# How can radioactive half-life be used to date fossils?

Mar 16, 2018

$A = {A}_{0} {e}^{- \lambda t}$, where:
$A$ = current activity (${s}^{-} 1$)
${A}_{0}$ = original activity (${s}^{-} 1$)
$\lambda$ = decay constant $\left(\ln \frac{2}{t} _ \left(\frac{1}{2}\right)\right)$ (${s}^{-} 1$)
$t$ = time ($s$, though sometimes uses different units like $y e a {r}^{-} 1$ when $\lambda$ is given in terms of years)

$\frac{A}{A} _ 0 = {e}^{- \lambda t}$

$\ln \left(\frac{A}{A} _ 0\right) = - \lambda t$

$t = - \ln \frac{\frac{A}{A} _ 0}{\lambda}$

$t = - \frac{{t}_{\frac{1}{2}} \ln \left(\frac{A}{A} _ 0\right)}{\ln} \left(2\right)$

$t$ $\left(\textrm{y e a r s}\right) = - \frac{5730 \ln \left(\frac{A}{A} _ 0\right)}{\ln} \left(2\right)$

If we know the current activity of the decay of $\text{^14"C}$ and use living material for a rough estimate for the original activity of $\text{^14"C}$, then we can put them into this equation to find its age in years.