Modelling Radioactive decay
We can model radioactive decay of atoms using the following equation:
N(t) = N0 e-λt
N0: is the initial quantity of the element
λ: is the radioactive decay constant
t: is time
N(t): is the quantity of the element remaining after time t.
So, for Carbon-14 which has a half life of 5730 years (this means that after 5730 years exactly half of the initial amount of Carbon-14 atoms will have decayed) we can calculate the decay constant λ.
After 5730 years, N(5730) will be exactly half of N0, therefore we can write the following:
N(5730) = 0.5N0 = N0 e-λt
0.5 = e-λt
and if we take the natural log of both sides and rearrange we get:
λ = ln(1/2) / -5730
We can now use this to solve problems involving Carbon-14 (which is used in Carbon-dating techniques to find out how old things are).
eg. You find an old parchment and after measuring the Carbon-14 content you find that it is just 30% of what a new piece of paper would contain. How old is this paper?
N(t) = N0 e-0.000121t
N(t)/N0 = e-0.000121t
0.30 = e-0.000121t
t = ln(0.30)/(-0.000121)
t = 9950 years old.
Probability density functions
We can also do some interesting maths by rearranging:
N(t) = N0 e-λt
N(t)/N0 = e-λt
and then plotting N(t)/N0 against time.
N(t)/N0 will have a range between 0 and 1 as when t = 0, N(0) = N0 which gives N(0)/N(0) = 1.
We can then manipulate this into the form of a probability density function – by finding the constant a which makes the area underneath the curve equal to 1.
solving this gives a = λ. Therefore the following integral:
will give the fraction of atoms which will have decayed between times t1 and t2.
We could use this integral to work out the half life of Carbon-14 as follows:
Which if we solve gives us t = 5728.5 which is what we’d expect (given our earlier rounding of the decay constant).
We can also now work out the expected (mean) time that an atom will exist before it decays. To do this we use the following equation for finding E(x) of a probability density function:
and if we substitute in our equation we get:
Now, we can integrate this by parts:
So the expected (mean) life of an atom is given by 1/λ. In the case of Carbon, with a decay constant λ ≈0.000121 we have an expected life of a Carbon-14 atom as:
E(t) = 1 /0.000121
E(t) = 8264 years.
Now that may sound a little strange – after all the half life is 5730 years, which means that half of all atoms will have decayed after 5730 years. So why is the mean life so much higher? Well it’s because of the long right tail in the graph – we will have some atoms with very large lifespans – and this will therefore skew the mean to the right.