MadSci Network: Physics

Re: What happens durring half lifes when there is only one atom left?

Date: Sun Mar 16 11:57:31 2003
Posted By: Sidney Chivers, Math and physics tutor
Area of science: Physics
ID: 1047516509.Ph

The half-life for decay of a sample of radioactive material is described, 
even in college textbooks, as being the time it takes half the 
radioactive atoms in a sample to decay.  See for instance  .  That 
definition does not apply to the decay of small numbers of atoms.  For 
instance, if a sample contains 10 atoms with a half-life of 1 minute, it 
is not certain that precisely 5 atoms will remain after a minute passes.  
Why not?

There is a probability that a radioactive atom will decay.  If you chose 
that atom's half-life as your unit of time, the probability of decay 
could be represented as 0.5 decays per half-life.  If you had 10 such 
atoms, however, and counted them again after one half-life had passed, 
the number of decayed atoms could be represented by any number from 10 to 
0.  Why?

Your question, though, was about 1 atom.  If you could really monitor a 
single specific atom, however, it might never be observed to decay.  To 
clarify these comments, try the following experiment intended to 
illustrate radioactive decay properties of a single atom.

Take a single coin and flip it.  The probability the coin will land with 
its 'head' side showing is equal to the probability the coin will land 
with its 'tail' showing.  The probability that something, heads or tails, 
will happen when you flip the coin is 1.0.  Since there are only two 
equal probabilities, the probability of heads is 0.5 and the probability 
of tails is 0.5.  You can simulate the decay of a single radioactive atom 
by assuming each coin flip represents the passage of one half-life.  Let 
heads represent no decay and tails represent decay.

Where probabilities are concerned, the outcome of your first coin flip 
would not be very helpful.  If you stopped after the first flip, you 
might conclude that a single atom will decay in a time less than a half-
life, as the probability of a decay (tails) is 0.5.  To address this 
difficulty, set up a series of bins labelled something like 'Decayed 
after 1 flip', 'Decayed after 2 flips', etc.  You could accomplish the 
same result by setting up a two column table with one column labelled 
Number of Flips and the second column labelled Decays.  Now, you flip 
your coin until the first occurrence of tails and place the coin in the 
appropriate bin or make a tick mark in the appropriate row of your 
table.  Try doing the experiment 10 times, depict your results on a 
Decays vs Number of Half-lifes graph.  Clear your bins or start a new 
table and do the flip-to-decay sequence 100 times, then depict the result 
on the same graph.  Examine your graph.  If the difference between a 
small number of atoms and a large number of atoms is still not evident, 
and you have the time, try again doing the flip-to-decay sequence 1000 
times.  What should be evident in the graph is the definition of half-
life better represents the decay of samples with larger numbers of 
atoms.  You'll also note that some coins took a larger number of flips to 
decay, maybe you'd even have one or more decays occurring after more than 
10 flips.  There's more of interest in such a graph, but that would 
further complicate this answer.

To summarize, radioactive decay of an atom is statistical rather than 
deterministic, you cannot state precisely when an atom will decay.  If 
there are a large number of atoms in a sample, however, the half-life can 
be used to effect a very accurate estimate of the amount of radioactivity 
that remains after a given time.

Additional references that may be helpful in understanding radioactive 
half-life would be any textbook on nuclear engineering.  Familiarity with 
the first several chapters of a statistics and probability textbook would 
also help.

Thanks for your question.


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