The cloth shroud from around a mummy is found to have a 14C 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 14C decay, 5730 yr, calculate the age of the shroud.

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Identify the given values: the activity of the shroud (9.7 disintegrations per minute per gram), the activity of living organisms (16.3 disintegrations per minute per gram), and the half-life of 14C (5730 years).

Use the formula for radioactive decay: \( N = N_0 e^{-\lambda t} \), where \( N \) is the current activity, \( N_0 \) is the initial activity, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.

Calculate the decay constant \( \lambda \) using the half-life formula: \( \lambda = \frac{\ln(2)}{\text{half-life}} \).

Rearrange the decay formula to solve for \( t \): \( t = \frac{\ln(N_0/N)}{\lambda} \).

Substitute the known values into the rearranged formula to calculate the age of the shroud.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radiocarbon Dating

Radiocarbon dating is a method used to determine the age of organic materials by measuring the amount of carbon-14 (14C) remaining in a sample. Living organisms continuously take in carbon, including 14C, until they die. After death, the 14C decays at a known rate, characterized by its half-life of 5730 years, allowing scientists to estimate the time since the organism's death based on the remaining 14C activity.

Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. For carbon-14, this period is approximately 5730 years. Understanding half-life is crucial for calculating the age of ancient organic materials, as it provides a consistent measure of decay that can be used to relate the remaining amount of 14C to the time elapsed since the organism's death.

Disintegration Rate

The disintegration rate refers to the number of radioactive decays occurring in a sample per unit time, typically measured in disintegrations per minute (dpm). In the context of radiocarbon dating, comparing the disintegration rates of a sample to that of a living organism allows for the calculation of the sample's age. The difference in activity levels indicates how much time has passed since the organism ceased to take in carbon.