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Magazine Chapter 23: Problem 9
Find the percent of \(\mathrm{a}{ }_{6}^{14} \mathrm{C}\) sample that will decay in the next 3000 years. Its halflife is 5730 years.
Short Answer
Expert verified
31.7% of the sample will decay in 3000 years.
Step by step solution
01
Determine the decay constant
First, we need to find the decay constant, represented by \( k \). The formula for the decay constant is \( k = \frac{\ln(2)}{\text{halflife}} \). For this problem, the halflife is 5730 years, so we have:\[k = \frac{\ln(2)}{5730} \approx 1.2097 \times 10^{-4} \text{ per year}\]
02
Calculate the number of half-lives that will pass
To find the fraction of the sample that will decay, we first determine how many half-lives will occur over 3000 years. We calculate this using:\[\text{Number of half-lives} = \frac{3000}{5730} \approx 0.5236 \text{ half-lives}\]
03
Compute the remaining fraction of the sample
The remaining amount of the sample can be calculated using the formula for exponential decay: \( N(t) = N_0 \cdot (0.5)^{\text{Number of half-lives}} \). Therefore:\[N(t) = N_0 \cdot (0.5)^{0.5236} \approx 0.683 \cdot N_0\]
04
Calculate the fraction that will decay
The fraction of the sample that will decay is the initial sample amount minus the remaining sample amount. Therefore, the decayed fraction is:\[1 - 0.683 = 0.317\]
05
Convert the decayed fraction to a percentage
Finally, convert the fraction of the sample that decays into a percentage by multiplying by 100:\[0.317 \times 100 = 31.7\%\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Half-life
The concept of half-life is central to understanding radioactive decay. It is defined as the time required for half of the radioactive atoms in a sample to decay. This measurement is crucial for scientists and researchers as it helps them predict how long a radioactive substance will remain active.
Understanding half-life can also aid in dating objects, assessing nuclear reactions, and medical diagnoses.
Understanding half-life can also aid in dating objects, assessing nuclear reactions, and medical diagnoses.
- For instance, if a substance has a half-life of 10 years, in 10 years only half of its atoms will remain radioactive.
- In another 10 years, half of the remaining will have decayed, leaving a quarter of the original radioactive atoms.
Decay Constant
The decay constant (\( k \)) is another fundamental component in understanding radioactive decay. It measures the probability of decay of a single atom per unit time. The decay constant can be calculated using the formula \( k = \frac{\ln(2)}{\text{halflife}} \).
This constant is pivotal in predicting how quickly a substance will decay over time.
This constant is pivotal in predicting how quickly a substance will decay over time.
- A higher decay constant signifies a faster rate of decay, meaning the substance is less stable.
- A lower decay constant indicates a slower decay rate, making the substance more stable over time.
C-14 Dating
Carbon-14 dating, or radiocarbon dating, is a technique used to determine the age of ancient organic materials. This method relies on the decay of Carbon-14, a naturally occurring isotope that is absorbed by living organisms. When the organism dies, it ceases to absorb Carbon-14, which then begins to decay at a consistent half-life of 5730 years.
By measuring the remaining Carbon-14 in a sample, scientists can calculate when the organism died.
By measuring the remaining Carbon-14 in a sample, scientists can calculate when the organism died.
- This is incredibly useful for archaeologists, paleontologists, and geologists to date ancient artifacts, fossils, and geological formations.
- C-14 dating is precise enough to estimate dates up to around 60,000 years ago.
Exponential Decay
Exponential decay describes the process by which quantities decrease at a rate proportional to their current value. In radioactive decay, the amount of a radioactive substance decreases exponentially over time.
This concept is represented mathematically by the equation \( N(t) = N_0 \cdot (0.5)^{\text{Number of half-lives}} \), where \( N(t) \) is the remaining quantity at time \( t \), and \( N_0 \) is the initial quantity.
This concept is represented mathematically by the equation \( N(t) = N_0 \cdot (0.5)^{\text{Number of half-lives}} \), where \( N(t) \) is the remaining quantity at time \( t \), and \( N_0 \) is the initial quantity.
- Exponential decay is characterized by a rapid decrease in the quantity initially, which slows down as time progresses.
- This differs from linear decay, where the subtraction occurs at a constant rate over time.
