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Chapter 21: Problem 39

The cloth shroud from around a mummy is found to have a \({ }^{14} \mathrm{C}\) activity of 9.7 disintegrations per minute per gram of carbon as compared with living organisms that undergo 16.3 disintegrations per minute per gram of carbon. From the half-life for \({ }^{14} \mathrm{C}\) decay, \(5715 \mathrm{yr},\) calculate the age of the shroud.

Short Answer

Expert verified
The age of the cloth shroud is approximately \(4382\) years.

Step by step solution

01

Calculate the decay ratio

First, we have to find the ratio of \({ }^{14} \mathrm{C}\) activity in the shroud as compared to living organisms. The formula for this ratio is: $$ \text{Decay ratio} = \frac{\text{Activity of the shroud}}{\text{Activity of living organisms}} $$ Now, we plug in the given values: $$ \text{Decay ratio} = \frac{9.7\text{ disintegrations per minute per gram of carbon}}{16.3\text{ disintegrations per minute per gram of carbon}} $$
02

Calculate the decay ratio value

Now, we can find the decay ratio value by doing the division: $$ \text{Decay ratio} = \frac{9.7}{16.3} \approx 0.5951 $$
03

Apply the half-life formula

Now that we have the decay ratio, we can use the half-life formula to determine the age of the shroud. The formula is: $$ \text{Age} = \frac{\text{Half-life} \times \log_{2}(\frac{1}{\text{Decay ratio}})}{\log_{2}(2)} $$ We plug in the given half-life and the calculated decay ratio: $$ \text{Age} = \frac{5715\mathrm{yr} \times \log_{2}(1/0.5951)}{\log_{2}(2)} $$
04

Calculate the age of the shroud

Now, we can calculate the age of the shroud: $$ \text{Age} = \frac{5715\mathrm{yr} \times \log_{2}(1.6805)}{1} \approx 5715\mathrm{yr} \times 0.767 \approx 4382\mathrm{yr} $$ So, the age of the shroud is approximately \(4382\) years.

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Most popular questions from this chapter

Charcoal samples from Stonehenge in England were burned in \(\mathrm{O}_{2},\) and the resultant \(\mathrm{CO}_{2}\) gas bubbled into a solution of \(\mathrm{Ca}(\mathrm{OH})_{2}\) (limewater), resulting in the precipitation of \(\mathrm{CaCO}_{3} .\) The \(\mathrm{CaCO}_{3}\) was removed by filtration and dried. A \(788-\mathrm{mg}\) sample of the \(\mathrm{CaCO}_{3}\) had a radioactivity of \(1.5 \times 10^{-2} \mathrm{~Bq}\) due to carbon-14. By comparison, living organisms undergo 15.3 disintegrations per minute per gram of carbon. Using the half-life of carbon- \(14,5715 \mathrm{yr},\) calculate the age of the charcoal sample.

Each of the following nuclei undergoes either beta decay or positron emission. Predict the type of emission for each: (a) tritium, \({ }_{1}^{3} \mathrm{H},(\mathbf{b}){ }_{38}^{89} \mathrm{Sr}\), (c) iodine-120, (d) silver-102.

The naturally occurring radioactive decay series that begins with \({ }_{92}^{235} \mathrm{U}\) stops with formation of the stable \({ }_{82}^{207} \mathrm{~Pb}\) nucleus. The decays proceed through a series of alpha-particle and beta-particle emissions. How many of each type of emission are involved in this series?

The half-life for the process \({ }^{238} \mathrm{U} \longrightarrow{ }^{206} \mathrm{~Pb}\) is \(4.5 \times 10^{9} \mathrm{yr}\). A mineral sample contains \(75.0 \mathrm{mg}\) of \({ }^{238} \mathrm{U}\) and \(18.0 \mathrm{mg}\) of \({ }^{206} \mathrm{~Pb}\). What is the age of the mineral?

Each statement that follows refers to a comparison between two radioisotopes, \(\mathrm{A}\) and \(\mathrm{X}\). Indicate whether each of the following statements is true or false, and why. (a) If the half-life for \(A\) is shorter than the half-life for \(X, A\) has a larger decay rate constant. (b) If \(X\) is "not radioactive," its half-life is essentially zero. (c) If A has a half-life of 10 years, and \(X\) has a half-life of 10,000 years, A would be a more suitable radioisotope to measure processes occurring on the 40 -year time scale.
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