Problem 46 - Thomas Calculus

Question

The sensitivity of carbon-14 dating to measurement To see the effect of a relatively small error in the estimate of the amount of carbon-14 in a sample being dated, consider this hypothetical situation: \begin{equation} \begin{array}{l}{\text { a. A bone fragment found in central Illinois in the year } 2000} \\ {\text { contains } 17 \% \text { of its original carbon-14 content. Estimate the }} \\ {\text { year the animal died. }} \\ {\text { b. Repeat part (a), assuming } 18 \% \text { instead of } 17 \% \text { . }} \\\ {\text { c. Repeat part (a), assuming } 16 \% \text { instead of } 17 \% .}\end{array} \end{equation}

Step-by-Step Solution

Answer

The animal likely died in approximately 1141 AD, 1089 AD, or 1204 AD if 17%, 18%, or 16% remains, respectively.

Step 1: Understanding Carbon-14 Decay

The decay of carbon-14 is exponential, given by the formula \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the remaining amount of carbon-14 at time \( t \), \( N_0 \) is the original amount, and \( k \) is the decay constant. For carbon-14, the half-life is approximately 5730 years.

Step 2: Calculate the Decay Constant

The decay constant is calculated using the formula \( k = \frac{\ln(2)}{T_{1/2}} \), where \( T_{1/2} = 5730 \) years. So, \( k = \frac{\ln(2)}{5730} \approx 1.21 \times 10^{-4} \) per year.

Step 3: Calculate the Original Time of Death (17%)

For 17% remaining carbon-14: Set up the equation \( 0.17 = e^{-k(t-2000)} \). Solving for \( t \): \[ t = 2000 - \frac{\ln(0.17)}{1.21 \times 10^{-4}} \approx 1141 AD \]

Step 4: Calculate the Original Time of Death (18%)

For 18% remaining carbon-14: Set up the equation \( 0.18 = e^{-k(t-2000)} \). Solving for \( t \): \[ t = 2000 - \frac{\ln(0.18)}{1.21 \times 10^{-4}} \approx 1089 AD \]

Step 5: Calculate the Original Time of Death (16%)

For 16% remaining carbon-14: Set up the equation \( 0.16 = e^{-k(t-2000)} \). Solving for \( t \): \[ t = 2000 - \frac{\ln(0.16)}{1.21 \times 10^{-4}} \approx 1204 AD \]

Key Concepts

Exponential Decay

Exponential decay is a fundamental concept in understanding how substances decrease in quantity over time. When discussing carbon-14 dating, exponential decay describes how the amount of carbon-14 in a sample decreases at a rate proportional to its current amount. This is expressed with the formula:\[\ N(t) = N_0 \ e^{-kt}\ \]where:

• \( N(t) \) is the quantity of carbon-14 remaining after time \( t \).
• \( N_0 \) is the original quantity of carbon-14.
• \( k \) is the decay constant.

Carbon-14 dating utilizes this concept to determine how long it has been since a plant or animal died, based on how much carbon-14 remains in its remains. The decrease in the carbon-14 amount is not linear; instead, it declines rapidly at first and then slows over time. This characteristic is why the decay is referred to as "exponential." Understanding this decay pattern is crucial for accurately dating archaeological and geological samples.

Half-life

Half-life is a key term in radioactive decay processes, such as carbon-14 dating. It refers to the amount of time it takes for half of the radioactive isotopes in a given sample to decay. For carbon-14, this half-life is approximately 5730 years. This means that every 5730 years, half of the original carbon-14 content in a sample decreases:

• At 5730 years, only 50% of the original carbon-14 remains.
• At 11,460 years, 25% remains (half of the remaining 50%).
• This decline continues, halving each time a span of 5730 years passes.

The half-life concept is essential because it provides a predictable measure to calculate how long a sample has been around. If we know the half-life of a radioactive isotope and the remaining percentage in a sample, we can set up equations to determine the age of a sample using exponential decay formulas.

Decay Constant

The decay constant \( k \) is a crucial component in the exponential decay equation used for carbon-14 dating. It represents the probability per unit time that a single isotope will decay. Mathematically, the decay constant is determined by the formula:\[ k = \frac{\ln(2)}{T_{1/2}} \]where \( T_{1/2} \) is the half-life. For carbon-14:

• The half-life is 5730 years.
• Thus, \[ k = \frac{\ln(2)}{5730} \approx 1.21 \times 10^{-4} \text{ per year} \]

This constant helps quantify how quickly a sample loses its radioactive isotopes and is integral in predicting the age of a sample through carbon dating. Accurately calculating \( k \) is vital to the precision of the date arrived at through carbon-14 dating.

Radioactive Dating

Radioactive dating, particularly carbon-14 dating, is a valuable tool for archaeologists, paleontologists, and geologists. It allows us to determine the age of an ancient object by measuring the remaining amount of carbon-14. Here's a brief overview of how it works:

• All living organisms absorb carbon, including a small amount of radioactive carbon-14, throughout their lives.
• Once the organism dies, it no longer absorbs carbon, and the carbon-14 it contains begins to decay at a known rate (its half-life).
• By measuring how much carbon-14 remains compared to a living organism's balance, scientists can estimate when the organism died.

This dating method is particularly effective for dating objects that are up to around 50,000 years old. Beyond that, the remaining carbon-14 is often too scarce to measure accurately. Radioactive dating provides a very reliable timeline, making it an important tool in historical and prehistorical research.