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Half life - Calculate the length of time

  1. Mar 19, 2017 #1
    1. The problem statement, all variables and given/known data
    A source having a half life of 5.27 years is calibrated and found to have an activity of 3.5*10^5 Bq.The uncertainty in the calibration is +- 2%

    Calculate the length of time in days after the calibration has been made for the stated activity to have a maximum possible error of 10%

    2. Relevant equations
    [tex]A=A_oe^{-\omega t} [/tex]
    [tex]t_{1/2} = \frac{ln2}{\omega} [/tex]
    3. The attempt at a solution
    I can't find a possible correlation between the uncertainty and activity.
     
  2. jcsd
  3. Mar 19, 2017 #2

    mfb

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    Currently the actual activity can be up to 2% above or 2% below the given value.

    Can the actual activity get 10% higher than the given value?
    Can the actual activity get 10% lower than the given value?
    If yes, when does that happen the earliest (what is the worst case)?
     
  4. Mar 19, 2017 #3
    One way to approach this is to plot the activity over time using the measured ##3.5*10^5Bq## value and then also plot the activity over time using plus and minus 2% of this value. Track how these curves diverge from each other over time.
     
  5. Mar 19, 2017 #4

    mfb

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    I'm quite sure the 2% uncertainty refer to the activity, not the half-life. The curves won't diverge.
     
  6. Mar 19, 2017 #5

    haruspex

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    I don't think @TJGilb thought it referred to the half-life. Rather, it looks like TJ thought the 10% referred to the discrepancy between the actual acitivity at a later time and the predicted activity at that time.
     
  7. Mar 19, 2017 #6
    Yeah, I think I misinterpreted what the problem was asking him to find. It looks like it may be asking him to find when the activity will be potentially 10% less than the original calibration. Of course, you still want to use the lower curve and the higher curve for the worst case scenario.
     
  8. Mar 20, 2017 #7
    The main problem I am having is I can't see why the activity and the uncertainty are related by the equation A = U + 2 rather than A = 2U
     
  9. Mar 20, 2017 #8

    haruspex

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    It's neither.
    If the measured activity is A and the uncertainty is ±2% then the actual activity is anything from A(1-.02) to A(1+.02).
     
  10. Mar 20, 2017 #9
    I know that
    What I am asking is, it is given in the answer that the source has to decay a further 8% to reach the uncertainty of 10%. My question is how is that relation derived?
     
  11. Mar 20, 2017 #10

    mfb

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    The source could have started at 98% the quoted activity, it will exceed the 10% deviation once it reduces its activity to 90% of the quoted activity, which is a ##\frac{0.08}{0.98} \approx 0.08## relative reduction.
     
  12. Mar 20, 2017 #11
    So for every percent decay, there is a 2% error?
     
  13. Mar 20, 2017 #12
    Oh so wait a minute are they asking length of time after which there would be a decrease of 10% from the original value?
    Due to which we will subtract 2% since there is already a deviation of 2% present?
     
  14. Mar 20, 2017 #13

    mfb

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    Not a 10% decrease the original activity (because that is not known exactly), but a decrease to 10% below the measurement value.
    It is not exactly 2%, but to a good approximation: Yes.
     
  15. Mar 20, 2017 #14
    Yes thank you very much.
    So if they want to ask "Calculate the length of time in days after the calibration has been made for the stated activity to have a maximum possible error of x%"
    All I have to do is use x-2/100 = e^-xt
    Right?
     
  16. Mar 20, 2017 #15

    mfb

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    What is x? No matter what it is it should not appear both in an exponential and outside.
     
  17. Mar 20, 2017 #16
    Oh sorry the first x was an arbitrary number, the second is decay constant
     
  18. Mar 20, 2017 #17

    mfb

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    With a lot of interpretation and a new variable name, that might lead to a possible solution.
     
  19. Mar 20, 2017 #18
    Okay thank you .
    Can you answer one more question about half life
     
  20. Mar 20, 2017 #19

    mfb

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    Which question?
     
  21. Mar 20, 2017 #20
    A radioactive source emits alpha particles at a constant rate 3.5x10^6. The particles are collected for a period of 40 days.
    BY reference to the half life of the source, suggest why it may be assumed that rate of emission of alpha particles remain constant?
     
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