Problem 96 - Calculus Single Variable

Question

Radiocarbon Dating Two isotopes of carbon, \({ }^{12} \mathrm{C},\) which is stable, and \({ }^{14} \mathrm{C},\) which decays exponentially with a 5700 -year half-life, are found in a known fixed ratio in living matter. After death, carbon is no longer metabolized, and the amount \(m(t)\) of \({ }^{14} \mathrm{C}\) decreases due to radioactive decay. In the analysis of a sample performed \(T\) years after death, the mass of \({ }^{12} \mathrm{C}\), unchanged since death, can be used to determine the mass \(m_{0}\) of \({ }^{14} \mathrm{C}\) that the sample had at the moment of death. The time \(T\) since death can then be calculated from the law of exponential decay and the measurement of \(m(T)\). Use this information for solving Exercises \(95-98\) In \(1994,\) a parka-clad mummified body of a girl was found in a subterranean meat cellar near Barrow, Alaska. Radiocarbon analysis showed that the girl died around ce 1200 . What percentage of \(m_{0}\) was the amount of \({ }^{14} \mathrm{C}\) in the mummy?

Step-by-Step Solution

Answer

About 91.85% of \\(m_0\\) remained.

Step 1: Identify the Half-Life Formula

The decay of \({ }^{14} \mathrm{C}\) is given by the exponential decay formula: \(m(t) = m_0 \times \left(\frac{1}{2}\right)^{t/5700}\), where \({m_0}\) is the initial amount of \(^{14} \mathrm{C}\), \(t\) is the time in years since the object's death, and 5700 is the half-life of \(^{14} \mathrm{C}\).

Step 2: Calculate the Time Since Death

The problem states that the girl died around CE 1200. Given that analysis was done in 1994, the time since death \(T\) is \(T = 1994 - 1200 = 794\) years.

Step 3: Substitute Values into Exponential Decay Formula

We substitute \(T = 794\) years into the decay formula: \(m(794) = m_0 \times \left(\frac{1}{2}\right)^{794/5700}\).

Step 4: Compute the Remaining Percentage of \\(^{14} \mathrm{C}\\)

Evaluate the value \(\left(\frac{1}{2}\right)^{794/5700}\) to find the remaining percentage of \(^{14} \mathrm{C}\). This gives: \(\left(\frac{1}{2}\right)^{794/5700} \approx 0.9185\), which means about 91.85% of \(m_0\) remains as \(^{14} \mathrm{C}\).

Key Concepts

Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. This is a common phenomenon seen with radioactive substances like carbon-14. In exponential decay, the rate of decrease is always proportionally linked to the existing amount of the substance. The mathematical model for exponential decay can be written as:\[ m(t) = m_0 \times e^{-kt} \]In this formula:

• \(m(t)\) is the amount of substance at time \(t\).
• \(m_0\) is the initial amount of the substance.
• \(k\) is the decay constant, which is related to the half-life.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

Exponential decay is characterized by a rapid decrease that slows over time. In real-world applications like radiocarbon dating, knowing the current amount and the initial amount allows scientists to calculate how long a substance has been decaying. This is an essential step in determining the time since death when analyzing archaeological samples.

Half-Life

Half-life is a key concept in the study of radioactive decay. It refers to the amount of time it takes for half of a quantity of a radioactive substance to decay. This concept helps us understand how long a radioactive isotope remains active or detectable. With carbon-14, the half-life is 5700 years, which means every 5700 years, half of the available carbon-14 will have decayed into nitrogen-14.
Half-life is critical in calculating the age of biological samples. The formula linking half-life and exponential decay is:\[ m(t) = m_0 \times \left(\frac{1}{2}\right)^{t/T_{1/2}} \]where:

• \(T_{1/2}\) is the half-life of the isotope.
• The fraction \(\left(\frac{1}{2}\right)\) raises to the power of \(t/T_{1/2}\), showing the exponential decay over multiple half-life periods.

In the case of the mummy found in Alaska, the half-life of carbon-14 allows scientists to accurately estimate the remaining percentage of carbon-14 and calculate the time lapse since the sample's initial state.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are versions of elements that have an unstable nucleus and decay over time by emitting radiation. Carbon-14 is a well-known radioactive isotope used in dating organic materials. These isotopes help us measure the passage of time in samples that are otherwise difficult to date. Radioisotopes are identified by their unique number of neutrons, which gives them different atomic weights from their stable counterparts. They lose energy by emitting particles until they transform into a more stable form.
The significance of radioactive isotopes like carbon-14 in radiocarbon dating lies in their predictable decay rates. This behavior is vital for dating artifacts and archaeological finds. By comparing the ratio of remaining radioisotopes to their stable forms in a sample, scientists can determine how long it has been since the organism, like the Alaskan mummy, stopped taking in new carbon. It's an insightful method to unravel historical timelines and understand ancient remains.