# Chemistry - If the half-life of an isotope exceeds the age of the Universe, then how is it measured?

## Solution 1:

Well, half-life describes the time after which half of the substance has decayed.

This is all probability and statistics. If you look at a single atom you cannot make any prediction when it will decay. It might be within the next 3 seconds or it might be here even after hundreds of billion years. The probability that it decays during the half-life is 50%, so if we look at a very large number of such atoms we can use statistics to predict how much substance will actually decay in a given time.

Now bismuth-209 has $1.9×10^{19}$ years half-life, So if we have a lot of Bismuth-209 after that time we will only have half of it. But that doesn't mean we don't have any decay right now. 209 grams of Bismuth-209 are 1 mole so $6.022×10^{23}$ particles, a lot of those. Now the decay goes down exponentially but if we just calculate the average decay rate during the first half-life if we start off with 1 mole and we keep in mind that the rate will be much faster in the beginning we see that we have an average of around 16000 decays every year, which means around 44 a day. Again, in the beginning it will be more, approaching the time of half-life it will be less.

Now how do we know the half-life? It's simple in theory, some french researcher put 31 grams bismuth-209 in a box with very sensitive equipment and measured the alpha particles which are send out during decay. This isn't a trivial task to do, but it's possible. They counted 128 particles over 5 days and then did the math to calculate the half-life.

The original paper can be found here.

## Solution 2:

It is not possible to measure the decay profile vs time as this is too long so the average number of disintegrations /time interval can be counted instead. From a consideration of the distribution of radioactive events the mean number of events in time $$t$$ is $$M=N_0(1-e^{-kt})$$ where $$k$$ is the decay constant and $$N_0$$ the initial number of atoms. For small values of $$kt$$, which is the case here as the decay lifetime is vast then, $$M = N_0kt$$. The rate and hence half-life (0.693/k) can be obtained if $$N_0$$ and $$t$$ are measured, however, there is uncertainty in this value.

In the case of measuring particles the binomial and poisson distributions have the property that the standard deviation in the number of events measure is $$\sigma=\sqrt{M}$$. Thus from a single measurement the std dev is obtained. This is not generally true, e.g. reading a thermometer or measuring a voltage. If $$1000$$ counts are recorded in $$10$$ seconds the standard deviation is $$\sigma= \sqrt{1000}\approx 32$$ and the mean count rate $$R=(1000\pm 32)/10 = 100\pm3.2$$ /sec. This can then be used to assess the uncertainty in the half-life using the propagation of error method.

As the count rate is inversely proportional to time, with longer counting times the std dev is reduced but only as $$1/\sqrt{t}$$