Question 14.5: A piece of cloth is discovered in a burial pit in the southw......
A piece of cloth is discovered in a burial pit in the southwestern United States. A tiny sample of the cloth is burned to CO _2, and the { }^{14} C /{ }^{12} C ratio is 0.250 times the ratio in today’s atmosphere. How old is the cloth?
Strategy We have the decay constant, k, for the relevant process from Example Problem 14.4. This will allow us to determine t, the time that has pased since the fiber in the cloth was harvested.
Recall from Example Problem 14.4
\begin{gathered}t_{1 / 2}=\frac{0.693}{k} \\k=\frac{0.693}{t_{1 / 2}}=\frac{0.693}{5730\text{ yr} }=1.21 \times 10^{-4}\text{ yr}^{-1}\end{gathered}
Equation 14.2 relates the amount of { }^{14} C to the time, which in this case is the age of the artifact:
\ln \frac{N_0}{N}=k t
Because the amount of { }^{12} C does not change over time, we can use the { }^{14} C /{ }^{12} C ratio in place of N here. We are told that the measured { }^{14} C /{ }^{12} C ratio is 25.0% of the current ratio in the atmosphere, so we can set N_0 / N=1 / 0.25. Solving for t gives
\begin{aligned}t & =\frac{1}{k} \ln \frac{N_0}{N} \\& =\frac{1}{1.21 \times 10^{-4}\text{ yr}^{-1}} \ln \frac{1}{0.250} \\& =11,500\text{ yr}\end{aligned}
Analyze Your Answer We can consider this answer with the same order of magnitude reasoning we used in Example Problem 14.4. The time is inversely proportional to the decay constant. (Units provide a good hint if you forget this.) With the decay constant on the order of 10^{-4}, our answer should be on the order of 10^4, which it is. We can also refer to the context of this problem: most carbon dating is carried out on archaeological artifacts, so a time frame on the order of 10,000 years seems reasonable.
Check Your Understanding We can also use the activity of a particular sample to solve a radiocarbon dating problem. The activity of { }^{14} C in the atmosphere (and therefore in all living things) today is 0.255 Bq/g of total carbon. Suppose that archaeologists found carbonized wheat grains in a fire pit at a dig site in the plains of northeastern Colorado. Measurement of the decay rate showed 0.070 Bq/g of carbon. How long ago was the wheat harvested?