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Help with radioactive decay

  1. Feb 25, 2015 #1
    1. The Problem

    Construct a mathematical model (system of differential equations) for a radioactive series of 3 elements. X,Y, and Z (Z is a stable element). (Note: W decays into X, X decays into Y, and Y decays into Z). At time zero there are 100e (approx. 271.828) moles of element X. After two hours there are exactly 100 moles each of elements X and Y. Solve the IVP and calculate the number of moles for each of the three elements after i) 1 hour and ii) 5 hours

    2. Relevant equations

    dw/dt = kW
    dx/dt = kW - cX
    dy/dt = cX - aY
    dz/dt = aY
    ( k ,c ,and a are all different constants)
    x(0) = 100e
    x(2) = 100
    y(2) = 100

    3. Attempt at solution

    Im posting this to see if i have my system set up correctly. Im not looking for an answer.
  2. jcsd
  3. Feb 25, 2015 #2

    Quantum Defect

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    Looks good to me, except for the first equation: dw/dt = -kw, which I suspect is a typo. [Note adding up all of the rates gives you conservation of atoms, d[W+X+Y+Z]/dt = 0, and you won't get this unless the first equation is -kw.]
  4. Feb 25, 2015 #3
    would kW in dx/dt need to be negative also? and would dz/dt need to be negative?
  5. Feb 25, 2015 #4


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    No and no. kW is the rate of formation of x from W. dz/dt is the rate of accumulation of z, in this case simply from Y without any further decay.

    You really should decide what symbol you are going to use for w, x, y, z, CAPS or lower case. - could that have helped confuse you?
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