Results 1 to 2 of 2

Math Help - radioactive/exponential decay

  1. #1
    Member
    Joined
    Jun 2008
    From
    Plymouth
    Posts
    120
    Thanks
    5

    radioactive/exponential decay

    Hi!

    I understand the concept of radioactive decay when you have only one element.
    <br />
\frac{dy}{dt}=-k\times y<br />

    But what happens when one element decays into a second element and the second decays into a third. What is the quantity of not decayed material of the second element? And what if you have a longer chain? For example:
    Element A has a half-life T_1=2 s.
    Element B has a half-life T_2=5 s.
    Element C is stable.
    A decays to B which decays to C.

    A(t), B(t), C(t) - is the quantity of A, B, C at time t
    A(t)=2^{-t/2}\times A(0)
    B(t)=?
    C(t)=?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Hi

    Between t and t+dt
    A goes from A(t) to A(t+dt) = A(t) - a A(t) dt
    B goes from B(t) to B(t+dt) = B(t) - b B(t) dt + a A(t) dt
    C goes from C(t) to C(t+dt) = C(t) + b B(t) dt

    It is clear that for every t, A(t) + B(t) + C(t) is constant

    A(t+dt) = A(t) - a A(t) dt

    \frac{A(t+dt) - A(t)}{dt} = -a A(t)

    \frac{dA}{dt} = -a A(t)

    A(t) = A_0 e^{-at}

    B(t+dt) = B(t) - b B(t) dt + a A(t) dt

    \frac{B(t+dt) - B(t)}{dt} = -b B(t) + a A(t)

    \frac{dB}{dt} = -b B(t) + a A_0 e^{-at}

    General solution of equation \frac{dB}{dt} = -b B(t) is
    B(t) = \lambda e^{-bt}

    Particular solution of \frac{dB}{dt} = -b B(t) + a A_0 e^{-at} is
    \frac{a A_0}{b-a} e^{-at} if a and b are different

    B(t) = \lambda e^{-bt} + \frac{a A_0}{b-a} e^{-at}
    B(t) = B_0

    B(t) = (B_0 - \frac{a A_0}{b-a})e^{-bt} + \frac{a A_0}{b-a} e^{-at}

    To be checked !!
    But I think that the general idea is not so far !
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Radioactive decay
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 22nd 2011, 12:45 AM
  2. Radioactive Decay
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: October 11th 2009, 01:12 PM
  3. Radioactive Decay
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: June 14th 2009, 05:22 PM
  4. Exponential radioactive decay while original amount increases.
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: June 27th 2008, 09:01 PM
  5. Radioactive Decay
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: May 17th 2008, 02:09 PM

Search Tags


/mathhelpforum @mathhelpforum