# Q3 How is the half life of an element determined? For something that takes 60 billion year

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>Q3 - How is the half life of an element determined?  For something
>     that takes 60 billion years to partially decay, how is an exact
>     measure of the decay rate determined in a few hours?  I know
>     nothing about the method.  My geology books are no help.  Can
>     anyone explain the procedure to a layman?

The fundamental relationship in radioactive decay is that the rate of
decay is directly proportional to the amount of the substance present.
In mathematics:

dN/dt = -lambda*N      (1)

where dN/dt is the rate of change of the number of atoms present (e.g.
number of decays / second), N is the number of atoms present (at that
particular time), and lambda is a constant of proportionality called
"the decay constant". The exponential decay law is a consequence of
this decay rate relationship.

Now the number of atoms of a material present at a given time is
related to the mass by Avagadro's Number and the atomic mass of the
material.

N = (M /A)*Nav         (2)

Where M is the mass of material (say in grams), A is the atomic mass
of the material (in grams/mole), and Nav is Avagadro's Number
(atoms/mole).

So, using equation (1), the decay constant can be found by dividing
the decay rate (number of decays / second) by the number of atoms
present (as derived from the mass).

lambda = -(dN/dt)/N     (3)

Everything on the right hand side of the equation can be
experimentally measured, so the decay constant can be determined.
The half-life is obtained by dividing the logarithm of 2 by the decay
constant.

t(1/2) = (ln 2)/lambda   (4)

Stating that the half-life of a material is, say, 1 yr DOES NOT mean
that you have to wait 1 yr before a decay occurs.  Instead, it means
that during that year you expect 1/2 of the atoms to decay (and for
macroscopic amounts of the material, that is a lot of decays).

As an example, suppose we have a material that has a half-life of 69
billion years (6.9 X 10^10 yr).  This a bit longer than the example
in your question.  The decay constant for that would be 10^(-11) /
yr.  If we had one sixth of a mole of the material, that's 10^23
atoms, so by equation (1) we find 10^12 decays per year or 3 X 10^4
decays per second.  In a week, 2.1 X 10^5 decays could be counted
(that's 210,000).  So, if you have a decent amount of the substance,
a long half-life does not present an insurmountable problem.

Returning to equation (1), recall that the exponential decay law is a
consequence of the equation; if the equation is wrong, the
exponential law is wrong.  If you wish to test the exponential decay
law, that can be done by testing equation (1).  And that can be done
simply:  Take a sample that has, say 13 times the number of atoms in
it, and look to see if the number of decays per second is 13 times as
great.