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 = $\frac{0.693}{t(1/2)} = \frac{0.693}{5750} = 1.205 X 10^-^4 hr^-^1$

Also, for 1st order reaction, we have
t = $\frac{2.303}{k} log \frac{\text{initial conc.}}{\text{final conc.}}$
 
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 = $\frac{2.303}{1.205 X 10^(-4)} log \frac{\text{100}}{\text{21.5}}$ = 1.2758 X 10^4 hours