Problem 60 - Fundamentals of Physics

Question

A \(5.00 \mathrm{~g}\) charcoal sample from an ancient fire pit has a \({ }^{14} \mathrm{C}\) activity of 63.0 disintegrations/min. A living tree has a \({ }^{14} \mathrm{C}\) activity of 15.3 disintegrations/min per \(1.00 \mathrm{~g}\). The half-life of \({ }^{14} \mathrm{C}\) is \(5730 \mathrm{y}\). How old is the charcoal sample?

Step-by-Step Solution

Answer

The charcoal sample is approximately 1605 years old.

Step 1: Calculate the Initial Activity

First, calculate the initial activity of the charcoal when it was part of a living tree. The initial activity is calculated using the activity per gram of a living tree. Multiply the activity per gram by the mass of the charcoal sample: \[ A_0 = 15.3 ext{ disint/min/g} \times 5.00 ext{ g} = 76.5 ext{ disint/min} \]

Step 2: Use the Decay Formula

Use the radioactive decay formula, which is given by \[ A = A_0 e^{-kt} \] where \( A \) is the current activity, \( A_0 \) is the initial activity, \( k \) is the decay constant, and \( t \) is the time in years. Here, \( A = 63.0 \text{ disint/min} \).

Step 3: Calculate the Decay Constant

The decay constant \( k \) can be calculated using the half-life formula: \[ k = \frac{\ln 2}{\text{Half-life}} \]Substitute the given half-life of \( {}^{14}C \):\[ k = \frac{\ln 2}{5730} \approx 1.2097 \times 10^{-4} \text{ year}^{-1} \]

Step 4: Solve for Time

Rearrange the decay formula to solve for \( t \):\[ t = \frac{1}{k} \ln \left(\frac{A_0}{A}\right) \]Substituting the known values:\[ t = \frac{1}{1.2097 \times 10^{-4}} \ln \left(\frac{76.5}{63.0}\right) \]

Step 5: Calculate the Time

Perform the calculation:\[ t = \frac{1}{1.2097 \times 10^{-4}} \ln \left(1.2143\right) \approx \frac{1}{1.2097 \times 10^{-4}} \times 0.1945 \approx 1605 \text{ years} \]

Step 6: Conclusion

The age of the charcoal sample is approximately 1605 years.

Key Concepts

Carbon-14 Dating

Carbon-14 dating is a technique used to determine the age of an object containing organic material. This method leverages the properties of carbon-14 ( ^{14}C), a radioactive isotope of carbon. Carbon-14 is continuously created in the atmosphere through interactions between cosmic rays and atmospheric nitrogen. Living organisms are in constant equilibrium with the atmospheric ratio of ^{14}C to ^{12}C while they are alive because they constantly absorb carbon via processes like respiration and feeding.
When an organism dies, it stops replenishing carbon, and the ^{14}C it contains begins to decay into nitrogen-14 with a known rate. This process is called radioactive decay. Scientists can measure the remaining ^{14}C activity in a sample and, through calculations using decay laws, deduce how long it has been since the organism died.

• The decay activity is measured in disintegrations per minute.
• By comparing the remaining activity with the initial activity expected in a living organism, we can estimate the age of the sample.
• This method is effective for dating materials up to around 50,000 years old.

Decay Constant Calculation

The decay constant (k) is a fundamental parameter in radiometric dating. It serves as a measure of how quickly a radioactive isotope decays. Calculating the decay constant involves the relationship between half-life, which is the time taken for half of the radioactive substance to decay, and the constant itself.To calculate the decay constant, you make use of the formula:

• \[ k = \frac{\ln 2}{\text{Half-life}} \]

This equation stems from the natural exponential function, where the natural logarithm of 2 (approximately 0.693) represents the decay of half of the radioactive atoms.
For example, to find the decay constant for carbon-14 with a half-life of 5730 years, you would substitute into the formula:

• \[ k = \frac{0.693}{5730} \approx 1.2097 \times 10^{-4} \text{ year}^{-1} \]

This decay constant can then be used in further calculations to determine the age of carbon-containing materials.
Understanding and calculating the decay constant is critical because it provides the necessary conversion factor in the decay formulas used for radiocarbon dating.

Half-life

Half-life is an essential concept in understanding radioactive decay. It is defined as the amount of time required for half of a sample of a radioactive substance to decay. Half-life allows scientists to predict how quickly a sample will reduce its radioactivity to a certain level and is pivotal in dating archaeological finds.
The half-life of carbon-14, specifically, is about 5730 years. This time span makes carbon-14 dating effective for dating historical artifacts back approximately 50,000 years. When half of the original quantity of ^{14}C has decayed in a sample, its activity is reduced, which can be measured to find the age of the sample.
To use half-life in decay calculations, especially when deriving the age of an archaeological find, these key points are vital:

• Recognize that after each half-life, half of the remaining ^{14}C will have decayed. For instance, after 2 half-lives, only 25% of the initial amount would be left.
• Use the formula relating decay constant and half-life to find the rate of decay ( k), which aids in calculating the age of a sample.
• This thorough understanding of the ^{14}C half-life allows scientists to accurately date organic materials by observing current decay levels compared to initial levels in living organisms.