# 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.

-(dN)/(dt)propto N

For different atoms different decay constants apply.

-(dN)/(dt)=\lambda N

The above differential equation is easily solved by separation of variables.

N=N_0e^(-lambda t)

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.

N_0/2=N_0e^(-lambda 5715)

e^(-5715lambda)=1/2

-5715lambda=ln(1/2)

lambda=-(ln(1/2))/5715

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

0.15N_0=N_0e^(-1.21times10^-4t)

Now we solve for t.

e^(-1.21times10^-4t=0.15)

1.21times10^-4=-ln 0.15

t=-(ln0.15)/(1.21times10^-4)

t=15678.68

According to our calculation the tree was burned approximately 15679 years ago.