Exponential Decay

Date: 06/09/2005 at 22:30:14
From: Adam
Subject: Exponential Decay Curve of a Radioactive Substance

In science we recently had a problem where we had to extend an 
exponential decay curve of a net counting rate of a radioactive 
substance by 1 day.  We were given the information:

  Time (minutes)    Net counting rate (counts / minute)
   0   (start)             171
  20                       110
  40                        70
  60                        49
  80                        31

When I extended the curve by one day I recieved 1.61 * 10^(-11) (which
is an estimate because the numbers that we were given would not always
turn out the same.)  My teacher insists that it would be zero.  I am
wondering who is correct.

I believe that I did my calculation correctly, but I am not sure if
the matter could reach that low a rate or not.  I don't think it can
reach 0 at all because there is always something left after each decay
cycle.

Could you please inform me whether my thinking is correct on this and
whether or not my answer is close to what the real one would be?

Date: 06/10/2005 at 23:22:33
From: Doctor Greenie
Subject: Re: Exponential Decay Curve of a Radioactive Substance

Hi, Adam --

I didn't check your calculations, but who is "right" depends on your 
interpretation.

In your last sentence, you ask whether or not your answer is close to 
what the real one would be.  The answer is that it is close.

And your teacher's answer is also close.  After all, there is not a 
lot of difference between "0" and "1.61 * 10^(-11)".

One thing that must be kept in mind in all of this is that rates of 
radioactive decay are statistical measurements.  If you start with 
4*10^10 atoms of a radioactive material with a half-life of 1 year, 
then after a year you can be pretty confident that the number of atoms 
remaining after a year is very close to 2*10^10.  But if you start 
with 40 atoms of the material, then after 1 year you can't have much 
confidence at all in the number of atoms remaining.

Since your measurements are "counts per minute", an answer of 
1.61*10^(-11) means the radioactivity is probably not measurable, so 
an answer of "0" is just as good.

I am reminded of an experience I had 30 years ago when I (as a math
student mildly interested in nuclear physics) took a nuclear 
engineering course with a group of students specifically studying to 
be engineers.  We had a problem on a test with a radioactive source 
and an absorptive shield and were supposed to determine the strength 
of the radiation on the other side of the shield.

Radiation strength on the other side of a shield, as a function of the 
thickness of the shield, is an exponential decay function just like 
radioactive decay.  When I put the numbers into the equation and did 
some quick mental approximate calculations, I got a mathematical 
result of something like e^(-60).  So without any further effort, I 
wrote down the answer "0".  Several of the engineering students I 
talked to after the test said they couldn't answer the question 
because, whenever they put the equation into their calculators, 
instead of getting an answer, their calculators kept showing 
"0"....!!!

I hope some of this helps.  Please write back if you have any further 
questions about any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/