Re: How is the half life of radioactive elements measured if it's very large?

Lance:

You are asking an excellent question.  You are correct in principle that 
it becomes much more difficult to measure the half-life of radioisotopes 
with extremely long half-lives.

However, to use U-238 as an example, a sample of 1 gram would contain 2.5 
x 10^21 atoms.  The decay constant, lambda, for a 4.5 billion year half-
life would be 4.88 x 10^-18 sec^-1.  To calculate the number of atoms that 
would decay (or "transform" and emit an alpha particle) you can multiply 
lambda by N (the number of atoms in the sample).  That calculation yields 
the result that one gram of U-238, with the unimaginably long half-life of 
4.5 billion years, still has more than 12,000 transformations every 
second.  Even though the probability of any specific atom decaying during 
your lifetime is exceedingly small, there are enough atoms present in a 
gram of material so that over 12,000 decays per second are occuring.  So, 
to use the phrase that you used, a one-gram sample size (10^21 atoms) is, 
statistically speaking, able to yield significant results in a matter of 
seconds to minutes.  Even a microscopic sample size (say, a microgram) of 
U-238 would emit about 45 alpha particles per hour.  Even a sample this 
small, if you accurately measured its mass, could allow a calculation of 
half-life to a precision of better than +/- 10% for a counting time of one 
day.

In summary, measurements of radioactivity can be incredibly sensitive, 
primarily due to the fact that our detectors can respond to a single 
atomic event.  There are so many atoms in any object large enough to be 
visible, that seemingly impossible tasks like measuring billion-year half-
lives are actually relatively easy.

Lance, I hope this helps to answer your question.  You might get some 
additional insights and perspectives from Herman Cember's 
book, "Introduction to Health Physics."