Re: Is the quantity of a radioactive isotope ever equal to exactly zero?

The half life of a radioactive material is the time it takes for one half 
of the original radioactive atoms to undergo radioactive decay. For large 
quantities of radioactive material, measured by the number of atoms of 
radioactive material present, the general decay equations are very 
accurate. Even though it may be very difficult or impossible to measure 
one billion radioactive atoms, that is still a lot statistically. However, 
after a number of half lives, we will eventually reach a point where there 
are only a few atoms left. Finally, after enough half lives have elapsed, 
we will be down to one atom of the radioactive material left. 

There is no concept of a half life for one atom. So, the decay equation

Atoms Remaining = (Atoms Originally Present) x exp-(ln(2)x time/half life)

which says that the number of atoms remaining at a time measured from some 
convenient starting point is equal to the number of atoms originally 
present times a decay factor. The decay factor is mathematical constant e 
(the basis of the natural logarithms) raised to the negative power of the 
natural logrithm of 2 times the quotient of the time since the start of 
measurement divided by the half life of the radioactive atoms in question. 
The time factor: (time divided by half life)gives the number of half lives 
the mass of atoms has been allowed to decay.

When the number of radioactive atoms reaches one, the equation breaks down 
because the exponential term may get very small, but it is never equal to 
zero. But that is the mathematical world. 

In reality, there will be some instant when that last atom decays, and 
there are none of the original atoms remaining. So there is a time when 
there are no original radioactive atoms remaining. This can be seen in 
nature where all that is left of radioactive fission product elements 
created when there were naturally occurring nuclear reactors in the 
earth's surface (about 2.3 billion years ago) are the stable elements that 
they decay to. In order for any atoms to remain after that time, they 
wouldd have to have a half life on the order of billions of years. Some of 
the Uranium and Thorium isotopes have such a half life, and are still 
present today as naturally occurring radioactive material (NORM) in the 
earth's surface.