Problem 40 - Chemistry Structure and Dynamics

Question

The \({ }^{14} \mathrm{C}\) in living matter has an activity of 15.3 disintegrations, or "counts," per minute (cpm). What is the age of an artifact that has an activity of \(4 \mathrm{cpm} ?\left({ }^{14} \mathrm{C}: t_{1 / 2}=5730 \mathrm{yr}\right)\)

Step-by-Step Solution

Answer

The artifact is approximately 8501 years old.

Step 1: Understand the Problem

We are given that living matter has an activity of 15.3 counts per minute (cpm), and an artifact has an activity of 4 cpm. We need to find the age of the artifact. The half-life of Carbon-14 ( ^{14}C ) is 5730 years.

Step 2: Determine the Ratio of Activities

Calculate the ratio of the current activity (artifact) to the original activity (living matter): \( \text{Ratio} = \frac{4}{15.3} \).

Step 3: Apply the Radioactive Decay Equation

The formula for radioactive decay is: \( N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/T} \), where \( N(t) \) is the activity at time \( t \), \( N_0 \) is the initial activity, and \( T \) is the half-life. Plug in the values: \( \frac{4}{15.3} = \left( \frac{1}{2} \right)^{t/5730} \).

Step 4: Solve for Time \( t \)

Take the natural logarithm of both sides to solve for \( t \): \( \ln \left( \frac{4}{15.3} \right) = -\frac{t}{5730} \ln(2) \). Calculate \( t \) using: \( t = - \frac{5730 \times \ln \left( \frac{4}{15.3} \right)}{\ln(2)} \).

Step 5: Calculate the Final Answer

Perform the calculations: \( \ln \left( \frac{4}{15.3} \right) \approx -1.3218 \) and \( \ln(2) \approx 0.6931 \). Substitute these values to find \( t \): \( t \approx - \frac{5730 \times (-1.3218)}{0.6931} \approx 8501 \) years.

Key Concepts

Carbon-14 Dating

Carbon-14 dating is a method used by archaeologists and scientists to determine the age of an object containing organic material. It relies on the principles of radioactive decay. Carbon-14, or ^{14} C, is a radioactive isotope of carbon that is present in the atmosphere and incorporated into living organisms. When an organism dies, it stops absorbing carbon-14, and the amount of ^{14} C begins to decrease as it decays into nitrogen-14. This technique is commonly used for dating artifacts, fossils, and historical relics. Carbon-14 has a half-life of approximately 5730 years, which allows for precise dating of objects up to around 50,000 years old. This makes carbon-14 dating a vital tool in archaeology for understanding the timeline of human history.

Half-life

The half-life of a radioactive isotope like carbon-14 is the time it takes for half of the radioactive atoms in a sample to decay. It's a constant and unique value for every radioactive isotope. In the case of carbon-14, the half-life is 5730 years. Understanding half-life is crucial for carbon-14 dating. It helps scientists calculate how many half-lives have passed since the organism's death. By measuring the remaining ^{14} C in an artifact and comparing it to the expected amount in a living organism, scientists can estimate the age of the artifact. Half-life provides a "clock" for measuring the time elapsed since death, making it indispensable in fields like archaeology and geology.

Exponential Decay Equation

The exponential decay equation governs how radioactive substances decay over time. For carbon-14, the equation used is:\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/T} \]Here:

• \(N(t)\): Activity at time \(t\)
• \(N_0\): Initial activity
• \(t\): Time since the organism's death
• \(T\): Half-life of the isotope

To find the age of an artifact, you rearrange the equation to solve for \(t\). Taking the natural logarithm helps simplify exponential equations:\[ t = - \frac{T \cdot \ln \left( \frac{N(t)}{N_0} \right)}{\ln(2)} \]This formula calculates the number of years that have passed since the death of an organism, based on the measured activity. It's essential for reducing uncertainty in dating ancient objects.

Activity Measurement

Activity measurement in radioactive decay refers to the number of disintegrations or decays per unit time. For carbon-14, it's measured in counts per minute (cpm). Living organisms have a consistent level of ^{14} C activity due to their interaction with the atmosphere. When we measure the activity of ^{14} C in an artifact and find it to be less than that in living organisms, it indicates decay over time. In the original exercise, the living matter has an activity of 15.3 cpm, while the artifact's activity is 4 cpm. By comparing these values, one can use the ratio of these activities in the exponential decay equation to determine the artifact's age. Activity measurement thus acts as a key step in carbon-14 dating, providing the crucial data necessary for accurate archaeological dating.