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Radioactive decay formula

  1. Feb 18, 2014 #1
    The formula for radioactive decay over time is N = N0eλt . The formula for how many number of atoms counted by the Geiger as distance (x) changes is C = k/x2. How can I merge these formulas to give one that accounts for distance and time? Secondly, how does the second formula work realistically given that C would tend to infinity as x decreases?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 18, 2014 #2

    tiny-tim

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    hi piisexactly3! :smile:
    you simply multiply them …

    the formula will be proportional to both eλt and 1/x2
    the geiger counter has a finite width w, which you assume is at a constant distance x from the source

    when 2πx = w, the counter would need to be wrapped all the way round the source (360°)

    and when 2πx = w/n the counter would need to be wrapped round it n times! :wink:
     
  4. Feb 19, 2014 #3
    Ok maybe I'm understanding this. You're saying that if the geiger counter was shaped like a sphere encasing the source and detecting every emmission, the k/x^2 formula would not hold true. But because it is a tube and cannot detect everything, the formula works and we don't get an infinte number of atoms detected?

    When I times N0eλt by k/x2, I get N2 = N0eλtkx-2 .
    So what exactly is the constant k, does it involve use of the actual numbers of atoms there, as I don't see how we can derive how much the distance x is decreasing the detections by without knowing how many atoms there actually are, and I don't see how the k/x2 formula uses that information... Unless k is N0eλt? So how can I incorporate information about the width of the tube to ensure that this formula doesn't tend to infinity as x tends to 0?

    I may have gone totally off track so sorry if I am fustrating you!
     
    Last edited: Feb 19, 2014
  5. Feb 19, 2014 #4

    tiny-tim

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    hi piisexactly3! :smile:
    no, i'm saying it would hold true …

    if its surface area was exactly enough to fit round the source once at that radius, it would count every emission

    if we choose a radius n times smaller, so that the geiger counter's surface area is enough to wrap round the source n2 times, it will count every emission n2 times (once for every layer) …

    so the k/x2 formula does work! :wink:
    i don't think the geiger counter is intended to be used that close to the source: if it is, the k/x2 formula certainly doesn't work (the count doesn't go off to ∞), and you'd need to do some horrible integration
     
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