t)
    [/tex]

    or

    [tex]
    \frac{dC}{dt} = bB(t) - cC(t)
    [/tex]
     
  2. jcsd
  3. Dec 29, 2008 #2

    Vanadium 50

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    Staff Emeritus
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    2015 Award

    The reason you are having trouble is because you are trying to do too many steps at once. Write each variable's differential equation first, e.g.

    [tex]\frac{dB}{dt} = - \frac{dA}{dt} - bB(t) [/tex]

    i.e. the amount added minus the amount lost. Then and only then start plugging in known quantities.

    I know it's tempting to take short cuts, but you are really much better off being methodical.
     
  4. Dec 29, 2008 #3

    KFC

    User Avatar

    Got it. Thanks
     
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