I'm Going Bananas!

It's a comment from the same Forbes guy who tried to compare the Fukushima leak with banana equivalents.

He made a common mistake and it gives me the opportunity to address a different aspect of health physics besides just radiation health risk.

The relevant part is this:

"One molecule of a metal is normally one atom of it. And Pu 239 has a half life of 24,000 years. Meaning that it’s got a 50% chance of decaying in that 24,000 year period. Now, if one atom of Pu decaying in your lungs will definitely cause lung cancer (something that is certainly not true anyway) then you’d still have to wait 24,000 years for there to be a 50% chance of this happening."

Do you see what he wrote wrong above?

Half-life is the amount of time for half (50%) of a given initial amount of a radioactive substance to decay. That doesn't mean that there's a 50% chance (probability) of an initial amount decaying in that time frame.

Some of the initial amount is always decaying that's how we detect the material, via its radioactive emissions.

You do NOT have to wait 24,000 years for there to be a 50% chance of decay!

Half life is a population statistic, it's not really meant to be applied to a single atom or molecule, but we can.

dA/dt = -kA

That equation is stating that the change in radioactivity (A) with time (t) is proportional (with negative proportionality constant, k.  In radioactive decay we call this the decay constant, it is the probability of decay) to the original activity.

When we employ calculus to solve that equation, we get another equation, called the decay equation:

A(f) = A(i) * e^(-kt)

where the (f) & (i) stand for final activity after time (t) and initial radioactivity respectively.

We can measure a sample of radioactive stuff as it decays to half of its original level and note how long it took.  That time period is the half life.  Or mathematically, rearranging the previous equation:

ln (A(f)/A(i)) = -kt

ln 1/2 = -kt

- ln 2 = -kt

ln 2/t = k

This tells us that the probability of decay (k) is equal to the ln 2 divided by the half life!

In the case of Pu-239, that's:

ln2 / 24,000 years = 0.003% per year (if I counted my zero's correctly).

So the chance of a single atom decaying is 0.003% per year.

It's highly unlikely to decay in any one year, but it could!