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Radioactive samples and differential equations

  1. Oct 29, 2013 #1
    1. The problem statement, all variables and given/known data

    A certain nuclear plant produces radioactive waste in the form of strontium-90 at the constant rate of 500kg. The waste decays exponentially with a half-life of 28 years. How much of this radioactive waste from the nuclear plant will be present after the following increments of time? Assume that initially 600kg of strontium-90 is in the nuclear plant.

    a. N years
    b. 140 years
    c. To perpetuity

    2. Relevant equations

    3. The attempt at a solution

    I found that the "rate in" was 500kg/year, and i'm not sure about the "rate out" (rate of depletion of the sample).

    We are given that the sample decays exponentially, so we should have that y(t) = yo e-0.02475t. We can assume that yo is just y (the radioactive sample that is decaying). Then, the rate of depletion would be (I doubt that this is true):

    y'(t) = -0.2475ye-0.02475t

    This is because this is the rate of change of the sample. Then, i set up my differential equation to be the following (k is just the constant k = (ln2)/(half life)):

    dm/dt = 500 - kme-kt

    I rearranged the terms into the form dm/dt - P(x)y = f(t), i came up that the integrating factor was actually e-e^(-kt)), but this is not true (while differentiating the function I realized that it it was a completely new expression that was different to the one shown).

    How can I proceed from this information or have i made a wrong set-up to the exercise?

    Thank you.
  2. jcsd
  3. Oct 29, 2013 #2

    Simon Bridge

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    The decay rate depends on the current amount in the store.
    That is what "exponential decay" means.

    If y is the mass of 90Sr then:

    $$\frac{dy}{dt} = -\lambda y$$


    The constant ##\lambda## is related to the half-life.
    Last edited: Oct 29, 2013
  4. Oct 29, 2013 #3


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    The differential equation has to be :


    as the amount of Sr-90 increase by 500 kg/year, and decreases proportionally with the amount present, that is by km. Solve, and fit the integration constant to the initial condition m(0)=600 kg.

  5. Oct 29, 2013 #4
    I see. So the only way of doing this exercise is knowing the definition? Or is there a non definition way?

    As well, the constant in the linear rate term (rate of decay) is just the k constant found in the exponent right?
  6. Oct 29, 2013 #5

    Simon Bridge

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    The only way to do a problem is to understand what the words mean - yeah.

    If you don't know, you should solve the differential equation to find out what the constant is.

    Have you not done a section of coursework that covers radioactive decay?
  7. Oct 29, 2013 #6


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    That is true with a sample decaying only - with no input. Now you have input at constant rate. It is true that the rate of decay is proportional to the amount present : -0.2475y, but there is also input 500 kg/year. So the net rate of change of the amount of Sr-90 is dm/dt=500-0.2475m.

  8. Oct 29, 2013 #7
    I have, but i just wanted to make sure if i'm correct or not (I'm sorry! I don't want to commit the same mistakes again).

    I was thinking, as the rate of depletion is ry0e-rt, can we just say that it should be ry? (consider that m, the mass of the radioactive sample, is defined as y = y0e-rt).

    After having this information it should just be a simple variables separable problem to get to the final answer. Say, if we had that the input wasn't constant (suppose it was 600t), then we would have to deal with a first order linear differential equation?

    Thank you for your great patience Simon.
  9. Oct 30, 2013 #8


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    The rate of depletion is is ry, but y is not equal y0e-rt.

    We deal with a first order linear differential equation anyway.

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