Question

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

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

Step-by-step solution

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: $$ \text{Decay ratio} = \frac{9.7\text{ disintegrations per minute per gram of carbon}}{16.3\text{ disintegrations per minute per gram of carbon}} $$

Calculate the decay ratio value

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

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: $$ \text{Age} = \frac{5715\mathrm{yr} \times \log_{2}(1/0.5951)}{\log_{2}(2)} $$

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.

Key concepts

Carbon-14 Decay

Carbon-14 decay is a naturally occurring process in which the radioactive isotope Carbon-14 (\(^{14}\text{C}\)) loses particles from its nucleus over time. All living organisms take in \(^{14}\text{C}\) through processes like eating and breathing. This isotope is unstable, meaning it will eventually decay to transform into a more stable form.
The decay process can be described as a type of exponential decay. This means it loses its radiocarbon activity at a predictable rate over time, which can be used to date organic materials.
The isotopic ratio becomes a crucial metric for scientists to compare between living organisms and ancient samples. For example, if a mummy's shroud shows lower disintegration rates compared to a living organism, it signifies that the \(^{14}\text{C}\) in the shroud has decayed over a period.

Half-life Calculation

Half-life is the time required for half of the radioactive nuclei in a sample to decay. For \(^{14}\text{C}\), the half-life is 5715 years. This concept is pivotal in radiocarbon dating because it helps determine how long it has been since an organism died and stopped replenishing its \(^{14}\text{C}\) intake.
During half-life calculation, scientists use the known half-life value to relate the current \(^{14}\text{C}\) level in a sample to its age.
This is done by comparing the \(^{14}\text{C}\) disintegration rate of a sample with that of a current living organism. By setting up the proportional equation between the past activity and known half-life, one can determine how many half-lives have passed, and thus estimate the age of the specimen.

Disintegration Rate

The disintegration rate, or the activity, of a sample in radiocarbon dating refers to the rate at which \(^{14}\text{C}\) atoms in a sample decay over time. It is generally measured in disintegrations per minute per gram of carbon.
In living organisms, the disintegration rate is consistent because they continually exchange carbon with the environment. However, in a non-living sample, this rate only decreases over time due to the lack of replenishment of \(^{14}\text{C}\).
By comparing the disintegration rate of an ancient sample with that of a living organism, scientists can determine how much \(^{14}\text{C}\) has decayed. This comparison allows for the calculation of the decay ratio, which in turn aids in determining the age of the sample.

Age Estimation

Age estimation using radiocarbon dating involves calculating how long ago a plant or animal died. This is done by applying the decay ratio to the half-life of \(^{14}\text{C}\).
The age estimation process begins by determining the decay ratio between the sample and living organisms. Following this, the half-life formula helps translate this ratio into an age for the sample. The logic is that the lower the \(^{14}\text{C}\) activity compared to a living equivalent, the older the sample.
Using logarithmic calculations, one can determine the number of half-lives that have elapsed and consequently, estimate the sample's age. For example, when the decay ratio deciphers to a certain value, mathematics will convert this into an actual timeline of years, such as approximating a shroud to be 4382 years old.