Calculus

An archeologist thinks that she has found the missing link and wants to radiocarbon date her specimen. She finds a formula in Wikipedia

Age in years = -8033 $ln (F_m)$

where $F_m=\frac{N(t)}{N(0)}$ is the remaining fraction of C-14 divided by what was there at the beginning. In this case, all but 20% of the C-14 remains, that is $F_m = 0.2$ and the age turns out to be 12,928 years.

But where does this come from? Calculus comes into existence just before 1700 and was put to good use with great haste. It is the equivalent of mathematical story telling, so here is the tale of how little things work.

The half-life of carbon-14 is 5568 years. That is all we know. Find the decay constant for C-14 and then determine how old the sample is. Radioactive material decays at a rate proportional to the amount present; we have.

$\frac{dN}{dt} = - kN(t)$

Multiply $N$ by the integrating factor $e^{kt}$ and apply the differentiation product rule to obtain

$\frac{d}{dt} \left(e^{kt}N\right)=ke^{kt}N+e^{kt}(-kN)=0$

Integrate to obtain

$\int_0^t \frac{d}{dt} \left(e^{kt}N(t)\right) dt = e^{kt}N(t)-N(0)=0$

$N(t)=e^{-kt}N(0)$

$t = -\frac{1}{k} \ln(F_m)$

The half-life occurs at the time $t=\tau = 5568[\text{yr}]$ such that one-half of the initial population of C-14 atoms have yet to decay. Hence $-k\tau=\ln(1/2)$ .

$t = \frac{\tau}{-k\tau}\ln(F_m)$

$t = \frac{\tau}{\ln(1/2)}\ln(F_m) = -8033 \ln(F_m)\;[\text{yr}]$

which is the same constant as the Wikipedia formula

contributed anonymously to raedwulf

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