Solution: Carbon-14 Dating

1) First, rearrange the decay function:

$N=N_0{e}^{-kt}$

$\frac{N}{N_0}=e^{-kt}$

By substituting $\frac{N}{N_0}=\frac{1}{2}$ and rearranging the equation, we get:

$k=\frac{\ln 2}{t_{\frac{1}{2}}}$

Then, substitute $t_{\frac{1}{2}}$ = 5730 ,

$k=\frac{\ln 2}{5730}$

$k=1.2097\times {10}^{-4}{\mathrm{y}}^{-1}$

∴ The decay constant is 1.2097 × 10⁻⁴ y⁻¹.

2)

$\frac{N}{N_0}=e^{-kt}$

$0.4=e^{-1.2097\times {10}^{-4}\times t}$

ln 0.4 = −1.2097 × 10⁻⁴ × t

t = 7575y

∴ The age of that piece of wood is 7575 y.

(Source: Science Focus, The Hong Kong University of Science and Technology)