Question

The radioactive element​ carbon-14 has a​ half-life of 5750 years. a scientist determined that the bones from a mastodon had lost 78.5​% of their​ carbon-14. how old were the bones at the time they were​ discovered?

Answer 1

Radioactive elements obey 1st order kinetics,

For 1st order reaction, relation between rate constant (k) and half life [t(1/2)] is,
k =
`(0.693)/(t(1/2)) = (0.693)/(5750) = 1.205 X 10^-^4 hr^-^1`

Also, for 1st order reaction, we have
t =
`(2.303)/(k) log \frac{\text{initial conc.}}{\text{final conc.}}`

Given that: the bones from a mastodon had lost 78.5​% of their​ C14,
∴ initial conc. of C14 = 100%, conc. of C14 left after time 't' = 21.5%

∴t =
`(2.303)/(1.205 X 10^(-4)) log \frac{\text{100}}{\text{21.5}}` = 1.2758 X 10^4 hours