Textbook Question
A sample of F-18 has an initial decay rate of 1.5⨉105/s. How long will it take for the decay rate to fall to 2.5⨉103/s? (F-18 has a half-life of 1.83 hours.)
Ch.21 - Radioactivity & Nuclear Chemistry
Chapter 21, Problem 55
An ancient skull has a carbon-14 decay rate of 0.85 disintegrations per minute per gram of carbon (0.85 dis/min/g C). How old is the skull? (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 years.)
Verified step by step guidance1
Identify the initial and final decay rates: The initial decay rate for living organisms is 15.3 dis/min/g C, and the final decay rate for the skull is 0.85 dis/min/g C.
Use the decay formula: The decay of carbon-14 can be described by the formula \( N = N_0 e^{-\lambda t} \), where \( N \) is the final decay rate, \( N_0 \) is the initial decay rate, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.
Calculate the decay constant \( \lambda \): The decay constant is related to the half-life \( t_{1/2} \) by the formula \( \lambda = \frac{\ln(2)}{t_{1/2}} \). Substitute the given half-life of 5715 years to find \( \lambda \).
Rearrange the decay formula to solve for \( t \): \( t = \frac{\ln(N_0/N)}{\lambda} \). Substitute the values for \( N_0 \), \( N \), and \( \lambda \) into this equation.
Calculate \( t \): This will give you the age of the skull in years.
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 maintain a constant ratio of carbon-14 to carbon-12 while alive, but upon death, carbon-14 begins to decay at a known rate, allowing scientists to estimate the time since death based on the remaining carbon-14.
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3 and 4 Carbon Alkyls
Half-Life
The half-life of a radioactive isotope is the time required for half of the isotope in a sample to decay. For carbon-14, this half-life is approximately 5715 years. Understanding half-life is crucial for calculating the age of a sample, as it provides a consistent measure of decay over time.
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02:17Zero-Order Half-life
Decay Rate
The decay rate of a radioactive substance indicates how quickly it disintegrates over time, typically expressed in disintegrations per minute per gram. In this question, the decay rates of the ancient skull and living organisms are compared to determine the age of the skull by calculating how many half-lives have passed since the organism's death.
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Related Practice