21.3 Radioactive decay  (Page 6/21)

Thus, the ${}_6^{14}\text{C}\text{:}{}_6^{12}\text{C}$ ratio gradually decreases after the plant dies. The decrease in the ratio with time provides a measure of the time that has elapsed since the death of the plant (or other organism that ate the plant). [link] visually depicts this process.

For example, with the half-life of ${}_6^{14}\text{C}$ being 5730 years, if the ${}_6^{14}\text{C}\text{:}{}_6^{12}\text{C}$ ratio in a wooden object found in an archaeological dig is half what it is in a living tree, this indicates that the wooden object is 5730 years old. Highly accurate determinations of ${}_6^{14}\text{C}\text{:}{}_6^{12}\text{C}$ ratios can be obtained from very small samples (as little as a milligram) by the use of a mass spectrometer.

Radiocarbon dating

A tiny piece of paper (produced from formerly living plant matter) taken from the Dead Sea Scrolls has an activity of 10.8 disintegrations per minute per gram of carbon. If the initial C-14 activity was 13.6 disintegrations/min/g of C, estimate the age of the Dead Sea Scrolls.

Solution

The rate of decay (number of disintegrations/minute/gram of carbon) is proportional to the amount of radioactive C-14 left in the paper, so we can substitute the rates for the amounts, N , in the relationship:

$t=-\frac{1}{\lambda }\text{ln}\left(\frac{N_t}{N_0}\right)⟶t=-\frac{1}{\lambda }\text{ln}\left(\frac{{\text{Rate}}_t}{{\text{Rate}}_0}\right)$

where the subscript 0 represents the time when the plants were cut to make the paper, and the subscript t represents the current time.

The decay constant can be determined from the half-life of C-14, 5730 years:

$\lambda =\frac{\text{ln 2}}{t_{1\text{/}2}}=\frac{0.693}{\text{5730 y}}=1.21\times {10}^{\mathrm{-4}}{\text{y}}^{\mathrm{-1}}$

Substituting and solving, we have:

$t=-\frac{1}{\lambda }\text{ln}\left(\frac{{\text{Rate}}_t}{{\text{Rate}}_0}\right)=-\frac{1}{1.21\times {10}^{\mathrm{-4}}{\text{y}}^{\mathrm{-1}}}\text{ln}\left(\frac{10.8\text{dis/min/g C}}{13.6\text{dis/min/g C}}\right)=\text{1910 y}$

Therefore, the Dead Sea Scrolls are approximately 1900 years old ( [link] ).

Check your learning

More accurate dates of the reigns of ancient Egyptian pharaohs have been determined recently using plants that were preserved in their tombs. Samples of seeds and plant matter from King Tutankhamun’s tomb have a C-14 decay rate of 9.07 disintegrations/min/g of C. How long ago did King Tut’s reign come to an end?

Answer:

about 3350 years ago, or approximately 1340 BC

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There have been some significant, well-documented changes to the ${}_6^{14}\text{C}\text{:}{}_6^{12}\text{C}$ ratio. The accuracy of a straightforward application of this technique depends on the ${}_6^{14}\text{C}\text{:}{}_6^{12}\text{C}$ ratio in a living plant being the same now as it was in an earlier era, but this is not always valid. Due to the increasing accumulation of CO ₂ molecules (largely ${}_6^{12}\text{C}{\text{O}}_2)$ in the atmosphere caused by combustion of fossil fuels (in which essentially all of the ${}_6^{14}\text{C}$ has decayed), the ratio of ${}_6^{14}\text{C}\text{:}{}_6^{12}\text{C}$ in the atmosphere may be changing. This manmade increase in ${}_6^{12}\text{C}{\text{O}}_2$ in the atmosphere causes the ${}_6^{14}\text{C}\text{:}{}_6^{12}\text{C}$ ratio to decrease, and this in turn affects the ratio in currently living organisms on the earth. Fortunately, however, we can use other data, such as tree dating via examination of annual growth rings, to calculate correction factors. With these correction factors, accurate dates can be determined. In general, radioactive dating only works for about 10 half-lives; therefore, the limit for carbon-14 dating is about 57,000 years.