Problem 9 - Applied Physics

Question

Find the percent of \(\mathrm{a}{\underline{\phantom{xx}}}_{6}^{14} \mathrm{C}\) sample that will decay in the next 3000 years. Its halflife is 5730 years.

Step-by-Step Solution

Answer

31.7% of the sample will decay in 3000 years.

Step 1: 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:\[k = \frac{\ln(2)}{5730} \approx 1.2097 \times 10^{-4} \text{ per year}\]

Step 2: 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:\[\text{Number of half-lives} = \frac{3000}{5730} \approx 0.5236 \text{ half-lives}\]

Step 3: 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:\[N(t) = N_0 \cdot (0.5)^{0.5236} \approx 0.683 \cdot N_0\]

Step 4: 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:\[1 - 0.683 = 0.317\]

Step 5: Convert the decayed fraction to a percentage

Finally, convert the fraction of the sample that decays into a percentage by multiplying by 100:\[0.317 \times 100 = 31.7\%\]

Key Concepts

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.

• 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.

The concept applies to all radioactive elements, including Carbon-14, which is vital in archaeological and geological dating.

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.

• 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.

Understanding the decay constant can greatly aid in calculating the exact time it takes for a particular fraction of a material to decay.

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.

• 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.

This method has revolutionized our understanding of historical timelines and evolutionary history.

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.

• 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.

Understanding exponential decay helps in predicting how long it will take for a substance to reach a certain level of decay, which is essential for applications in medicine, environmental science, and any field dealing with radioactive substances.