Problem 43 - Chemistry Structure and Dynamics

Question

The Lascaux cave near Montignac in France contains a series of remarkable cave paintings. Radiocarbon dating of charcoal taken from this site suggests an age of 15,520 years. What fraction of the \({ }^{14} \mathrm{C}\) present in living tissue is still present in this sample? \(\left({ }^{14} \mathrm{C}: t_{1 / 2}=5730 \mathrm{yr}\right)\)

Step-by-Step Solution

Answer

About 15% of the original carbon-14 is present.

Step 1: Understand the Problem

We need to determine the fraction of carbon-14 \( (^{14}\text{C}) \) remaining in a sample that is 15,520 years old compared to a living organism. The half-life \( (t_{1/2}) \) of carbon-14 is given as 5730 years.

Step 2: Calculate the Number of Half-Lives

To find the fraction remaining, we first calculate the number of half-lives that have passed. This is given by dividing the total time by the half-life:\[\text{Number of half-lives} = \frac{15,520 \text{ years}}{5730 \text{ years/half-life}}.\]

Step 3: Calculate the Exponential Decay Factor

Calculate \( n \), where \( n \) is the number of half-lives:\[ n = \frac{15,520}{5730} \approx 2.71. \]

Step 4: Apply the Decay Formula

The fraction of carbon-14 remaining is given by the equation:\[ f = \left(\frac{1}{2}\right)^n, \]substitute \( n = 2.71 \).

Step 5: Compute the Fraction Remaining

Calculate the remaining fraction:\[ f = \left(\frac{1}{2}\right)^{2.71} \approx 0.15. \]

Step 6: Conclusion

Approximately 15% of the original carbon-14 remains in the sample.

Key Concepts

Understanding Half-Life

The concept of half-life is crucial to understanding the principle behind radiocarbon dating. A half-life is the time required for half of a quantity of a radioactive substance to decay. In simpler terms, if you start with a certain amount of a radioactive material, after one half-life, only half of it will remain.

For carbon-14, which is used in dating archaeological artifacts such as the Lascaux cave paintings, the half-life is approximately 5730 years. This means that over 5730 years, half of the carbon-14 present in an organism when it was living will have decayed.

This predictable rate of decay allows scientists to calculate how long it has been since an organism, like the charcoal found in the cave, stopped exchanging carbon with the atmosphere, hence giving its age. This regularity in decay makes half-life a powerful tool for dating ancient artifacts.

The Process of Carbon-14 Decay

Carbon-14 decay is the process by which carbon-14 is transformed into stable nitrogen-14 over time. This decay occurs at a steady rate, allowing it to serve as a kind of clock for dating purposes.

• All living organisms absorb carbon-14 from their environment.
• When an organism dies, it stops absorbing carbon-14, and the existing carbon-14 in its system begins to decay.
• By measuring the remaining carbon-14 in a sample and comparing it to what would be expected in a living counterpart, the time since the organism's death can be estimated.

In the case of the Lascaux cave paintings, the amount of carbon-14 found in the charcoal allows scientists to determine its age by using the half-life formula. This calculation tells us that the carbon-14 levels in the sample have decayed to about 15% of what would be present in a living organism, indicating that the paintings are approximately 15,520 years old.

Exponential Decay and Radiocarbon Dating

Exponential decay describes the process by which the quantity of a radioactive isotope decreases at a rate proportional to its current value. This concept is exemplified in the decay of carbon-14.

The decay process follows an exponential function, which is mathematically expressed as: \[ f = \left( \frac{1}{2} \right)^n \]where \( f \) is the remaining fraction of the substance, and \( n \) is the number of half-lives elapsed.

In practical terms:

• As the number of half-lives increases, the remaining amount of carbon-14 decreases exponentially.
• This decay pattern means that after two half-lives, only a quarter of the original carbon-14 remains.
• More than a mere counting process, radiocarbon dating involves understanding the mathematical relationships governing exponential decay.

This precise nature of exponential decay allows for accurate predictions and calculations required for dating ancient artifacts, such as those found in the Lascaux caves. Understanding this helps to grasp why only about 15% of the original carbon-14 remains after 15,520 years.