Question

(Radiocarbon dating) Carbon extracted from an ancient skull contained only one-sixth as much 14C as carbon extracted from present-day bone. How old is the skull?

Answer 1

Answer:

14734 years

Step-by-step explanation

We know the decay rate with time is given by;

$\frac{dN}{dt}=-kN$

Where K = 0.0001216 (radioactive decay constant)

Also we know the number of atoms remaining to decay in proportion to original number of atoms is given by;

$Nt=N_{0} e^{-kt}$

Where,

$N_{0} = N(0)$

Here we will find  $t_{0}$  such that  $N(t_{0} )=\frac{1}{6}N_{0}$

So we can say;

$N(t_{0} )=\frac{1}{6}N_{0}$ ⇔  $N_{0}e^{-kt_{0} } =\frac{1}{6} N_{0}$

⇒ $e^{-kt_{0} } =\frac{1}{6}$

Taking natural log  on both sides. we get

$ln(e^{-kt_{0} }) = ln (\frac{1}{6} )$

⇒ $-kt_{0} = ln(1)-ln(6)$

$-kt_{0} =-1.79175946923$

$t_{0} = \frac{1.79175946923}{0.0001216} = 14734 (approx)$

So, the skull is approximately 14734 years old