Question

The cloth shroud from around a mummy is found to have ${\mathrm{a}}^{14}\mathrm{C}$ 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 yr, calculate the age of the shroud.

Short answer

The age of the shroud is approximately 3600 years old, calculated using the given carbon-14 activity values, half-life of carbon-14, and the decay constant formula. The ratio of the activities is 0.5951, and the decay constant is approximately $1.210\times {10}^{-4}{\text{yr}}^{-1}$.

Step-by-step solution

Calculate the ratio of the C-14 activities

First, we should calculate the ratio of the given activity of the cloth to the activity of living organisms. This ratio can be calculated as: $$Ratio=\frac{Activit{y}_{shroud}}{Activit{y}_{living}}$$

Substitute the activity values of the shroud and organisms

Now, let's substitute the given activity values of the shroud (9.7 disintegrations per minute per gram) and living organisms (16.3 disintegrations per minute per gram) into the formula: $$Ratio=\frac{9.7}{16.3}$$

Calculate the ratio

Now, we can calculate the ratio by dividing the activity of the shroud by the activity of the living organisms: $$Ratio\approx 0.5951$$

Use the half-life formula

Now, we will use the half-life formula to determine the age of the shroud. Tthe relationship between the ratio of the activities, time passed (t), and the half-life of carbon-14 (T_half) is given by the following equation: $$Ratio=e^{-\lambda t}$$ Where $\lambda$ is the decay constant, and it has a relationship with the half-life: $$\lambda =\frac{ln(2)}{T_{half}}$$ Now let's substitute the half-life of carbon-14 decay, 5715 years, into the decay constant formula: $$\lambda =\frac{ln(2)}{5715}$$

Calculate the decay constant

By calculating the decay constant, we get: $$\lambda \approx 1.210\times {10}^{-4}{\text{yr}}^{-1}$$

Replace the ratio and decay constant in the formula

Now, we will substitute both the ratio and the decay constant into the half-life formula: $$0.5951=e^{-1.210\times {10}^{-4}\times t}$$

Solving for the age (t)

To calculate the age, t, we need to solve the equation for t. To isolate t, we will first take the natural logarithm (ln) of both sides: $$ln(0.5951)=-1.210\times {10}^{-4}\times t$$ Now, we need to divide both sides by -1.210×10^{-4} to get the value of t: $$t=\frac{ln(0.5951)}{-1.210\times {10}^{-4}}$$

Calculate the age of the shroud

Finally, we can calculate the age of the shroud by performing the division: $$t\approx 3600\text{yr}$$ The age of the shroud is approximately 3600 years old.

Key concepts

Carbon-14 Decay

Carbon-14 decay is a fascinating process that occurs in isotopes of carbon. As an unstable isotope, Carbon-14 (¹⁴C) undergoes radioactive decay over time. This is a natural event where the Carbon-14 atoms emit radiation, and transform into a more stable nitrogen-14 (¹⁴N) isotope. This decay process happens at a predictable rate, allowing scientists to use it as a clock to determine the age of organic materials. The activity—or the number of disintegrations per minute—is a measure of how many Carbon-14 atoms decay in a given amount of time.

In the case of the shroud mentioned in the exercise, the reduction in activity indicates that it contains less ¹⁴C compared to what is found in a living organism. This decrease in activity is tied directly to the time that has elapsed since the shroud stopped exchanging carbon with the environment—typically the time since the death of the organism from which it came.

Half-life Calculation

Half-life is a crucial concept in understanding the decay of radioactive materials. It is defined as the time required for half of the radioactive isotopes in a sample to decay. For Carbon-14, the half-life is approximately 5,715 years. Knowing this value allows archaeologists and scientists to calculate the age of organic remains by comparing the amount of Carbon-14 left in a sample to what would have been present when it was part of a living organism.

The half-life is constant, no matter the size of the sample or the remaining amount of Carbon-14, and is unaffected by environmental factors like temperature or pressure. This constancy makes it an invaluable tool in dating archaeological finds, as highlighted by the fact that the calculation of the age of the mummy's shroud was based on the known half-life of Carbon-14.

Exponential Decay Formula

The mathematical representation of radioactive decay follows an exponential decay formula. Exponential decay means that as time progresses, the quantity of the isotope decreases by proportions rather than a fixed amount. The general formula for the decay of a substance is given by:

e^{- t}

where represents the remaining quantity of the substance, is the initial quantity, is the decay constant (related to half-life), and t is the time elapsed. This model is crucial as it gives a precise method to calculate the time since an organism's death when applied to measure the decay of Carbon-14. In the exercise, the explication of the natural logarithm was used to solve for the elapsed time, which is a typical step when dealing with exponential equations.

Archaeological Dating Methods

Radiocarbon dating is just one component of archaeological dating methods. As a relative dating method, it is employed alongside other techniques to estimate the age of artifacts and environmental contexts. Other methods include dendrochronology, which uses tree ring patterns to date events and environmental changes, or stratigraphy, which analyzes the layering of rocks and soil.

Importance of Cross-Dating

To confirm dates and improve accuracy, archaeologists often use multiple dating methods together, a process known as cross-dating. This can help paint a clearer picture of the past, ensuring that a singular dating technique doesn't lead to misconceptions about the timeline of historical or pre-historical artifacts. By integrating results from Carbon-14 dating with other archaeological evidence, scientists can construct a more detailed and accurate history of the object and the civilization it came from.