91Ó°ÊÓ

This is a continuation of Exercise 10. We rewrite the decay formula as $$ \frac{A}{A_{0}}=0.5^{t / H} . $$ This formula shows the fraction of the original amount that is present as a function of time. Carbon 14 has a half-life of \(5.77\) thousand years. It is thought that in recent geological time (the last few million years or so), the amount of \(\mathrm{C}_{14}\) in the atmosphere has remained constant. As a consequence, all organisms that take in air (trees, people, and so on) maintain the same level of \(\mathrm{C}_{14}\) so long as they are alive. When a living organism dies, it no longer takes in \(\mathrm{C}_{14}\), and the amount present at death begins to decay according to the formula above. This phenomenon can be used to date some archaeological objects. Suppose that the amount of carbon 14 in charcoal from an ancient campfire is \(\frac{1}{3}\) of the amount in a modern, living tree. In terms of the formula above, this means \(A / A_{0}=\frac{1}{3}\). When did the tree that was used to make the campfire die? Be sure to explain how you got your answer.

Step by step solution

01

Identify Known Values

We are given that \( \frac{A}{A_{0}} = \frac{1}{3} \) and the half-life of Carbon 14 (\( H \)) is 5.77 thousand years. These values will be used in the decay formula to find \( t \), the time when the tree died.

02

Set Up the Equation

Insert the known values into the decay formula. Therefore, we have \( \frac{1}{3} = 0.5^{t / 5.77} \). Our goal is to solve for \( t \).

03

Apply Logarithms

Take the natural logarithm of both sides to unravel the exponent: \( \ln\left(\frac{1}{3}\right) = \ln\left( 0.5^{t / 5.77} \right) \). By using the logarithmic identity \( \ln(a^b) = b \ln(a) \), the equation becomes \( \ln\left(\frac{1}{3}\right) = \frac{t}{5.77} \ln(0.5) \).

04

Solve for Time \( t \)

Rearrange the equation to isolate \( t \): \( t = 5.77 \frac{\ln\left(\frac{1}{3}\right)}{\ln(0.5)} \). Calculate the values using a calculator. Firstly, \( \ln\left(\frac{1}{3}\right) \approx -1.0986 \) and \( \ln(0.5) \approx -0.6931 \). Substitute these into the equation to find \( t \).

05

Perform the Calculation

Compute \( t = 5.77 \times \frac{-1.0986}{-0.6931} \approx 9.13 \). This calculation gives the time in thousands of years.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-life

The concept of half-life is essential in understanding radioactive decay, such as that of Carbon-14. Simply put, half-life is the time required for half of the radioactive atoms in a sample to decay.
This is a constant for a given isotope, meaning it does not change. For Carbon-14, the half-life is 5.77 thousand years. This means that every 5.77 thousand years, half of the Carbon-14 in a sample will have decayed.
Even though the half-life is a fixed value, the fraction of decayed substance is predictable. Knowing the half-life allows scientists to estimate how long it has been since the organism or object stopped exchanging Carbon-14 with the environment.

Decay Formula

The decay formula is a mathematical expression used to determine the remaining amount of a substance over time. For Carbon-14, this formula is expressed as \( \frac{A}{A_{0}}=0.5^{t / H} \). Here, \( A \) is the amount of Carbon-14 remaining, \( A_{0} \) is the initial amount, \( t \) is the time elapsed, and \( H \) represents the half-life.

• The formula shows an exponential decay, as the substance decreases by half every half-life period.
• This decaying pattern allows us to quantify the rate at which isotopes decay.
• To find \( t \), knowing \( A \) and \( A_{0} \), logarithms can be applied, effectively making it easier to handle the exponential aspects of the equation.

Using the decay formula is crucial in determining the ages of archaeological finds, where the half-life serves as a natural clock for dating purposes.

Archaeological Dating

Archaeological dating is a method used to determine the age of ancient artifacts or remains. Carbon dating, specifically, is a form of radiocarbon dating that utilizes Carbon-14.
Carbon-14 dating is effective for samples that are less than 50,000 years old. It relies on measuring the remaining amount of Carbon-14 in a sample to estimate how long since the organism or material stopped intaking carbon through atmospheric exchange.

• Once the organism dies, no further Carbon-14 is absorbed, and its decay starts becoming significant.
• By comparing the remaining \(C_{14} \) in the sample with the known \(C_{14} \) atmospheric ratio, scientists can estimate the date of death or creation.

This method is invaluable in archaeology, providing insights into past civilizations, climatic data, and cultural paradigms, allowing us to timeline historical and ancient events efficiently.