In a cave in Oregon, archaeologists found bones, plant remains, and fossilized feces. DNA remaining in the feces indi-cates their human origin but not their age. To date the remains, the decay rate was measured and found to be 2.71 disinte-grations/min per gram of carbon. Currently living organisms have a decay rate of 15.3 disintegrations/min per gram of carbon, and the half-life of 14C is 5715 years. How old are the remains? (a) 1460 years(b) 9900 years(c) 14300 years(d) 18600 years

Verified step by step guidance

1. Identify the problem as a radioactive decay problem involving carbon-14 dating.
2. Use the formula for radioactive decay: \( N = N_0 e^{-\lambda t} \), 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.
3. Calculate the decay constant \( \lambda \) using the half-life formula: \( \lambda = \frac{\ln(2)}{\text{half-life}} \). Substitute the given half-life of 5715 years into the formula.
4. Rearrange the decay formula to solve for \( t \): \( t = \frac{\ln(N_0/N)}{\lambda} \). Substitute the given values for \( N_0 = 15.3 \) disintegrations/min and \( N = 2.71 \) disintegrations/min.
5. Calculate \( t \) using the rearranged formula to find the age of the remains.

Key Concepts

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This decay occurs at a predictable rate, characterized by the half-life, which is the time required for half of the radioactive atoms in a sample to decay. Understanding this concept is crucial for dating ancient remains, as it allows scientists to estimate the age of organic materials based on the remaining concentration of radioactive isotopes.

Half-Life

Half-life is a specific time period in which half of a given quantity of a radioactive substance decays. For carbon-14 (14C), the half-life is approximately 5715 years. This concept is essential for calculating the age of archaeological finds, as it provides a framework for determining how many half-lives have passed since the organism died, based on the ratio of remaining 14C to stable carbon isotopes.

Carbon Dating

Carbon dating, or radiocarbon dating, is a method used to determine the age of organic materials by measuring the amount of carbon-14 they contain. Since living organisms continuously exchange carbon with their environment, the ratio of 14C to 12C remains relatively constant during their life. After death, 14C decays while 12C remains stable, allowing scientists to calculate the time since death by comparing the current 14C levels to those in the atmosphere.