# Homework Help: Radioactive decay formula

1. Feb 18, 2014

### piisexactly3

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. Feb 18, 2014

### tiny-tim

hi piisexactly3!
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!

3. Feb 19, 2014

### piisexactly3

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
4. Feb 19, 2014

### tiny-tim

hi piisexactly3!
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!
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