The half-life for radioactive decay of ${ }^{14} \mathrm{C}$ is 5730 years. An archaeological artifact containing wood had only $80 \%$ of the ${ }^{14} \mathrm{C}$ found in a living tree. Estimate the age of the sample.

Question

The half-life for radioactive decay of ${ }^{14} \mathrm{C}$ is 5730 years. An archaeological artifact containing wood had only $80 \%$ of the ${ }^{14} \mathrm{C}$ found in a living tree. Estimate the age of the sample.,

MARKS App · Accepted Answer

Radioactive decay follows first order kinetics. Decay constant $(k)=\frac{0 \cdot 693}{t_{1 / 2}}=\frac{0.693}{5730}$ year $^{-1}$ $\begin{aligned} t &=\frac{2.303}{k} \log \frac{\left[\mathrm{A}_0\right]}{[\mathrm{A}]} \ &=\frac{2.303}{\left(0.693 / 5730 \text { years }^{-1}\right)} \log \frac{100}{80} \ &=\frac{2.303 \times 5730}{0.693} \times 0.0969=1845 \text { years. } \end{aligned}$

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