Problem 30 - Chemistry: Principles and Reactions

Question

Sandals found in a cave were determined by carbon-14 dating to be \(3.9 \times 10^{2}\) years old. Assuming that carbon from living material gives \(15.3\) disintegrations/min of C-14 per gram of carbon, what is the activity of the C-14 in the sandals in disintegrations/min/g of carbon? ( \(t_{1 / 2}\) of \(\mathrm{C}-14=\) 5730 years)

Step-by-Step Solution

Answer

Answer: The activity of the C-14 in the old sandals is approximately 12.3 disintegrations/min/g of carbon.

Step 1: Identify given information

We are given the following information: - The age of the sandals: 3.9 x 10^2 years - The activity of C-14 from living material: 15.3 disintegrations/min/g of carbon - Half-life of C-14: 5730 years

Step 2: Consider the radioactive decay formula

The radioactive decay formula can be used to find the activity of a radioactive material at any time. The formula is given by: \(A_t = A_0 e^{-\lambda t}\) Where: - \(A_t\) is the activity at time \(t\) - \(A_0\) is the initial activity - \(\lambda\) is the decay constant - \(t\) is the time

Step 3: Calculate the decay constant

The decay constant, \(\lambda\), can be calculated using the half-life, \(t_{1/2}\), with the following formula: \(\lambda = \frac{ln(2)}{t_{1/2}}\) Plugging in the half-life of C-14: \(\lambda = \frac{ln(2)}{5730}\)

Step 4: Calculate the activity at 3.9 x 10^2 years

Now, we have to find the activity of C-14 in the sandals. We know the initial activity of the C-14, which is 15.3 disintegrations/min/g of carbon. We can plug in the values into the radioactive decay formula to find the activity of C-14 at 3.9 x 10^2 years: \(A_t = A_0 e^{-\lambda t}\) \(A_t = 15.3 e^{-\frac{ln(2)}{5730}(3.9 \times 10^2)}\) Calculating the activity: \(A_t \approx 12.3\) disintegrations/min/g of carbon

Step 5: Conclusion

The activity of the C-14 in the sandals is approximately 12.3 disintegrations/min/g of carbon.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process in which unstable atomic nuclei lose energy by emitting radiation. A nucleus will emit this radiation in the form of particles or electromagnetic waves until it becomes stable. This process is spontaneous, meaning it happens on its own without any external influence. It is important in various fields such as archaeology, geology, and even medicine.Carbon-14 dating is a useful application of radioactive decay, helping determine the ages of ancient objects. This dating method is based on the decay of carbon-14 (C-14), a radioactive isotope of carbon. Since living organisms constantly exchange carbon with their environment, the ratio of C-14 to other carbon isotopes remains constant while the organism is alive. Once it dies, the C-14 in its tissues decays over time.We calculate the decay using the formula:\[A_t = A_0 e^{-\lambda t}\]where:

• \( A_t \) is the remaining activity at time \( t \)
• \( A_0 \) is the initial activity
• \( \lambda \) is the decay constant
• \( t \) is the time elapsed since the material was alive

Understanding radioactive decay is essential to perform carbon-14 dating and to interpret the results correctly.

Half-Life

The half-life of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. It is a constant property of each radioactive element and doesn’t depend on the initial quantity of the substance or environmental conditions. This concept is vital to accurately determine the age of archaeological samples using carbon-14.For carbon-14, the half-life is approximately 5730 years. This means that after 5730 years, half of the initial C-14 atoms would have decayed, reducing their number by 50%. After another 5730 years, half of the remaining C-14 atoms would decay again.With this information, we can understand that as time passes:

• The number of C-14 atoms decreases,
• At the same time, the activity level (disintegrations per minute) declines.

The formula linking half-life and the decay constant is expressed as:\[\lambda = \frac{\ln(2)}{t_{1/2}}\]This equation is crucial to calculate the decay constant, which helps predict the C-14 activity in samples of known age.

Decay Constant

The decay constant, denoted as \( \lambda \), is a fundamental parameter in the study of radioactive decay. It represents the probability of a nucleus decaying per unit time. A higher decay constant means that the substance decays faster.The decay constant is calculated using the half-life \( t_{1/2} \) with the formula:\[\lambda = \frac{\ln(2)}{t_{1/2}}\]For instance, for carbon-14 with a half-life of 5730 years, the decay constant is calculated as \( \frac{\ln(2)}{5730} \), which helps us determine how quickly the activity of a sample decreases over time.This constant allows us to predict the amount of radioactive material that remains after a certain period, which is crucial for carbon dating. Understanding how to compute and apply \( \lambda \) can help in:

• Determining the age of archaeological finds.
• Understanding the behavior of radioactive substances.
• Applications in nuclear physics and other scientific fields.

By integrating the decay constant into predictions, scientists can accurately track the decay of carbon-14 in ancient objects, helping uncover their historical timelines.