Carbon 14 dating math problem
: 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. 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. Therefore the following integral: will give the fraction of atoms which will have decayed between times t1 and t2.
In the previous article, we saw that light attenuation obeys an exponential law.
To show this, we needed to make one critical assumption: that for a thin enough slice of matter, the proportion of light getting through the slice was proportional to the thickness of the slice.
Exactly the same treatment can be applied to radioactive decay.
However, now the "thin slice" is an interval of time, and the dependent variable is the number of radioactive atoms present, N(t). If we have a sample of atoms, and we consider a time interval short enough that the population of atoms hasn't changed significantly through decay, then the proportion of atoms decaying in our short time interval will be proportional to the length of the interval.
We end up with a solution known as the "Law of Radioactive Decay", which mathematically is merely the same solution that we saw in the case of light attenuation.We get an expression for the number of atoms remaining, N, as a proportion of the number of atoms N, where the quantity l, known as the "radioactive decay constant", depends on the particular radioactive substance.Again, we find a "chance" process being described by an exponential decay law.We can easily find an expression for the chance that a radioactive atom will "survive" (be an original element atom) to at least a time t.