A mammoth skeleton has a carbon-14 decay rate of 0.48 disintegration per minute per gram of carbon (0.48 dis/min • g C). When did the mammoth live? (Assume that living organisms have a carbon-14 decay rate of 15.3 dis/min • g C and that carbon- 14 has a half-life of 5715 yr.)

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Identify the given values: the current decay rate of the mammoth skeleton is 0.48 dis/min \( \cdot \) g C, the decay rate of living organisms is 15.3 dis/min \( \cdot \) g C, and the half-life of carbon-14 is 5715 years.

Use the formula for radioactive decay: \( N = N_0 \times e^{-\lambda t} \), where \( N \) is the current decay rate, \( N_0 \) is the initial decay rate, \( \lambda \) 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 time \( t \) when the mammoth lived.

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Carbon-14 Dating

Carbon-14 dating is a radiometric dating method used to determine the age of organic materials by measuring the amount of carbon-14 they contain. Living organisms absorb carbon-14 from the atmosphere, and when they die, the carbon-14 begins to decay at a known rate, characterized by its half-life. This technique is particularly useful for dating materials up to about 50,000 years old.

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Half-Life

The half-life of a radioactive isotope is the time required for half of the isotope in a sample to decay into a different element or isotope. For carbon-14, the half-life is approximately 5,715 years. Understanding half-life is crucial for calculating the age of a sample based on the remaining amount of carbon-14 compared to its initial concentration in living organisms.

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Disintegration Rate

The disintegration rate refers to the number of radioactive decays occurring per unit time, typically expressed in disintegrations per minute (dis/min). In this context, the disintegration rate of carbon-14 in living organisms is higher than in a deceased organism, allowing scientists to estimate the time since death by comparing the current decay rate to the known rate in living organisms.

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