Carbon-14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of absorbed by a tree that grew several...
Carbon-14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of absorbed by a tree that grew several centuries ago should be the same as the amount of absorbed by a tree growing today. A piece of ancient charcoal contains only 15% as much of the radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal? (The half-life of Carbon-14 is 5715 years.)
It is impossible to predict when a particular atom will decay. However, it is equally likely to decay at any instant in time. Therefore, given a sample of a particular radioisotope, the number of decay events `−dN` expected to occur in a small interval of time `dt` is proportional to the number of atoms present `N,` i.e.
For different atoms different decay constants apply.
The above differential equation is easily solved by separation of variables.
where `N_0` is the number of undecayed atoms at time `t=0.`
We can now calculate decay constant `lambda` for carbon-14 using the given half-life.
`lambda=1.21 times 10^-4`
Note that the above constant is usually measured in seconds rather than years.
Now we can return to the problem at hand. Since the charcoal contains only 15% (`0.15N_0` ) of the original carbon-14, we have
Now we solve for `t.`
According to our calculation the tree was burned approximately 15679 years ago.