Question 43.9: Radioactive Dating A piece of charcoal containing 25.0 g of ...
Radioactive Dating
A piece of charcoal containing 25.0 g of carbon is found in some ruins of an ancient city. The sample shows a {}^{14} C activity R of 250 decays/min. How long has the tree from which this charcoal came been dead?
Conceptualize Because the charcoal was found in ancient ruins, we expect the current activity to be smaller than the initial activity. If we can determine the initial activity, we can find out how long the wood has been dead.
Categorize The text of the question helps us categorize this example as a carbon dating problem.
Analyze Solve Equation 43.7 for t and incorporate Equation 43.8:
R=\left|\frac{d N}{d t}\right|=\lambda N=\lambda N_0 e^{-\lambda t}=R_0 e^{-\lambda t} (43.7)
T_{1 / 2}=\frac{\ln 2}{\lambda}=\frac{0.693}{\lambda} (43.8)
(1) t=-\frac{1}{\lambda} \ln \left(\frac{R}{R_0}\right)=-\frac{T_{1 / 2}}{\ln 2} \ln \left(\frac{R}{R_0}\right)
Evaluate the ratio R / R_0 using Equation 43.7, the initial value of the { }^{14} C /{ }^{12} C ratio r_0, the number of moles n of carbon, and Avogadro’s number N_{A} :
\frac{R}{R_0}=\frac{R}{\lambda N_0\left({ }^{14} C\right)}=\frac{R}{\lambda r_0 N_0\left({ }^{12} C\right)}=\frac{R}{\lambda r_0 n N_{A}}Replace the number of moles in terms of the molar mass M of carbon and the mass m of the sample and substitute for the decay constant λ:
\frac{R}{R_0}=\frac{R}{\left(\ln 2 / T_{1 / 2}\right) r_0(m / M) N_{A}}=\frac{R M T_{1 / 2}}{r_0 m N_{A} \ln 2}Substitute numerical values:
\begin{aligned} \frac{R}{R_0}&=\frac{\left(250 \min^{-1}\right)(12.0 g/ \text{mol})(5730 yr)}{\left(1.3 \times 10^{-12}\right)(25.0 g)\left(6.022 \times 10^{23} \text{ mol}^{-1}\right) \ln 2}\left(\frac{3.156 \times 10^7 s}{1 yr}\right)\left(\frac{1 \min}{60 s}\right)\\ &=0.667\end{aligned}Substitute this ratio into Equation (1):
t=-\frac{5 730 yr}{\ln 2} \ln (0.667)=3.4 \times 10^3 yrFinalize Note that the time interval found here is on the same order of magnitude as the half-life, so { }^{14} C is a valid isotope to use for this sample, as discussed in Conceptual Example 43.8.