A bone suspected to have originated during the period of the Roman emperors was found in Great Britain. Accelerator techniques gave its ^14C/^12C ratio as 1.10 × 10^-12. Is the bone old enough to have Roman origins? Strategy Remember that the initial ratio of ^14C/^12C at the time of death was

Question

A bone suspected to have originated during the period of the Roman emperors was found in Great Britain. Accelerator techniques gave its [latex]{ }^{14} C /{ }^{12} C ext { ratio as } 1.10 imes 10^{-12}[/latex]. Is the bone old enough to have Roman origins?

Strategy Remember that the initial ratio of [latex]{ }^{14} C /{ }^{12} C[/latex] at the time of death was [latex]R_{0}=1.2 imes 10^{-12}[/latex]. We use the radioactive decay law to determine the time t that it will take for the ratio to decrease to [latex]1.10 imes 10^{-12}[/latex].

Accepted Answer

The number of [latex]{ }^{14} C ext { atoms decays as } e^{-\lambda t}[/latex].

[latex]N\left({ }^{14} C ight)=N_{0} e^{-\lambda t}[/latex]

The ratio of ions is given by

[latex]R=\frac{N\left({ }^{14} C ight)}{N\left({ }^{12} C ight)}=\frac{N_{0}\left({ }^{14} C ight) e^{-\lambda t}}{N\left({ }^{12} C ight)}=R_{0} e^{-\lambda t}[/latex]

where [latex]R_{0}[/latex] is the original ratio. We can solve this equation for t.

[latex]\begin{aligned}e^{-\lambda t} &=\frac{R}{R_{0}} \t &=\frac{-\ln \left(R / R_{0} ight)}{\lambda}=-t_{1 / 2} \frac{\ln \left(R / R_{0} ight)}{\ln (2)} \&=-(5730 y )\left(\frac{-0.087}{0.693} ight)=720 y\end{aligned}[/latex]

where we have inserted the known values of [latex]t_{1 / 2}, R_{0}[/latex], and R to find the age of the bone. The bone does not date from the Roman Empire, but from the medieval period.

Question 12.19: A bone suspected to have originated during the period of the...

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